G eometry ampersand T opology 10 open parenthesis 2006 closing parenthesis 1 endash 26 .... 1 Quadrisecants give new lower bounds for the ropelength of a knot ELIZABETH DENNE YUANAN DIAO JOHN M SULLIVAN U .. s .. in .. g .. t h .. e .. e .. x .. i s .. t .. e n .. c .. e .. o f .. a .. s .. p .. e .. c i .. a n o .. n .. r .. i .. v .. i .. a .. l .. k .. n .. o .. t i s .. a .. t .. l .. e .. a s .. t .. 1 5 .. period .. 6 o .. f .. 1 .. 2 .. period .. N .. u m .. e .. r i .. c .. a .. l .. e .. x .. p .. e .. r .. m .. e .. n s .. o .. o .. u .. r .. n .. e .. w .. b 5 .. 7 .. M .. 2 .. 5 .. semicolon 1 .. I .. n .. t .. r T .... h .... e .... r .... o .... p .... e .... l .... e .... n .... g .... t .... h .... p .... r .... o .... b m .... a .... i .... n .... t .... a .... i .... n .... i .... n .... g .... a .... n .... e .... m .... b .... e m .. o .. d .. e .. l .. o .. f .. t .. y .. i .. n .. g .. t .. h M .... o .... r .... e .... t .... e .... c .... h .... FS-n K slash-period .... c f .... a Vdash l s l p y K i .... t-s c h u d e Vdash Vdash e-t .... n h Vdash i .... d c a .. d .. d 1 Vdash one-t sub c Vdash .. kb s Vdash .. t .. w .. i .. c .. t-r period a semicolon b semicolon c-.notdef slash i f n .. c Vdash i m .. c a .. l .. r t i .. s .. t .. i .. n .. c K-t .. p I .. i-t n i t s .. k .. f n .. n .. f .. r .. Vdash .. m .. t .. h .. Vdash .. w 4 t .... h FS K-a slash-period t D 0 .... n .... K i .... s-C-s to the power of 1 semicolon 1 .... m .... a .... n .... i .... n .... th a .... t .... i t .... Vdash .... t a .... n .... Vdash .... n u .. n .. c t .. i Vdash .. n .. Vdash .. a r K-c i .. s-C to the power of 1 .. w .. period c W-a h .notdef-n n-d .. n Vdash n .. Vdash r .. m a-K Vdash .. a t .. h .. Vdash .. n FS K-i slash-period n .. s d t Vdash h Vdash e d .. p .. r .. e .. Vdash .. d .. i .. Vdash .. m .. e .. t .. er .. o .. f o .. t .. e .. h .. Vdash .. t .. n .. t .. h .. e .. e .. x .. i .. s .. t .. i .. n .. Vdash .. l i .. t .. e .. r .. Vdash .. t .. r e t .. h .. a .. n .. d .. i .. a .. m .. e W .. e .. d .. e .. fi .. n .. e .. r .. o .. p .. e l .. e .. n .. g .. t .. h .. t .. o .. b .. e B .... e .... c .... a .... u .... s .... e .... t .... h .... i .... s .... i C-s to the power of 0 .... s .... p e Vdash .... m Vdash g i y hyphen Vdash c n .... o-c .... n .... Vdash t-.notdef .... d h .. a .. r .. d open square bracket 4 bracketright-t t .. o .. h a-s .. ht o awn .. y .. open parenthesis .. t .. a .. m .. e .. closing parenthesis .. k .. n .. o C .. a .. n .. t .. a .. r .. e .. l .. l .. a .. comma .. K .. u .. s .. n .. e .. r .. a .. n .. d .. S .. u t .. h .. e .. s .. e .. a .. r .. e .. s .. h .. a .. r .. p .. i .. n .. c .. e .. r .. t .. a .. i .. n open square bracket 4 closing square bracket to the power of period-s H .. o .. w .. e .. v .. e .. r .. comma .. t .. h .. e .. s e .. e .. x .. a .. m .. p .. l .. e .. s .. a b .... y .... C .... a .... n .... t .... a .... r .... e .... l .... l a .... comma open square bracket 2 closing square bracket d .... ue .... comma s .... K-c u-r is .... b-n .... ee .... sr .... comma-a P .. u .. b .. l .. i s .. h .. e .. D-a .. r O .. y I .. colon-two 0 1 0 0 six-period 2 .... Elizabeth Denne comma Yuanan Diao and John M Sullivan t .. i .. g .. h .. t .. open parenthesis .. r .. o .. p .. e l .. e .. B-n sub 0 g f .. t-t .. h Vdash hyphen-B .notdef-c r r r i Vdash t-m .. i a c n a r l .. parenright-n Vdash m Vdash w h .. a .. t .. d .. i f f .. Vdash r .. Vdash .. n .. t G Vdash h r .. i .. n .. g .. n Vdash B-i sub 0 .. hn Vdash u .notdef-f .. td h Vdash .. tci k a .. n .. d .. p r Vdash Vdash .. ma .. b .. y .. m .. i n im i Vdash i .. n g comma Vdash .. r .. t .. Vdash .. d i c .. Vdash m .. p Vdash .. n .. B sub zero-n it Vdash Vdash Vdash f t i .. Vdash .. p .. a n .. a .. r comma .. a .. n d .. i .. t Vdash Vdash .. m Vdash Vdash i .. g e .. x .. p .. l .. i .. c .. i t .. l .. y N .. u .. m .. e .. r .. i .. c .. a .. ln s open square bracket 1 x-k-bracketright sub comma 7 i-p .. Se u bracketleft-r 1 l closing square bracket l-a-nine m n e-v-d a-n R-bracketleft to the power of 1 t n a closing square bracket s-s-eight .. wub d gy o t .. h .. e .. m .. i .. n .. i .. m .. u .. m .. r .. o .. p .. e .. l .. one-e : 3 n-six 7 g .. a-two .. n h .. d f .. t o a .. n .. Vdash .. t .. h .. Vdash .. r .. t .. i .. g .. h .. t .. t .. r .. Vdash .. f .. Vdash .. i .. l 1 .. : 7 8 iFt Vdash .. hr .. dc .. i n Vdash .. m .. Vdash .. r .. i .. c .. a .. l .. s i .. m .. Vdash .. l .. a .. t .. i .. Vdash .. n .. s .. f 2 t .. 1 h-.notdef b .. s .. t .. l .. Vdash .. w 4 .. ra 1 b sub s-colon 70 2 .. t n-six .. hd .. ii sn w .. a .. s .. eight-i .. mw .. hp r-.notdef .. s .. vh a .. n .. y .. k .. n .. o .. t .. h .. a .. one-s open parenthesis r-two mo .. e p .. ae .. nl .. i-e n-n g g .. t-t h-h .. am .. to o .. f .. o .. n .. e H .. e .. r .. e .. comma .. w .. e .. u .. s .. e .. t .. h .. e .. i .. d .. e .. a .. o .. f .. q .. u .. a t .. o .. g .. e .. t .. b .. e .. t .. t .. e .. r .. l .. o .. w .. e .. r .. b .. o .. u .. open square bracket one-n closing square bracket six-s d h-s .. o u .. s .. i .. n .. g .. p .. o .. l .. y .. g .. o .. n .. a .. l .. k n .. o .. t .. s .. comma .. t .. h .. a h .. a .. v .. e .. a .. q u .. a .. d-bracketleft 1 closing square bracket e-three .. ixs t e e .. cn a d .. n e t .. d period .. tK h .. ui ps .. er g .. e .. n .. e .. r .. i .. c .. k .. n .. o .. t .. s .. h .. a .. v .. e .. e .. s .. s .. e .. n .. t i a .. r .. t .. i c .. e .. b .. bracketleft-y 1 closing square bracket period M-five closing parenthesis .. oW r .. e-t .. o w .. n i la .. ln .. dd .. eM .. fi o n n e d q .. u .. a .. d .. r .. i .. s .. e .. c .. a .. n .. t .. s .. a .. r .. e .. e .. x .. a .. c .. t .. l .. y t .. h .. a .. t .. a .. c .. u .. r .. v .. e .. a .. r .. b .. i .. t .. r .. a .. r .. i .. EM-l .. y c .. Vdash c .. l n .. o .. n .. e .. s .. s .. e .. n B .. y .. c .. o .. m .. p .. a .. r .. i .. n .. g .. t .. h .. e .. o .. r .. d .. e .. r .. i .. n .. g w .. e .. d .. i .. s .. i .. n .. g .. u .. i .. s .. h .. t .. h .. r e .. e .. t .. y .. p .. e .. s a .. r .. g .. u .. m .. e .. n .. t s .. t .. o .. o .. b .. t .. a .. i n .. a .. l .. o .. w .. e o .. f .. t .. h .. a .. t .. t .. y .. p .. e .. period .. T .. h 1 colon-e 9 3 three-period I .... n .... h .... e .... r .... d .... o .... c-bracketleft sub 5 closing square bracket comma .... oD .... r e .... anl n .... d ei ss hs .... oe w r t-s .... at t hi q .. u .. a .. d .. r .. i .. s .. e .. c .. a .. n .. t .. s .. o .. f .. a .. l .. tT eh .. re .. n o comma a .. r-s te t .. h a .. t .. a .. n .. y .. n .. o .. n .. t .. r .. 1 colon-v 65 period N .. o .. n .. t .. i .. v .. i .. a .. l .. l .. i .. n .. k .. s .. a .. l .. s .. o .. n .. e .. c .. e b .. o .. u .. n .. d .. s .. o .. b .. t .. a .. i .. n .. e .. d .. f .. o .. l .. i .. n .. k .. s i .. n .. t .. e .. r .. p .. r .. e .. t .. a .. t .. i .. o .. n .. o .. f .. t .. h .. e .. a .. r .. four-g sub EM comma u i .. n .. t .. h .. e .. G .. e .. h .. r .. i .. n .. g .. l .. i .. n .. k .. p .. r .. o .. b .. l .. e r .. o .. p .. e .. l .. e .. n .. g .. t .. h G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l .. y-u Ropelength boundsfrom quadrisecants .... 3 2 .. Definitions and lemmas Definition .. A k .. n i .. os .. ta .. n .. o .. r .. i .. e .. n .. K i e H11938-n to the power of 3 s-A n i y .. m w p-.notdef l-a-p a .. n-b .. d n .. a .. K d .. ni .. Vdash i-t .. d .. i .. t .. i .. n .. t .. Vdash CR-t a .. a n CR d-c-b .. H Vdash m CR a p i .. Vdash .. t h Vdash .. ar a t .. b-.notdef Vdash Vdash m l .. l .. Vdash .. w .. i .. n .. g .. t .. h .notdef-K .. g p two-f v-CR a .. n-w Vdash Vdash m .. Vdash t i m .. Vdash CR a .. r D sub p b-CR t-a .. b .. Vdash m .. p .. h .. a .. i Vdash Vdash .. t .. Vdash Vdash r .. d K .. r .. Vdash b p i grave sub i-a l t .. o .. d .. e .. n .. o .. CR-t a .. semicolon-e b .. b-t y-h .. e c .. l j a e b n-j d-p g a-e t .. nhi o ost o-f e .. n-s t-a t .. b-o hline i .. n .. s .. p .. a .. c .. e .. comma .. t .. h a .. e-period D .... e .... fi A-n i-n notdef t-s .notdef-i c-o n-a Vdash n-r .... t K ik s-i n na Vdash Vdash n-t Vdash .... r H11938 to the power of 3 Vdash Vdash .notdef-n Vdash t i .... n d t Vdash i .. K-h h .. a .. s .. n c .. .notdef-t m-l Vdash .. p a Vdash n-s nts Vdash Vdash n ic ts a-s a period-n .. n A-n notdef Vdash ntr .. d p .. l r Vdash .notdef-K n .. t .. w Vdash Vdash .. f .. hi .. c .. h .. l .. i .. K .. nw ah .. i .. c .. m-h .. lm .. i a .. n n s .. Vdash .. c .. a .. n .. t .. l s i Vdash .. nc t Vdash ar period n-i W a s-t n-q-w-.notdef c-i d .. l-a-a .. l n-d .. t r-t .notdef-s i Vdash m 2 Vdash s a-c Vdash a 3 .. s .. Vdash .. c .. four-a .. sn Vdash t c a .. a n n .. dt .. comma r-m Vdash i sd .. p s .. Vdash c to the power of g-a tmqi .notdef-.notdef v .. a-n-a-.notdef s sub l b-d i y-r csi hline s .. e .. g b .. m-period T .. h .. e .. o .. r .. i .. e .. n .. t .. a .. t .. o .. n .. o .. f .. a .. t .. r .. i .. s .. e .. c t .. h .. e .. t .. h .. r .. e .. e .. a-p .. o c i-c .. c n .. Vdash t .. r s .. i o .. n f .. t a .. h t .. a r .. t i l s .. i .. e .. n c b .. Vdash t .. m .. a .. y .. Vdash .. c .. c .. Vdash .. r .. i .. n .. Vdash .. i t .. h .. Vdash .. r .. c .. y .. c .. l t .. h .. Vdash .. c .. C-y sub 3 i c n .. l .. h Vdash c .. y g .. m r Vdash S sub three-r i p c-T h .. r Vdash Vdash u cp .. u .. Vdash .. d .. b .. Vdash Vdash Vdash x .. i c Vdash .. r .. a .. p .. i c .. a a Vdash .. a y c n-a-.notdef Vdash c-d sub a comma b .. b Vdash .notdef-.notdef Vdash m-t Vdash .. n .. Vdash c open parenthesis h d Vdash i-.notdef s-a r r .. Vdash .. v .. t .. rr .. s-i .. s d Vdash .. c .. a .. n .. F-t .. s i .. g period r uC Vdash r s-h e .. a-p 1 n .. c g .. t i .. i n v g Vdash l t y k .. n .. o .. t .. o .. r .. t .. h .. e .. t .. r .. i .. s .. a e .. b-i sub c s-c a .. d n .. i-t .. rc e h .. b-c 2 CR-n a .. i-period ba .. c .. ba F i .. g .. r .. Vdash 1 Vdash ah .. a-.notdef c r-s Vdash Vdash .. tr r Vdash vi Vdash sr Vdash sc Vdash ad .. n open parenthesis tl s Vdash f t .. h .. c .. y .. c .. l .. i .. c .. Vdash .. K d-i .. asr a .notdef-b n-a t-b-d r-h-c Vdash .. sp .. p Vdash Vdash i cn t .. h .. e .. o .. r .. i .. e n .. t .. a .. t .. i .. o n .. o .. f .. t .. h S i .. m .. i .. l .. a .. r .. l .. y .. comma .. t .. h .. e a .. f b o c-c sub u .. d r-c Vdash p r o .. i .. n-n .. t t .. s h o a f t c .. a .. n .. t .. l .. i .. n .. comma .. b .. t .. m .. a .. K-y Vdash O cf .. cc .. Vdash Vdash .. r ri s n Vdash K-a i .. s .. Vdash .. n .. l .. y .. a .. c .. y .. c .. l .. i .. c .. Vdash .. r .. d-K .notdef-i .. r .. comma i .. as .. n-r .notdef-d .. ai r .. d Vdash .. r .. comma .. m .. Vdash .. a .. n .. i .. n .. g .. Vdash .. n .. Vdash Vdash f .. D-t sub 4 i-h S-.notdef sub 4 .. P h i c r Vdash Vdash Vdash .. x i c .. Vdash g .. r .. a .. p .. hi .. c .. a Vdash Vdash y .. Vdash a .. Vdash t .. Vdash Vdash Vdash m .. Vdash .. n a-t .. i b n ac .. a-b G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l .. y-u 4 .... Elizabeth Denne comma Yuanan Diao and John M Sullivan a .... a .... d c .... b .... Vdash d .... i .... l .... l .... c .... a .... l .... l .... t .... h .... Vdash .... c .... Vdash .... s .... r-i Vdash m sipp alp a .... l .... t .... r-r .notdef-n .... sa .... p-t i-.notdef F cn i gt g sub period-i .... u N r .... l-o e y t .... 2 e .... a t .... s-h i-a n-t .... t .... h .... i .... s o .. K-f sub comma .. a .. n .. d .. s .. w i .. t .. c .. h .. i .. n .. g .. t .. h .. e .. o .. r .. i .. e a .. b .. c .. d .. a .. b .. c .. d .. a .. b .. c .. d F i .. g .. r .. 2 .. H .. Vdash .. r Vdash a-w Vdash b sc n .notdef-d Vdash q-a Vdash ca hd Vdash f-r it s-h t .. o .. r .. i .. g .. h .. t .. comma .. t .. h .. e .. s .. e .. a .. r .. e .. s i .. m .. p .. l e o .. f .. t .. h .. e .. K io a-s i .. n-b sub comma a-c t .. s-d-b a d a a c-l d o-c sub comma b-n r d ge .. s W h .... e .... n .... d .... i .... s .... c .... u .... a s .... b-s c-i w n-.notdef d .... g i a .... l q l u .... a-K s d Vdash Vdash r a t-b-i l-two s-CR a e a y t-c sub h Vdash a .. n .. t h .. c .. y .. c Vdash i .. c .. Vdash .. K d Vdash i r l a f-l p-b b a-.notdef c i b d t a-r .. c a-c .notdef-b d n d .notdef-g .. p .. Vdash t .. h .. e .. t .. y .. p .. e .. o I b e to the power of n .... e s .... o n m c e a l .... s sub l e e n sub d s .... N to the power of e comma S a sub N S to the power of l t NUL to the power of e c q .... d-a u r b-NUL-n sub a to the power of n a d a-a sub s r to the power of t Case 1 i n g Case 2 t h-s e-.notdef hline S .. Vdash Vdash .. t .. h .. n .. d s .. b Vdash .. cftt h .. h Vdash a-.notdef n .. b-m i .. h-i s d-d .. ans .. n Vdash g-a .. lm .. t Vdash Vdash n .. t-K i .. s .. i .. t .. s .. t .. h .. Vdash .. s .. Vdash Vdash .. n .. d .. s .. a .. l .. t a .. c-r b-n I a-d t tw .. al C .. a .. n .. t .. a .. r .. e l .. l .. a .. comma .. K .. open square bracket 3 u closing square bracket t .. ph .. ea .. rt b teh .. re g .. m comma .. iK .. du q .. u .. a .. d .. r K i .. s s e a c Vdash a t .. n-.notdef .. m t s a-f Vdash o-t-c i-r .. cn K-a sub d l h-D .. l Vdash y-l sub n .. l n 5 n t .. t-s n .. Vdash .. n .. t .. r .. i .. v .. i .. a .. l .. k .. n .. Vdash .. t .. s .. h .. a .. v .. a .. l t .. Vdash r .. 1 h .. a .. v .. Vdash .. s .. h .. Vdash .. w .. n .. t .. h .. a .. t .. t .. h .. Vdash .. n .. 2 c t a Vdash n-hyphen .. tb .. y .. p c .. o .. u .. n .. t .. i .. n .. g .. a .. l .. t .. e .. r .. n .. a .. t .. i .. n 3 .. K .. n .. o .. t .. s .. w B .. e .. c .. a .. u .. s .. e .. t .. h .. e .. r .. o .. p .. e .. l .. e .. n a .. n .. K-y t .. Vdash k h .. n a .. v o .. t t .. h .. 1 .notdef-c h-k .. in s .. si K i-m open parenthesis p-C to the power of 1 semicolon 1 c i l-u to the power of r-.notdef Vdash s a c .. u .. r .. v .. a .. t .. u .. r .. 2 sub e-period F .. o .. r .. a a two-n H11938 to the power of 3 y .. Vdash .. B period o-a slash .. i e .. n o t t e .. t .. h .. e .. o .. p .. e .. n .. a .. n O i Vdash t r b Vdash r-l a .. b .. o .. u .. t .. t .. h .. e .. l .. o .. c .. a .. l .. s .. t .. r .. u .. c .. t .. open square bracket u-eight comma .. rLee .. bracketright-o a .. n-bracketleft 4 comma d L .. e .. closing square bracket period m L e .. m .. m-L e-K-a b t-three Vdash period a 1 .. k .. n .. Vdash .. t .. a 2 Kn .. i-t B-h-t period a .notdef-slash .. n h-o .. n i t c Vdash .. k n n k .. n .. o .. t K .. d t .. Vdash h .. i c .. s .. f a r .. c EM .. h n a-d .. i s s l t .. r .. n Vdash .. n g s .. v-t e h r .. a-s G e o .. m .. T-e sub o t p r sub o-comma .. yl V .. ampersand-o-o .. g l y u Ropelength boundsfrom quadrisecants .... 5 hline c .... e .... n .... t a e .... I-r-a i-e b s d .... a a .... t s K .... ci j a-a-t NUL b-h-n j less-t 1 f-t .... h .... Vdash .... n .... t h .... Vdash ab i .. n .. t .. K r-i s .. Vdash a c .. t-s i-s .. n .. g .. l .. Vdash CR .notdef-a .. n r CR-k b .. a wh Vdash Vdash t .. Vdash t Vdash Vdash .. n d g a-t h a .. r .. j a-c NUL s-b j period P .... r .... Io .... f-o f-t .... h .... e .... r .... K-e t .... a-w .... e n .... r g e .... n a .... c n-t .... a a .... t r n s Vdash Vdash .notdef-o mf f Vdash t .... ph a i t B period h-a slash .. i t n .. e .. t .. period a semicolon c sub p-semicolon c slash eo .. su n .. d .. Vdash h Vdash .. v e .. r .. 1 divided by 2 .. F i Vdash r sub hline t h l Vdash e t a .. t .. Vdash .. m .. K h t .. a i d f .. a .. t .. h .. i .. r c d i it .. nh .. t t .. h r Vdash s s Vdash a-c p .. t-b hti w .... e .... w .... o r period-u a semicolon b semicolon c slash d 1 divided by less-two h-period T-a .... vh e e .... l .... e .... n .... g .... t .... h .... b .... o .... line-line A .. n .. i .. m .. m .. e .. d C .. o .. r .. o .. I-l l-f-K h a .. a r .. s y .. Vdash 3 n period a-i sub semicolon b-two 2 K-t ah .. p-n 2 d-CR-c a .. k i n a semicolon b s two-B period s-p slash .. h t .. h .. e .. c .. o .. m .. CR-p b .. l-l sub a .. e .. e m .. s B-e period o-p n-slash period T .. h .. e .. f .. o .. l .. l .. o .. w .. i .. n .. g .. 8 comma .. eem .. m am .. m-n a comma d-a sub sLeh m sub o-comma .. u m l a-u w .. e .. g .. i .. v .. e .. a .. s .. l .. i .. g .. h .. t .. l L .... e .... m .... m-L e-K-a b-t sub 3 Vdash period a 3 .... k .... n .... t .... f a 2 n-K .... it .... th .... t-.notdef .... hn t-i .... ch .... k Vdash K X .. a g-f Vdash .. e .. u .. at i-B period a slash .. s 1 .. p L .. Vdash i .. r p e .. s .. c .. h .. i .. t .. Vdash comma .. t .. h P .... r .... C-o .... oo .... n-f .... s .... i .... d .... e .... r .... w .... h .... a .... t .... t .... h .... i .... s .... p-b 2 K .... L hline d D .. a j b j .. n .. d .. DC2 .. e t .. h .. e .. b b-n Vdash .. t w-K a-.notdef t n n d .. t .. a-h .. b Vdash s .. h .. Vdash g .. pm s .. t .. r .. Vdash t .. c .. h-one = d ns Vdash b a-b b-y Vdash a t f d a Vdash c .. s-t Vdash nr .. t .. s .. Vdash Vdash .. t .. h t .. K-.notdef Vdash .. h .. s .. t .. h .. Vdash .. l .. Vdash .. c .. K l-i period s-s L i-DC2 i to the power of d-n-slash p = Ns .. w-c h-c .. i .. n t .. .notdef-s t .. h r a a-.notdef Vdash n g dh .. t a-K a .. n-b t g .. l-.notdef .. na .. tn .. t .. g F Vdash i Vdash .. g parenright-m .. us .. r h .. e r o y 3 w open parenthesis s i .. r D d = 2 s-period i-DC2 n-slash sub b .. u .. i .. t .. i .. s .. Vdash .. l i m .. i t .. o .. f .. K i .. r s c .. l .. b .. s y p .... Vdash .... i .... n .... t .... c .... h .... a .... r .... a r ic .... s t .... Vdash a 1 divided by two-r t .... t-i l Vdash a Vdash a t a t sit t h n .... L Vdash i f 1 equal-t 2 c i .... s .... a .... 1 period line-line C .. o .. r .. o .. l-S lu ap r-K p-h sub y a-o 3 s s period e 4 n .. i .. t p semicolon t-a sub semicolon b 2 i-K c i p-t n two-CR-h a Vdash s-L Vdash dagger a .. p .. b .. b Vdash .. t .. h .. Vdash .. a .. n .. g .. l .. a b p a .. t n-b d p .. .notdef-n h t-grave sub a Vdash n b Vdash a v-dagger to the power of b p .. c p .. a .. r .. t a i .. ic .. b-u .. al .. a .. re comma v .. i f-K sub comma-r t-s grave sub e-h-a NAK b EM period P .. r .. B-o L sub y-o .. ef .. m .. grave sub m-a .. i a Vdash three-a t .. period Vdash Vdash 3 a Vdash .. t h .. Vdash Vdash .. n .. t h .. f B-t period p slash .. p w r h i .. s .. a .. t .. l .. e .. a .. s .. t dagger a-t .. p b e e-b s-t .. p w .. he .. ee .. n i i c t as l a ed w-n i h .. a dagger a-v .. p e D b-EM comma .. a p n CR-two a d e .. x .. a .. c .. t .. l .. y .. w .. h .. e .. line-line O .... b .... s .... e .... r .... v .... e .... a .... h b i a c n-t .... dc a l q Vdash .... u d a .... a-d .... r b a i c-.notdef b r-a d-e a-t c d n-b sub a i sub c-d S .. i .. m .. p .. l .. Vdash comma notdef .. i .. p .. p .. Vdash .. d .. a .. n .. d .. a .. l .. t .. Vdash .. r .. n .. a c .. a .. n .. t .. s .. period .. C W .. Vdash o .. c r .. a o t n .. ol a .. a .. p h-r .. p ey .. l s y e .. four-t .. r .. i s .. e .. c .. a r .. o .. p .. e .. l .. e .. n .. g .. t .. h .. f .. o G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l .. y-u 6 .... Elizabeth Denne comma Yuanan Diao and John M Sullivan a .. r .. DC2 to the power of d sub DC2 .. r .. b .. DC2 .. K F i g .. r .. 3 Vdash L .. I e n .. m t .. comma m-h t Vdash a h p 3 e period-r c 3 i .. K-r a c b a-l e n .. t d a t a h sub n r g a b 2 b-l K e-e to the power of g-h a-t h-a K a to the power of s e r sub a to the power of NUL e sub period-d to the power of D to the power of a-d = two-d sub period-s i i-DC2 Vdash n-slash-s h e .. r-d e D .. a NUL b j .. n DC2-d .. s T .. h .. e .. o .. T-r e h .. e-m r-three period-o .. p 5 e .. l .. e .. n .. g .. t .. h .. o .. f .. a .. k .. n .. o c .. a .. n .. t .. i-EM comma two-s sub EM a-o 3 EM comma l re e a s s P .. r .. o-R .. oe f K c-t Vdash a h l .. a-e .. v Vdash Vdash .. n .. i .. t .. t .. h .. i .. c .. k .. n .. Vdash .. s .. grave sub period-s a .. b bc .. Vdash d t .. h .. Vdash .. q .. a .. d .. K-r ii .. ns .. t-.notdef .. c ha Vdash .. n Vdash s-t .. b a-two CR a l n .. wd c .. a .. Vdash Vdash Vdash .. a .. Vdash i m .. p .. Vdash .. q .. u a .. d d r .. i Vdash a-s c r a .. n .. t-v grave-comma NAK t h-grave sub d s NAK a EM-r d s .. C n .. Vdash o .. r .. o period .. l I .. l na .. r .. h y e 3 .. period .. a 4 .. s .. e .. o .. f .. a .. c-fl .. a p b-n .. p a .. r .. Vdash .. r .. Vdash grave-v Vdash NAK grave sub c r a-C grave sub b Vdash d d 2 EM period .. s I .. Vdash .. t .. h .. c .. Vdash .. e .. o .. f .. Vdash .. n .. l t .. r .. i .. a-s comma-b-e b .. c-c aa d n-d-n d as a-r .. e .. r .. grave v NAK grave sub a e c-C grave-r sub b .. C grave sub d-e d a 3 EM-comma period line-line O .. f .. c .. o .. u .. r .. s .. e .. comma .. a .. n .. y .. c .. l EM o .. i s n e-d .. e d .. c n u d e-r .. v n .. o .. t .. t e .. d .. o r .. h .. s .. Vdash .. n y .. q .. u .. d r i .. c .. two-n sub EM period s .. B .. ubt e r .. b .. r .. r .. i .. C to the power of 1 .. c .. l Vdash o C-.notdef-s to the power of 1 .. e .. n i s n .. t h o e a r .. o u n d c i .. cl e q .. u .. a .. d .. r .. i .. s .. e .. c .. a .. n .. t .. s .. comma .. s .. o .. a .. t .. l A .. l .. t .. h .. o .. u .. g .. h .. K .. u .. p .. e .. r .. b .. e r .. g .. h .. a .. s .. s .. h a .. n d .. D .. e .. n .. n .. e .. s .. h .. o .. w .. s .. t .. h .. a .. t .. i .. n .. f .. a T .. h .. e .. o a r r .. e e .. mn .. 3 o .. period t .. 5 a .. s .. g .. o .. o .. d .. a .. s-bracketleft 8 closing square bracket t-o h-r bracketleft-e-e 4 closing square bracket period p i .. m p .. r .. o .. v .. eS .. o-e u-c w to the power of t-r ei b owo niu .. n-five ld .. cs .. o comma nis n .. i .. d .. e .. r q .. u .. a .. d G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l y u Ropelength boundsfrom quadrisecants .... 7 4 .. Length bounds in terms of segment lengths G .. i .. v .. e .. K-n .. i a t .. h t .. q h .. a i ab .. c-c-d w r k d c .. s k a .. n-n c .. a-b o Vdash n t-.notdef tn .. d t .. h .. Vdash .. d .. i .. s .. t .. a .. n .. c .. s .. a .. l .. n .. g .. t .. h .. Vdash .. q .. Vdash Vdash .. i .. l .. l .. a .. b .. b .. r .. Vdash v .. i r a-W .. t a-D Vdash b-j j t s W h .. .notdef-b-D s c j .. n t W h .. D-c NUL j-.notdef-d j W .. e .. s .. t .. a .. r .. t .. w .. i .. t r h-comma s as .. t-n f d m-o r-e .. lq .. ou .. w a .. d-e .. r b-i L .. e .. m .. m-I .. a a-f .. b four-i .. s .. d a-one notdef .. i .. p .. p .. Vdash .. d .. q .. a .. d .. r .. i .. s .. Vdash m .. i d .. s .. Vdash .. g .. m s NAK Vdash 1 n .. tF .. Vdash h rat .. sh r .. l NAK one-r-.notdef .. tnmh g Vdash Vdash n-r h-.notdef t ih CR c .. a i o .. u .. t-B period to the power of s-b slash semicolon is die m t i NAK 1 t-l a-h CR b e-r l sub d n-l .. y .. e is B-f period o-c slash period P .. r .. O-o r-o f-i .. e .. n .. t .. t .. h .. e .. k s D n .. o-b NUL t-c j .. less 1 .. n .. t .. h .. .notdef-L e n e .. u-m-b e .. i .. grave-t sub c h less e .. two-EM = r-grave sub b .. d less c .. two-EM = B .. Vdash t c .. s aia n-b nd a-c d r .. Vdash .. r .. Vdash Vdash .. r Vdash .. h .. grave sub a-c v a .notdef-EM .. n grave sub b NAK d EM period .. T .. h .. i .. s .. i .. s .. Vdash .. c o .. grave sub c r a D grave sub c i C-a-c grave sub t-a a n grave sub b d sub d D grave sub b .. C grave sub d .. T .. h Vdash Vdash .. c Vdash .. n d .. Vdash t a .. t Vdash Vdash .. n .. t f .. Vdash C Vdash Vdash o-w Vdash r .. o period line-line L e .. m .. m-I .. a a-f .. b four-i .. c period-s d 2 a .. n .. a .. l .. t Vdash .. r .. n .. a .. t .. i .. n .. g .. q .. Vdash r NAK 1 .. a .. n-t NAK d 1 .. W i .. t .. h .. t .. h .. Vdash .. s .. Vdash .. a .. l .. CR .notdef-d .. r a .. i-u Vdash Vdash .. n i Vdash t Vdash B period b slash open square bracket B period c slash period s-I NAK 1 a .. s .. w .. CR a .. l sub c l .. comma e ts h-B period to the power of b-o slash aun t-CR-n b d l sub d i .. de .. e-s B period o-c slash period P .. r .. I-o Row 1 o Row 2 D . j b j less 1 .. t .. h .. L Vdash .. n e .. m-b y-e m .. ia grave-t sub a 3 less e .. two-EM-one = r-grave sub b .. d less a .. two-EM S .. i .. m .. t D l .. c j d r-j less 1 .. y .. t comma .. h i Vdash n-grave sub c .. b less i .. EM-two-t = grave-r sub d Vdash less-r c .. two-EM = B p .. r .. Vdash T .. f h Vdash e f o comma .. r we e m-grave sub a h 3 c EM-a period v n-five e-grave sub b NAK d EM .. c .. o n .. t .. r .. Vdash .. i c .. t h .. e .. i .. n .. e .. q .. u .. Vdash .. l .. i .. t .. r semicolon t NAK 1 .. B e .. Vdash period c-a aa h u-.notdef-d-n a d-s .notdef-w-r Vdash .. h B-.notdef period b t-slash a .. B period c-d slash comma .. t .. h .. e .. r .. e .. m .. a .. i .. n .. C .. n o .. g r .. s o period three-line-line-e A .. s .. s .. u .. g .. g .. e .. s .. t .. e .. d .. b .. y .. t .. h .. e .. d .. i .. s .. c .. u .. s w .. h .. e .. r .. e .. w .. e .. h .. a .. v .. e .. a .. n .. a .. r .. c .. o .. f .. a .. k .. n e .. x .. a .. c .. t .. l .. y .. t .. h .. e .. m .. i .. n .. i .. m .. u .. m .. l .. e .. n D .. e .. fi F-n i-o-r NAK one-r .. i l o-f Vdash r slash n-W t .. NUL-line sub r 2 1 C a .. r .. period to the power of one-c = r s-slash period .. i F r sub n-semicolon s NAK r 1 a .. DC2-n 2 d-OE 0 semicolon EM t .. h .. e .. m .. i .. n i .. m .. u .. m .. l m period r semicolon s semicolon DC2-W Case 1 line-NUL-r-c sub r 2 C s to the power of 2 2 s-o-DC2 i DC2 DC4 a r period to the power of one-c = r slash-c C o-a s-r period to the power of one-c = s slash-c Case 2 f r slash C f s slash C period DC2 EM i DC2 NAK a r period to the power of c-one = r c-slash C a-o r-s period to the power of c-one = s c-slash h .. e .. f .. u f r slash-c t i .. lol .. Vdash .. i .. s .. e .. Vdash .. i .. n .. n .. o .. t m e a .. s-s .. i d e .. x .. a .. c .. t .. l .. y .. t .. o .. m .. a G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l .. y-u 8 .... Elizabeth Denne comma Yuanan Diao and John M Sullivan a .. a to the power of 0 sub r ogonek check from p to b to the power of 0 .. s .. b Figure 4 : .. If points a semicolon b a-r e at distances r semicolon s NAK 1 open parenthesis respectively closing parenthesis from p comma then t .. h .. e .. s h .. o r a t-t .notdef-e-b a s v t Vdash .. B period d-p-u slash i r s n v p l-e-g Vdash f n r-.notdef .. o period m E .. i t e Vdash m .. n .. t .. r .. open parenthesis i .. n .. t .. e .. c Vdash .. e .. i l .. at-l sub B period p slash period .. I Vdash n t .. i dt i .. t .. s .. l m-e period r semicolon s-n sub semicolon dagger g-a .. t slash-p sub period L e .. m .. Aa .. n 4 CR f-a sub 3 r o-a .. t c b .. s .. t .. a .. y .. B-i period p slash-n g-h Vdash Vdash s .. t .. s n i m period j a NUL p j semicolon j b NUL p j semicolon dagger a .. y p-slash period P .. r .. L-o e-r-o W ft .. D-a NUL j-p j .. n s W .. D-b NUL j-p j .. Vdash .. e .. d .. i p-s t .. w a .. r semicolon t-c-s NAK e h-one s .. t a l .. DC2 W t D a .. dagger b .. b Vdash .. t .. h .. Vdash .. a .. n-a NUL p-g al .. n-b-.notdef NUL p-b-d Vdash Vdash t hw Vdash Vdash Vdash sn h .. Vdash r a t-t b .. s .. t .. a .. y .. i B n-W g-B-D period p slash-.notdef Vdash i t t .. s .. e i r d i s .. t .. h e .. s .. t .. r .. C to the power of 1 j Vdash Vdash i t n t r .. a .. ig .. Vdash Vdash a t Vdash at-.notdef sub B-n .. a r .. g m .. r .. Vdash .. a .. at B .. a-c i-n .. rd .. ca ls Vdash .. t ar at B t .. b-.notdef .. I .. n .. Vdash .. i .. t .. h .. Vdash .. r .. c .. a .. s .. Vdash comma .. w .. Vdash .. s Vdash a Vdash p t b .. h .. sah i F n .... i .... g closing parenthesis u-period .... rI .... n-e four-t .... h .... i .... s .... p .... a a a .... n-b-n t d-e a comma n .... at-d sub g-B r Vdash a-L ogonek W w t D-t sub t-t-a Vdash to the power of 0 n .. check W .. D b .. p to the power of 0 dagger w b h e a to the power of 0 .. n .. b to the power of 0 .. e t h .. e .. p o .. t s o .. t a c-n e-o-ogonek n D 1 equal-y r a T c .. caron-o D 1 = s .. C .. l .. ogonek a-C check NAK r DC2 l h y e i .. f-t .. h .. e .. s .. h .. r .. t e .. s t p .. Vdash t a t .. b-.notdef .. w .. i .. t .. c-r-n-NUL-.notdef-l-line r to the power of 2 C s to the power of 2 2 .. g .. t DC2-o-s h-period .. s ogonek-I f-C check DC4 DC2 .. h .. e .. t .. h .. s h .. r o .. f .. C 1 jh o e i .. n .. d .. e .. s .. c .. r line-NUL sub r 2 1 C NUL DC2 NUL period ogonek-C check SOH C-slash NUL-line sub s 2 1 D f r slash C f s slash C period NUL EM line-line n .. i .. m .. p .. o .. r .. t .. Vdash .. n .. t .. s p DC2 D EM period .. Hl ec .. Vdash e s ee .. Vdash .. r .. w .. h Vdash ogonek-l w-C check DC4 DC2 s .. o .. w .. e .. g .. e .. t .. t .. h C .. o .. r .. o .. l-I l-a a .. a n-b l d i-y Vdash 4 a-period .. t r ad .. s-i-n asd l-t Vdash an n g .. c .. Vdash s-p p-p Vdash s t .. h dagger a a .. D p EM-b closing parenthesis .. t .. h .. e .. n .. t .. h .. e .. l a e-t .. b a g v to B-h i-period sub p slash o-d-i s-i-f .. n a-a f r slash C f s slash D line-NUL sub r 2 1 C a .. r .. period to the power of c-one = r slash-s C NUL-n-line sub s 2 1 C a .. r .. period to the power of one-c = s slash s e .. n .. t .. e .. t .. h Vdash .. t .. e .. s p .. e c .. Row 1 D Row 2 l . a .. Vdash s s e .. o .. a-f .. t p .. p i Vdash b .. y .. D .. Vdash .. m .. i .. t .. r .. Vdash .. s .. c .. Vdash comma Vdash .. b .. b .. Vdash nine-r 1 s .. i-zero B-n .. va Vdash Vdash s .. m G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l .. y-u Ropelength boundsfrom quadrisecants .... 9 g .. e .. o .. m .. e .. t r .. i .. c .. d .. i .. la .. t .. i .. o .. n .. o .. r .. d .. i s .. t .. o f .. o .. r .. t .. h .. e .. d .. i .. s .. t .. o .. r .. t .. i .. o .. n .. o .. f .. a .. c .. one-l .. o 4 semicolon s open square bracket to the power of i-nine semicolon 1 n bracketright-d s .. h .. a .. r .. p .. e .. r .. b .. o .. u .. n .. d .. s .. i .. n .. t .. e .. r .. m .. s .. o .. f m .. i n .. i .. m .. u .. m .. hyphen .. l .. e .. n .. g .. t .. h .. a .. r .. c .. a .. v .. o .. i .. d p .. l .. a .. n .. e .. c .. u .. r .. v .. e .. s .. t .. h .. e .. y .. a .. p .. p .. l .. y .. e .. q a .. n .. d .. S-bracketleft 6 closing square bracket u-f o-l r-l .. if vu ar nt .. h .. e .. r .. d .. e .. v .. e .. l .. o .. p .. m h .. a .. v .. e .. d .. i .. s .. four-t sub period L e .. m .. m-L a-e t-four-b .. b c-period Vdash 5 da .. n .. a .. l t .. Vdash .. r .. n .. a .. t .. i .. n .. g .. q Vdash .. r .. i .. Vdash .. n .. t .. Vdash .. d .. i r W n .. D-a t b j-h .. s W .. D-b NUL c j-j a-n t W l .. w-c-D NUL a-d j .. y .. period Vdash .. e L f .. t .. Vdash .. e g a .. m b .. c-n Vdash t grave-d sub a Vdash NAK d f r slash C s C f t slash period-period .. T .. h .. s .. m a e .. b i o a .. s .. i .. m .. p .. l .. e .. q .. u r semicolon t a NAK 1 d-period P .. r .. Io .. no .. f e .. i .. t .. h .. e .. r .. c .. a .. s .. L e .. e-parenleft a-m f-n sub m .. o d a r a 4 t s period h 2 w e e w .. e .. fi .. n .. C .. d o comma .. r u .. o comma slti CR d-h-n sub a g-a i r t-.notdef sub y Vdash u-three B-t .notdef-period-period b slash-i bracketleft-two B period-.notdef c slash period .. i .. n .. t .. h L .. e .. m .. comma m .. t .. ah .. 4 e .. period s 3 h .. d-o t r .notdef-a t .. Vdash e t s .. s t i .. a d .. Vdash c .. t .. h .. C to the power of 1 j m i b a .. r .. i Vdash .. p i Vdash .. c Vdash Vdash .. th Vdash Vdash .. a .. Vdash t Vdash r .. n .. a .. t .. Vdash b .. Vdash t w Vdash .. n .. Vdash a r .. c .. Vdash .. i n .. th .. b Vdash u .. n .. d a .. r .. i .. Vdash Vdash f .. h .. Vdash b .. a .. Vdash Vdash Vdash period H .. r Vdash comma a .. Vdash Vdash Vdash n .. g t .. s W x D-a b NUL t-j-c j .. y A .. s C i n o .. r .. o comma .. lt l ha .. e r .. o-y .. v 4 .. e period .. 4 l .. e f a r slash C-s s C f t slash a .. s .. d .. line-line 5 .. E .. s .. s .. e W .. e .. h .. a .. v .. e .. s .. e .. e .. n K t-i .. s h .. n a .. t t Vdash n-t Vdash h Vdash e b .. Vdash Vdash .. n .. d .. s .. Vdash .. n .. r .. Vdash .. p .. Vdash l .. Vdash .. n .. g .. t .. h .. comma .. b .. c .. a K .. K .. Vdash .. p .. 1 Vdash three-i-r b-n t-.notdef r r .. g .. d .. Vdash .. c .. Vdash s .. d s .. .notdef-t-s n-.notdef h c .. i-a n to the power of a-n Vdash t-l-t .. is .. Vdash w .. h .. i .. c .. h .. h .. Vdash .. c .. a .. l .. l .. d .. t .. Vdash .. p .. Vdash .. l .. Vdash .. g .. i .. c .. a b .. e .. t .. t .. e .. r .. r W .. e .. e .. x .. t .. e .. n .. d .. K .. u .. p .. e .. r b .. CR e-a .. r f g a .. quoteright k n-s-K i Vdash d-t s .. e .. fi s c .. a .. p .. t .. Vdash .. r .. i .. n .. g .. p .. K-a .. rG .. .notdef-t n-.notdef .. f .. rt b sub h-i .. K a-t sub k .. ln .. gl Vdash to the power of t-y t t .. h hline s .. Vdash g .. S D m a .. n-b Vdash tr .. m .. s .. quotesinglbase a Vdash k .. Vdash n p .. h t i t n .. Vdash s .. d .. Vdash c e .. t .. Vdash .. t o n .. n .. c .. i .. Vdash .. t .. h .. e .. s .. Vdash .. m .. e t .. w o .. e .. t .. c .. e .. s .. closing parenthesis .. period k .. n .. o-quotesinglbase sub endash t t-g .. r D .. e .. fi S-n i-u .. t ogonek i p-caron oo n d-CR e-r .. e .. h .. r .. e .. d i .. s .. jo p t n .notdef-q .. s Vdash i .. m a .. k .. n quotesinglbase Vdash .notdef-t r t .. Vdash p h-d period .. T .. h e .. n .. e .. s .. Vdash .. y .. period t ogonek-h Vdash semicolon t .. check tn h-semicolon ees CR slash r h .. r e .. D b Vdash Vdash d Vdash .. n s .. d .. Vdash d ogonek open square bracket b-caron h y Vdash v t i h n .. Vdash .. k .. o n .. Vdash n .. t .. r .. o n .. ogonek bracketleft-t CR period open parenthesis .. W .. e .. Vdash .. l .. l .. o .. w .. D f .. hyphen-a .. i n n .. d .. e i r n .. e t c Vdash .. r check i semicolon n-.notdef sub t l .. a .. t .. t .. e .. r .. a .. r .. e .. ogonek bracketleft-c check i-e r-s .. tk .. a G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l .. y-u 10 .... Elizabeth Denne comma Yuanan Diao and John M Sullivan n .. e .. q .. u i .. v .. a .. l .. e .. n .. F .. d i .. e g comma u-i n .. s-r i .. e-t-a s-five o X-f W o D-H11938 to the power of 3 i period-l-l l-ogonek-o-bracketleft CR SOH-u w n .. o .. n .. s i .. d .. e .. i p-ogonek open square bracket a-caron .. r X-a l-H .. r p .. c a b .. r to the power of y-v e-a m-l .notdef-l a-ogonek l-bracketleft check n i .. o b .. o .. u .. n d .. n .. Vdash n .. n .. X .. W .. s .. c-e .. h b Vdash e Vdash d-s .. e .. d t i h n .. Vdash .. p t .. r .. i X-v .. i .. aS .. li inn c .. Vdash .. t .. h Vdash .. h Vdash .. m .. X l-i H11946-s-.notdef .. g .. h y i s s-f i t w .. e .. dotlessi-t ta .. o k .. he .. a .. v .. e .. l .. i .. n .. k-ogonek open square bracket CR-i period n T .. gh .. ni us .. m-dotlessi d-u e-b n e t-i e-r h .. o .. m .. o .. x-t sub 0 2 i .. p e .. .notdef-y-p b-r sub period closing parenthesis .. a L b e .. a X s .. Vdash a pn Row 1 d Row 2 D . period l n ogonek Vdash t semicolon-t 2 caron-EM sub one-.notdef period to the power of semicolon-r-X semicolon x sub 0 CR-slash b .. e .. t .. h .. e .. h .. o i period m .. To period h t ogonek e o semicolon p-i i-caron sy n semicolon c e CR .. l s-i slash h-a i s e ss n t W .. e .. s .. period-a y-ogonek semicolon caron-h s semicolon a .. t CR .. n slash f .. i Vdash .. li .. s .. n .. t .. i .. n .. e .. h period ogonek .. n semicolon n .. o .. n N o .. w-NAK b .. l e .. at .. m .. e .. r .. i .. d-ogonek open square bracket CR-i nae x-n sub 0 a-parenright l .. i-r o .. n-o .. p .. h open parenthesis el .. i-k X sub n-n-period t .. h .. e .. c .. OE h period .. m ogonek .. m semicolon .. u check .. s semicolon a n CR o t slash-n o t semicolon r .. i-NAK .. v c141 period Vdash ogonek l .. semicolon caron-s et semicolon r .. CR e slash Vdash s l period ogonek p CR check to the power of i q F i Vdash .. u .. : .. I .. quotesinglbase t h r .. ogonek bracketleft-k h-caron open square bracket CR .. t .. e-h e .. r .. d .. period r e-ogonek d semicolon t check s s .. e .. n .. i .. Vdash .. l period .. To .. s .. e .. t i D h period-s comma ogonek semicolon t check ogonek semicolon-bracketleft check CR-t hh slash e cp l .. i .. n .. k i .. n .. g n ogonek bracketleft-u CR comma m a bn e dr .. n z .. e-o tr eo .. t w hia t t h i .. n .. t .. h e .. k n .. H11938 to the power of 3 X period bracketleft-c-ogonek CR o period slash m .. n p .. t l h .. i-e s m .. check .. n u t ht er s .. e g .. hline p m comma q e sn o tw .. e .. e .. q .. u .. a-ogonek ol fl .. t-y ogonek-s e-bracketleft CR i y-k .. s nt e h-o .. s T .. h .. i .. s .. n .. o .. t .. i .. o .. n .. i .. s .. c .. l .. e .. a r .. l .. y .. a .. o .. p .. o k .. n .. o quotesinglbase t Vdash t r .. e .. d .. h .. period .. F .. o .. r .. Vdash .. n .. i n t .. r .. o .. d u .. c .. t .. o .. n .. t .. o Vdash .. u .. f .. 2 f-semicolon n-m .notdef-o .. e .. t .. h .. Vdash .. s .. i .. n .. quotesinglbase e Vdash rh Vdash p .. he hr Vdash .. v e c .. d u .. r .. s .. i .. t .. u .. Vdash .. t i .. o .. t .. h e .. r e .. i s .. n .. o .. d .. i .. s .. t .. n .. t i .. o e r .. t .. i .. c .. e s period .. T h .. e .. t .. r .. e e .. r c .. s .. Vdash .. p p .. r .. o .. Vdash c .. h i a .. f .. f .. e .. c .. t .. i .. n .. g .. t .. h .. e .. quotesinglbase-t sub endash o g .. p L e .. m .. m-I .. n a .. a 5 k-period-quotesinglbase 1 to the power of n-.notdef r o .notdef-ogonek bracketleft-t check t-bracketleft CR e .. t h .. e .. period t ogonek-r .. i semicolon s-l-caron .. e semicolon t .. CR-r .. o slash n i .. f .. a .. n .. period d CR .. o semicolon in P .. r .. T-o h-o f-e .. h .. o .. m Row 1 o Row 2 D . t-period ogonek-o p-semicolon caron-y-n d-h to the power of 0 D c h-CR period CR-slash-a s-semicolon p s r-caron o e semicolon s ogonek-e e slash d f r x sub 0 a-m Vdash Vdash n-caron .. g n d .. h e .. n .. r e t .. u r ogonek CR period-.notdef c Nk ow t Vdash e .. t hs .. h to the power of NUL 1 t-h to the power of 0 G e o .. m .. T-e sub o t p r sub o-comma .. yl V .. ampersand-o-o .. g l y u Ropelength boundsfrom quadrisecants .... 1 1 i .. s .. h .. o .. m .. o .. t .. o .. p .. i .. c .. ogonek open square bracket CR-o period .. a S .. p n a c r .. a .. l t .. o-l .. r e u l s .. o h f-.notdef g .. r .. o .. u .. p .. comma t .. h .. i .. s .. p .. a .. r .. NAK a-period .. lI lt .. e f .. lo OE .. c NAK-h-l c141 D o OE m o h-NAK 0 sub c141 sub period-m-w line-line T .. h .. i .. s .. c OE .. o h-NAK period m ogonek .. m semicolon .. u .. h-caron .. t semicolon a-t CR t slash o .. c141 r e s .. u .. p .. i .. n .. t .. e .. d i .. l .. l .. l .. Vdash .. t e .. r .. e .. o .. l-o sub check f .. o-e check semicolon .. i d .. t .. o Vdash .. n sb e .. h .. e .. p .. e .. s f .. o .. l .. o w .. s check to the power of 0 .. f check .. rt .. e .. l-n ello .. ogonek p-bracketleft CR-s .. l .. o o u Vdash n .. Vdash d .. e .. q .. d-t f .. o .. check to the power of 0 1 comma o .. t wh .. se .. n .. l .. o .. o .. p .. s .. b .. a p comma .. k a .. F w-s .. iai g period CR ogonek p sub check l to the power of check to the power of q to the power of OE to the power of h-NAK period ogonek i Vdash .. e NAK : Vdash .. f e .. i .. d i .. Vdash n .. ogonek-c u-bracketleft CR-r .. teh e-l sub i .. nthi n .. c OE .. NAK-h period semicolon ogonek r .. i-semicolon s-caron .. h slash c141-o .. m .. o .. t l caron-o a pl .. o-caron period W .. e .. a .. p .. p .. l .. y .. t .. h .. e .. d e .. fi .. n .. i .. t .. i o hline D .. e .. fi n-I f-i-K i t .. s-i a-o .. n k-a sub semicolon b 2 .notdef-K .. tl S D a a-t-n d-b CR-s a .. i-a b Vdash y-.notdef t .. r .. n g .. Vdash .. y i .. n-K i .. f .. Vdash r-quotedbl zero-greater-.notdef tv Vdash h .. .notdef-r y-r Vdash .. quotedbl notdef x p i e r s t t .. s S s Vdash w Vdash t m i t o Vdash h n .. Vdash .. f n t .. Vdash .. a .. S-c to the power of 0 .. u c-r h .. K open square bracket S to the power of 0 .. a n .. e .. b e quotesinglbase notdef e Vdash d r .. Vdash .. p .. h .. period n a CR semicolon S to the power of 0 sub semicolon-h i-b sub CR slash a open parenthesis .. s .. t .. r .. o .. n R .. e .. m .. A .. rl kl .. o-five .. quotedbl period w p 2 e-i r n t g .. r t .. b h .. e .. i .. e .. n .. s .. u .. r .. e s l .. o .. s .. e .. d .. i .. n .. t .. h e .. s K-e .. t a .. n f .. d Vdash l l l Vdash s t e s c Vdash Vdash n s t-S ih .. n-o a f tn .notdef-K sub d Vdash .. c .. Vdash l .. d .. a .. l .. l .. Vdash .. w .. t .. h .. p .. r .. t .. Vdash .. r .. b .. a .. t .. i 7 .. s .. h .. t .. h .. r .. Vdash .. s Vdash .. l .. t .. i .. n .. g .. d .. Vdash Vdash S to the power of 0 .. e quotedbl notdef t c .. i l S i .. n .. C to the power of 0 .. h .. n-.notdef semicolon .. it .. t hu .. Vdash c .. Vdash u Vdash .. d .. b .. Vdash Vdash .. c .. a .. Vdash y .. k .. n a .. b .. o .. u .. t .. t .. h .. h comma h-a n-o m-d n-o t-o .. ot I .. open square bracket sub n-three closing square bracket i .. t .. w .. a .. s .. s K i-h o-s .. w a .. n n .. .notdef-t .. n h .. k-a CR-n-t a .. i .. Vdash f i n t Vdash Vdash h .. .notdef-.notdef n-n c .. Vdash n .. Vdash x t comma .. th .. i Vdash f Vdash Vdash Vdash .. w Vdash .. i m .. m Vdash .. di .. a .. Vdash b sub y comma .. b .. Vdash .. c a .. Vdash X W .. D-H11938 to the power of 3 X K a .. r .. Vdash Vdash .. q .. Vdash .. a .. f Vdash .. r .. dotlessi-a .. n Vdash v Vdash i n n .. k .. n i Vdash n t comma i i .. K i .. s .. h .. Vdash .. m .. t .. Vdash X p Wi Vdash c .. ca .. a-l l-n .. y .. st .. r .. Di Vdash h-i .. an .. quote-l s .. t .. a G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l y u 12 .. Elizabeth Denne comma Yuanan Diao and John M Sullivan Theorem 5 period 3 .. I .. a sub f-semicolon b 2 K a .. n .. d CR a .. a .. n-.notdef CR b-t sub d h-a .. r .. Vdash i .. n .. Vdash Vdash Vdash .. K i .. s-i a Vdash Vdash .. n hline P .. r .. L-o e-S-o b f-t Vdash .. t .. h .. .notdef-a b-s p .. c-.notdef r a .. t n Vdash tr .. b .. Vdash .. d .. i .. f .. n .. Vdash .. c .. Vdash i K-t .. h .. Vdash k .. n .. CR .notdef-a open square bracket S a h n-CR b a bracketleft-d S b .. Vdash .. n .. d .. d .. i .. s .. k .. s .. w r K-.notdef .. mG .. l .. Vdash .. t .. h .. Vdash .. s .. Vdash .. t .. S t .. d f i Vdash s r k-D m-s .. spat a K n i Vdash s-.notdef-n h-t i-k i m .. a .. y .. h .. a .. v .. s .. l .. f .. i .. n .. t .. Vdash K-r .. sw Vdash h c .. it ci .. h .. i-n .. s D comma B .. y .. D .. Vdash .. h .. n .. quote .. s .. l .. D-.notdef bm .. y m .. aa n .. .notdef-w Vdash m .. cb a K n-d-i r-d-s .notdef-.notdef-.notdef d-n-p line-line-k-l D .. e .. fi A-n .. is t .. a i b-c o K-a i-n .. n s s-t .. s .. i .. n .. t-b .. i ta CR h-a l a s n .notdef-CR b-d .. a .. a .. r Vdash Vdash c Vdash A .. s .. a .. c-i sub b s sa t n r t o .. n .. i g CR a-f l b-parenleft y .. o e .. r s comma s .. e .. CR b-q n-parenright a u ii i sv a a l tl .. re T .. o .. c .. a .. l .. l .. a a .. b q cs u ds a .. Vdash d n r t .. i s a e l c comma .. a .. n c t Vdash Vdash .. l .. d s .. Vdash .. c a .. a b-b sub n act c-n s d-a d .. l .. l .. b .. Vdash .. s .. s .. n .. t .. i .. a .. l .. period .. B t .. h .. Vdash .. q .. Vdash .. a .. d .. r .. i .. s .. Vdash .. c .. a .. n .. t .. comma .. w Vdash .. r .. q .. i .. r Vdash b .. Vdash .. b .. Vdash Vdash .. n T .. d h .. e .. d .. o-comma r-a n s e .. i mm .. n three-e period l .. y-five .. t .. h .. o .. s .. e .. s .. e a .. l .. o .. n .. g .. t .. h .. e .. k .. n .. o .. t .. period .. T .. h .. a .. t .. i .. s .. comma .. f .. o f .. o .. r .. fl i .. p .. p .. e .. d .. q .. u a a ab d n-c r d-m-d sub i Vdash s s e .. t c .. b a Vdash n Vdash .. t s .. s s .. t q .. Vdash .. a .. d .. r .. i s .. Vdash .. c .. a bn m-c sub t Vdash ss comma t t .. bh .. F Vdash Vdash i sm g i-s s u h-d-.notdef o-n d e l w seven-i-.notdef s a e .. s .. s .. e .. n .. t .. i .. a .. l .. a .. n Case 1 d Case 2 b c Case 3 a Case 4 A i .. g .. r .. 7 Vdash .. h .. is .. t .. r .. f .. i l .. k .. n a Vdash b-t ic a .. l .. t .. Vdash .. r .. n .. a .. t .. i n .. g .. a b .. n-i sub c d-s Vdash Vdash ss .. ss Vdash Vdash n-n .. tt .. i-i a-a a .. l as ab Vdash n-c d-a-d .. r .. Vdash .. s .. s .. Vdash .. n .. t Ai B-a l-i sub C s-parenright .. D s Qi .. Vdash m a p d .. l r s .. i A-n a-c-B e C d D-a .. r .. e .. i .. n .. e .. s .. B-s e i C ns t ei s a M .. o .. r .. e .. f .. o .. r .. m .. a .. l .. l .. y .. comma .. w .. e .. c .. a-n endash ns g e D .... e .... fi n-A .... n nt s i Vdash a-o sub c-one a two-n a-colon colon-n-a sub n i t-colon Vdash Vdash Vdash .... n i .... fi .... .notdef-a h-.notdef a i-a a sub i-v C 1 Vdash .... n .... t i a i s .. u .. c .. h .. t .. h a CR a t-i a sub i C one-o .. an .. CR e-a sub i d-C 1 a-o sub i if .. nt c hl e .. u .. d-a sub j period K .. u .. p .. e .. r .. b .. e .. r .. g .. i .. n .. t .. r .. o .. d .. u .. c .. e .. d .. t .. h .. e G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l y u Ropelength boundsfrom quadrisecants .... 13 Theorem 5 period 4 .. I .. K-f i .. s .. a .. n .. n .. t .. r .. i .. v .. i .. a .. l .. k .. n .. Vdash .. t .. K-p h .. a .. s .. a .. n .. e .. s .. s A .. s .. m .. e .. n .. i .. o .. n .. e .. d .. a .. b .. o .. v .. e .. comma .. K .. u .. p .. e .. r h .. e .. a .. c .. t .. u .. a .. l .. l .. y .. a o .. b b c h-t d a .. i Vdash n .. a e l d .. l a-a t .. b-b-q sub h c-a u-r .notdef-a-c-n a .. r .. Vdash Vdash .. s .. s .. Vdash .. n .. t i .. a .. l .. period .. S .. i .. n .. c .. t .. h .. i .. s .. i .. s d .... Vdash Vdash .... n .... i .... t .... i .... Vdash .... n .... Vdash .... f .... s .... s .... Vdash .... n .... t .... i .... a .... l .... Vdash .... t .... h p .. r .. o K .... u .... p .... e .... r .... b .... e .... r .... g .... u .... s .... e .... d .... t .... h .... e .... f .... a .... c .... t .... t .... h .... a i .. n .. o .. r .. d .. e .. r .. t .. o .. s .. h .. o .. w .. t .. h .. a t .. e .. v .. e .. r .. y q .. u .. a .. d .. r .. i .. s .. e .. c .. a .. n t .. f o .. r .. e .. v .. e .. r .. y .. k .. n .. o s .. u .. c 6 .. L .. i .. m .. i .. t .. s .. o B .. e .. i .. n .. g .. e .. s .. s .. e .. n .. t quotesinglbase .. i Vdash r .. a .. p l h period-i .. s .. n .. a Vdash h .. Vdash .. t .. l .. m .. i .. t .. o .. f .. e .. s .. s .. t .. i .. Vdash .. l .. e .. c Vdash .. t .. s .. quotesinglbase e r .. i s .. o .. t .. o .. p .. c .. period .. I n .. f .. Vdash c .. t .. w .. e .. Vdash .. v .. u-h .. e r-s h .. w .. t .. h .. Vdash t .. Vdash .. i .. e .. n .. Vdash .. n .. y .. k .. o .. t .. t .. d .. Vdash .. r .. Vdash .. p .. o .. f u .. f .. Vdash .. c .. C to the power of 1 .. y .. n-c Vdash .. o closing parenthesis i e Vdash parenleft-i .. i .. n t Vdash Vdash p .. i c period .. O u .. r .. d Vdash Vdash .. n c .. Vdash .. n .. d .. i .. i Vdash n .. comma Vdash .. t .. Vdash .. c .. a .. Vdash Vdash .. h Vdash .. r Vdash .. a .. Vdash .. c .. a .. n .. t .. h .. a .. Vdash t .. h .. u .. s .. n .. o .. t .. a .. k .. n .. o H .. e .. r e .. comma .. h .. o .. w .. e .. v .. e C to the power of 1 semicolon 1 comma .. a o n .. u-d .. r Vdash g-k i n v .. o a t .. i s m .. p-a f .. o .. r .. t .. h .. e .. l .. i .. m L e .. m .. m-L e-K-a b t-six Vdash period a 1 .. k .. n .. Vdash FS 0 greater f .. a Kn h e d C to the power of 1 k n-k Vdash n t .notdef-w .. h i c K i n .. t .. h .. f .. Vdash .. l .. l .. Vdash w i .. n .. p-g a .. s p-n to the power of 0 .. dn e s-w i Vdash .. i .. c .. .notdef-quotedbl r less t to the power of four-r a .. n .. d .. t .. h .. Vdash .. i r .. t .. a .. n .. EM-g 6 = .notdef-n h K a-v .notdef-n n-K-c to the power of 0 dt .. e Vdash m-r b s .. e a-n Case 1 i Case 2 r t .. h .. e .. i .. s .. o .. t .. o .. p .. y .. c .. a .. n .. b .. e .. c .. h .. o quotedbl period N .. o .. t .. e .. t .. h .. a .. t .. t .. h e .. c .. o .. n .. s t .. a .. n .. s .. h .. e f .. r K-o t .. .notdef-K-m to the power of 0 .. x c e .. FS two-s = Vdash .. c .. a .. n .. h .. a .. v .. Vdash .. s .. t .. r .. a .. n .. d t .. a .. n .. g .. e .. n .. t EM v-two we .. ec .. t-c o-a .. rn .. sh .. ea .. x K to the power of 0 sub period-e P .. r .. R-o e-o f-s c-K h a .. a l .. s e .. Vdash s n o .. i .. quotedbl-t less-t-one = 4 h .. C-i l-c Kk .. a i n r s .. ls .. y-s-t .. i a ub e .. ar .. K-u Vdash .. a .. c .. h-p to the power of 0 2 K-p to the power of 0 .. or i r .. e-n t-p .. o n .. ds t .. o .. o-p two-m e-K .. o b .. n Vdash t t a .. Vdash .. n .. i .. q .. Vdash Vdash .. p zero-n 2 K a .. wr h .. s .. t-c .. hp .. i Vdash i j p to the power of 0 NUL p-i-t j t-less .. h quotedbl f-p-i to the power of 0 .. n e n-d i t h 2 quotedbl .. f-p .. B L .. y .. e .. m .. comma .. m .. a h .. 3 e .. period a 1 p 0 c .. l-p i .. e s n a .. g-a sub t t r .. h-two-m sub c-quotedbl f-s s r-i G e o .. m .. T-e sub o t p r sub o-comma .. yl V .. ampersand-o-o .. g l y u 14 .... Elizabeth Denne comma Yuanan Diao and John M Sullivan a .... n .... g .... l e .... b .... e .... t .... w .... e .... e .... n .... t .... h .... e 2 a-t .... a r .... n-two sub c-quotedbl g less i-three-e EM-n-n Vdash h-equal .... v .... p-e to the power of 0 p c hu .. s i .. the .. n .. p 0 a d n is d k h a a t Vdash .. t a .. n g Vdash n-EM t 2 = Vdash f c t .. p-h r-zero w-a sub I n t th a Vdash r .. K Vdash comma .. t r .. a n .. v .. e .. s e .. o .. h e f o .. i a t .. o n .. f .. t .. e .. t hi c .. k s o .. o py K-f to the power of 0 .. o K a .. s .. t .. h .. Vdash Vdash .. n .. i .. Vdash .. n .. f .. i .. s .. Vdash .. t Vdash .. p p to the power of 0 .. p 0 .. c .. Vdash .. n .. i .. n .. g .. th .. Vdash .. t w a r .. d Vdash .. t Vdash .. t .. h .. Vdash x .. Vdash d .. b .. Vdash u .. n t .... h p to the power of 0 w .... n h i .... c .... h .... m .... o .... v quotedbl period line-line P .. r .. o .. p .. I s f C-i to the power of 1 semicolon 1 i-k-h sub n eo K sub i n-.notdef a 6 Vdash period Vdash Vdash 2 Vdash Vdash n CR t-a sub i to the power of b-i sub i a-a n Vdash i-a f r h-K sub c-i-e .notdef-c Vdash n i .. C-n to the power of 1 Vdash Vdash Vdash t hi c .. k K .. i w m .. a-i-i sub i exclam-t sub a n-a Vdash n-b sub i d-exclam b .. t .. h .. Vdash CR-n a .. i t Vdash h Vdash Vdash Vdash Equation: f .. o-K sub period P .. r .. W-o .. oe .. fc .. a .. n .. r .. e a sub i D d a u b sub i D c b b .. t y .. o .. a .. t p .. h p .. e l .. c y a i s n open parenthesis .. a .. p .. p .. r .. o .. a .. c .. h K-i sub i period hline G .. i .. v .. quotedbl e zero-greater sub n .. w a Vdash n p .. y r .. Vdash 2 quotedbl .. p t e h r .. Vdash t .. r .. .notdef-a m-i-b t-s a i a k-o .. i o n f .. g .. a quotesinglbase period .. T .. h .. e .. n .. CR a .. i sub b s-d .. ee .. fi F .. o .. r .. l .. a .. i r .. g-t e-h K-e sub i i-k n Vdash n o .. i u quotedbl-t g i h-n-K-f .. L I sub i b t .. t h Vdash .. a .. b .. i .. Vdash d Vdash Vdash .. c .. r .. i L .. b e Vdash d m i w-m n .. Row 1 D Row 2 6 . sub i period K sub i one-slash period .. S .. K sub i i .. Vdash Vdash Vdash Vdash .. n .. t i .. a .. Vdash comma .. b c a .... n .... Vdash n-quotedbl notdef d p .... e-a r n .... t .... S sub i to the power of 0 b-f S sub i .... t c-i o .... quotesinglbase sub i W a K-D sub i open square bracket S sub i to the power of 0 i .... Vdash .... a n underbar Vdash m .... b Vdash d .... d quotesinglbase Vdash r .... p .... h .... S-period sub i to the power of 0 W to the power of 0 S e-D-I sub i period to the power of S-t sub i to the power of 0 slash .... it .... Vdash i .... s .... i .... s 2 t-quotedbl pheer d .... eu s .... a-i .... b t-B di .... y-o .... d K sub i open square bracket S sub i to the power of 0 i .... s .... i .... s K-o open square bracket t-S sub i to the power of 0 comma o .... sp .... oi .... c-t t-h .... e o .... l .... a .... line-line-a C .. o .. r .. o .. E-l .. l v a e .. r r .. y y C to the power of 1 semicolon 1 o-period-k n-three n o t .. tr .. i-h v-a s-i .. aa .. ln P .. r .. A-o o C to the power of 1 semicolon 1 k n .. .notdef-K i .. C to the power of 1 a .. i .. i .. t Vdash .. f .. p .. Vdash .. y .. n .. Vdash m i a .. Vdash .. k n .. Vdash b .. Vdash g Vdash .. n .. Vdash r .. i .. c .. i .. n-one 3 h Vdash h Vdash .. n n-T b Vdash h y Vdash e .. o t rh .. ee .. my .. 5 h .. period a 4 v .. e .. e S .. o .. m .. e .. s .. u .. b .. s .. e .. q .. u .. e .. n .. c .. e .. o .. f .. t .. h K e .. w s h e e .. s .. s .. e .. P n r .. t .. oi .. p a .. period line-line 7 .. A .. r .. c .. s .. b .. e W .. e .. s .. C .. h .. o .. o-r .. o t w l .. h l-e aa d-t r .. e i-y .. v six-n-C to the power of 1 semicolon 1 three-r-k sub n y Vdash t n h o-a Vdash n a t n .. r Vdash i Vdash c a .. n t period O .. u .. r .. a .. im .. i Vdash .. t Vdash .. n .. d .. t h Vdash Vdash a-CR sub p Vdash t sub q Vdash r n ag thhi c-K .. ka a s .. t .. h .. i .. s .. t .. Vdash .. g .. Vdash t .. b .. Vdash .. t .. t .. Vdash .. r .. l .. Vdash .. w .. Vdash .. r .. b .. Vdash w .. h .. a .. t .. h .. a p .. p e .. n .. s .. w .. h .. e .. n .. a T .. h .. e .. o .. Sr ue p m-CR a .. i 7 Vdash o i-period s 1 .. e h .. Vdash b .. Vdash u .. n .. d .. a .. r .. y .. Vdash f .. t h Vdash Vdash Vdash .. K Vdash .. h .. CR-a a .. t-i sub c Vdash i-.notdef Vdash s Vdash .. n .. c sub i a .. Vdash comma .. b u .. t .. t h Vdash r .. .notdef-K a i-r Vdash t i h n .. Vdash Vdash n Vdash Vdash n-d t c .. l .. o a as .. n-e-c period t-d closing parenthesis o T-K h-m .. e u .. n-s .. t .. i .. n .. t .. e .. r .. s .. hline e-a sub comma c-c .. a t .. nt .. dh e-i i .. s .. s .. o .. m .. e .. e .. a s .. b-period G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l .. y-u Ropelength boundsfrom quadrisecants .... 15 Proof .. I .. K-f d .. i .. d .. n .. Vdash .. t .. i .. n .. t .. Vdash .. r .. s hline a .. cc .. tn .. ta .. h .. c .. im n r a-.notdef am .. c-n .. dt .. h .. Vdash .. n .. a .. l .. l .. n .. Vdash a .. r b .. y .. s .. Vdash .. c .. quotesinglbase-a sub notdef Vdash n r .. t Vdash n .. d .. h .. u .. s .. w .. o u .. l .. d .. Vdash .. l .. l .. a a .. e n-c a s d r e .. t t i .. .notdef-h .. l .. period .. r T h s e s .. Vdash .. m .. Vdash .. a .. r b .. i L-c S .notdef-a t-c S-n to the power of 0 d e n tw o .. p e .. t .. u .. b .. hline a t .. c on s r quotesinglbase-f sub m Vdash i i .. K h .. s .. Vdash .. c period a-h CR semicolon S sub semicolon-t .. c h-CR slash a a s t e .. s .. e period n a CR semicolon i-S to the power of 0 sub semicolon Vdash c CR slash a b su .. o .. t .. period .. open parenthesis .. T b .. e .. CR a a i sub c us .. s e e .. s s .. e .. n .. t .. i .. a .. l .. comma .. a .. n .. d .. t .. h F .. o .. r .. c .. l .. a .. r .. i .. t b y-i s-comma w-t .. h e Vdash fi Vdash r-K i s ln t y .. t a .. p-h s Vdash Vdash hline ai c-a-u nn m-n tt S a .. n-S to the power of 0 .. d i f .. e .. m e .. e l .. y .. go .. g t o .. th .. e .. w-K n-o d Vdash b a .. ar F s .. i-n i-t g period open parenthesis .. W .. e .. w .. i .. l .. l .. r .. e .. t .. u .. r .. n .. t .. o .. t .. h .. e .. m .. o .. r .. a-e i .. s .. a .. s .. t .. r .. Vdash .. n .. g .. l .. y .. Vdash .. s .. s .. n .. t .. b-i c-a a l ns ad .. b-i sub t-c c-s-a e .. s .. s .. e .. n .. t .. i .. a a .. S to the power of 0 b S c i .. g .. r Vdash a-eight c-i .. sS .. .notdef-.notdef .. sc .. a-s Vdash nnt t .. i .. a l .. b .. Vdash t .. Vdash T .. h .. e .. o tr he e mr .. 7 .. period m 1 .. u .. s .. t .. b .. e-a .. b .. c-b n Vdash e c s a s Vdash e p .. Vdash r .. t .. S a b n-S to the power of 0 d f hline t-a i-c .notdef-h .. n .. c-s .. h .. a .. r Vdash Vdash .. s .. s .. Vdash .. n .. t B .... y .... t .... h .... Vdash .... d .... Vdash Vdash .... n .... i .... t .... i Vdash .... n .... Vdash f .... h W s D-h period-s .... a CR semicolon S-n sub semicolon c-t CR slash .... a .... l .... o-comma .... n EM sub 1 period H11938 to the power of 3 X K slash .. b .. h-u 0 W .. D-h period .. a CR semicolon S to the power of 0 semicolon .. c CR slash a .. s .. t .. i .. v .. i .. Vdash l .. period .. S i .. n .. c z .. e .. r .. K-o sub comma .. wt .. ih .. t e .. h y .. d c .. f .. f .. e .. r .. n o .. K noe b-n open parenthesis a-l y-r-s .. e .. be c .... h .... a .... n S t .... S-e-.notdef to the power of 0 .... f .... u r .... a l .... m .... by .... e .... r i di .... a K s o .notdef-o p to the power of m-a r Vdash .... w .... n h .... Vdash a .. CR-r a .. c c Vdash a .. y n-a Vdash L-a i Vdash t-n i to the power of 0 .. e .. t .. sta .. nd .. a r d .. l .. p s .. r e p .. re se l .. s e .. open parenthesis .. a s i .. n .. t .. e .. Vdash .. n i .. ti .. n .. o f .. s .. e .. nt i .. i closing parenthesis h comma i .. n-c d h c hline l .. b n ac .. n .. d .. t .. h .. CR a n .. T-b sub h Vdash a .. c-a dotlessi-k to the power of 0 .. a n a l .. t l hyphen n h .. o g .. ot .. o .. pi .. c .. m hline u .. b a r .. c .. is h .. o .. o t op .. c .. t b-o p-a Vdash a .. h l-i l e-dotlessi-s l .. m-t-s sub o h .. a m .. o-n .. t s .. l sub a to the power of hline c hline a .... Vdash a-n .... b .... a .... F s .... i i g period n uT .... l-r sub a h to the power of hline e-r u Vdash 9 s-p .... r .... Vdash Vdash Vdash .... n .... t Vdash .... h Vdash .... n .... Vdash n .... t .... h .... i s i hline d .. e .. fi a n i b ist .. si to r .. n o I .. n .. f .. u .. l .. l .. g .. e hline a m-c e a .. r y a .. l-K i-n i n t Vdash mr comma as t .notdef-n h y e tp s .. e i c n B .. Vdash .. t .. s .. t .. i .. l .. l .. comma .. t .. h .. t .. w h a .. f n-h-.notdef to the power of 0 d-n .. f d .. e r .. b-m y s Vdash o n e t Vdash G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l .. y-u 16 .... Elizabeth Denne comma Yuanan Diao and John M Sullivan i a to the power of 0 i to the power of b .. c .. a .. l to the power of hline from ab to b .. c b sub a c F i g .. r .. Vdash 9 Vdash .. A .. t .. t .. h .. dotlessi-.notdef .. t n d-.notdef-dotlessi 0 p re l-p r e-.notdef f e t-n t ing Vdash t s-h e l a s .. h-e s-a n-h 0 d i s .. n .. ro m .. th e .. ss e nt ia .. l an d i n es s S a .. S to the power of 0 d f hline a a-c .. s F i i n .. g period .. u Br ee i to the power of 0 8 .. a n u u l l-s hyphen e o .. to .. i c .. a am .. m o ti .. t o .. pa i t h o .. w .. s .. it .. i .. s .. h o .. o .. l sub a to the power of hline o b t i-.notdef c-p .. r hline a .. l .. a-o period n-b Ag .. t .. t .. h .. e .. b .. o .. t t .. o .. m .. w .. e .. s .. e .. e t .. h .. e .. d .. i .. f f .. S-e a r .. e-S-n to the power of 0 d-n .. c c a e t u-b .. e .. t .. b w y w e-r .. a e p p-n .. ng a .. d .. ff o .. u K a .. t .. s .. Vdash .. m .. Vdash .. n .. i .. t .. Vdash .. n .. b sub m-one semicolon .. b .. b sub .notdef-k .. r a .. Vdash semicolon-F sub f-i n-i i g n period I .. n .. p .. a .. r .. t .. i .. c .. u n sub i l .. w a .. r-h a comma v Vdash S h-o a-r r t n-a sub s-i p o s-r .. m .. e m i Vdash .. n b-a sub i a S n .. o e .. period .. W .. c .. an .. c .. h a S t .. S-.notdef to the power of 0 .. o P n sub i .. t Vdash .. p .. Vdash comma .. a t .. Vdash a .. c .. h .. Vdash Vdash Vdash i .. mp .. Vdash .. k i .. n .. d .. f F c .. i .. g a period n u A Vdash r .. t h e Vdash w 8 a-n .. i t n l .. e .. a .. s .. t c .. h .. a n .. g .. e .. f .. r .. o .. m .. e .. s .. s .. e .. n .. t .. i .. a .. l .. t .. o .. i .. h hline j u .. s .. t .. b .. e .. f .. o .. r .. e .. s .. u .. c .. h l sub a to the power of hline a sub i .. a slt .. oe .. np gc .. s a-a i-o period-n N .. o .. t .. e .. t .. h .. a .. t .. b .. e .. c .. a .. u b sub 1 semicolon s .. e b sub i : o-one Vdash f f semicolon t h i-h .. e Vdash .notdef-i g .. .notdef-K t .. h .. Vdash l sub a to the power of hline l sub i i-.notdef Vdash no t-p p .. io .. uiiqu e .. yde Vdash n e .. semicolon it .. h .. l b e i .. t .. rp ret .. e ro .. un .. t .. o se p rev io us i nt ers .. e .. ti onp .. i n .. s .. h esa .. S-e d ay Vdash t e-.notdef s .. r c .. Vdash .. n .. v Vdash .. n .. t .. i .. Vdash .. n .. l sub a to the power of hline sub i h ep r w s Vdash .. v ts t r e comma on w t r-.notdef iv s ia Vdash Vdash ho t-m h .. h Vdash y-a d .. Vdash Vdash .. n .. i t .. i .. Vdash .. n .. Vdash .. f .. s .. t .. r .. Vdash .. n .. g .. l .. y .. Vdash .. s .. s .. n .. t G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l y u Ropelength boundsfrom quadrisecants .... 17 a Case 1 b sub 1 b sub 2 b sub 3 b sub k Case 2 c Case 3 S S to the power of 0 i gu .. e .. 1 0 : hline a n i-c c a n s e t .. K a t-s Vdash m-c t-a .. sn .. y .. p .. Vdash .. i .. n .. t i n .. Vdash .. s .. s .. Vdash n .. t .. S-i aa .. l-S to the power of 0 d-a p n Vdash t r .. t s .notdef-e .. r .. b fe a .. y i n i t-n e a .. r .. o .. u .. n .. d .. i n b sub i period hline hline i .. m .. m .. e .. d .. a i sub i b-i-a Vdash t Vdash e .. r Vdash y n g t Vdash y-h .. a Vdash t .. Vdash .. n .. b sub i i-c i a .. s period .. B a y .. l .. y-s Vdash m .. s-.notdef T .... h b W u b-D sub i-s i .... s .... t .... h .... e .... d .... e .... s .... i .... r .... e .... a .... b-i .... n s-c .... t e e .... line-line-n 8 .. M .. i .. n .. i .. m .. u .. m .. a .. r .. c .. l .. e .. n W .. e .. w .. i .. l .. l .. i .. m .. p .. r .. o .. v .. e .. o .. u .. r e .. s .. s .. e .. n .. t .. i .. a .. l .. a .. r L e .. m .. m-I .. f a .. s a-eight e period-i sub b c s-one .. a e n s .. t s e .. n .. t .. i .. a .. i j a n b j .. one-NAK comma-k n-a i .. f .. CR a-a .. ir .. sc .. e .. s .. s grave sub a e NAK b EM-n period hline P .... r .... Io j a-f o b j less 1 t .... h .... L .... n .... e .... b m .... y t m ha .... B .... b period a d-one l i l-a a cbm Vdash Vdash n t t Vdash a r hline n .. k .. n .. Vdash .. t CR a-t .. b Vdash d K .. aNr .. wc .. f open parenthesis Vdash sr a .. a y .. n S y .. ap b .. h r it .. c Vdash r-h r CR-.notdef a .. m w .. c .. a n CR a .. p-bracketleft S b n .. y .. a .. n .. Vdash .. m .. b .. B .. dw .. dh .. Vdash Vdash d s .. d .. i-i d .. i .. s .. j K-o sub period i T n .. ht .. i f .. sr CR a o-m b-parenleft e-m a .. an .. n-a d closing parenthesis b t it hshu ia s K .. n .. o .. w .. i .. n .. g .. t .. h .. a .. t .. s .. u .. f .. fi .. c a a-i r-e Vdash n i t .. n-l Vdash y .. s s n Vdash .. w .. t .. h .. Vdash CR-s a .. h h .notdef-i .. c r h t i Vdash Vdash Vdash s Vdash t Vdash Vdash a-T-n .. t r h i e-a-c Vdash o t F r h-r ee m r-m .. e 7 m-period 1 a .. p .. iq .. t .. h .. b .. a-.notdef .. aph p-n s-d q s .. cs .. .notdef-a .. n n tt .. i .. a .. l .. comma .. i .. m-a ap .. q-n a d Vdash .. t .. B-s period p i-slash period .. d S Vdash a n i-p c .. e-s Vdash .. s .. s .. n .. t .. i .. a-q .. l .. comma h .. p-b a y-CR-two-v a .. t m h a Vdash n t .. h a .. a i .. q-s .. r .. e .. v Ce .. r-o s-r o-e w to the power of d-l l-e period grave sub a-a-g F NAK b grave-r sub a y-t o q m-three-EM period line-line I .. n .. t .. u .. i .. t .. i .. v .. e .. l .. y CR a-comma .. b w .. a e k .. e .. x-.notdef t p .. Vdash .. e w .. r .. t-a a-p .. a n .. t .. Vdash e Vdash m .. p .. i .. n .. Vdash .. n .. th .. Vdash c Vdash m .. CR-p b .. A Vdash Vdash tn .notdef-t agrh .. y-j-w a NUL b j-r-n .. c D-two Vdash .. c G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l .. y-u 18 .... Elizabeth Denne comma Yuanan Diao and John M Sullivan h .. a grave sub a v b sub D EM .. w h j a-e NUL b j less 2 Vdash Vdash .. x .. p .. Vdash .. c t .. a .. b .. grave sub a .. t E t v .. Vdash n r t i .. n .. f .. a .. c .. a .. CR-n a .. Vdash i Vdash g Vdash hnt tii .. na Vdash .. a .. d .. w .. r .. a .. p .. a .. r .. Vdash un .. d .. Vdash d .. e .. r .. i .. v .. e .. t b L e .. m .. I .. CR-a-f a .. i 8 .. period a n-two Vdash Vdash Vdash Vdash n .. t i .. a .. Vdash .. a r .. c i .. n .. j a NUL n-b j .. less 2 t h grave sub a e NAK b n-two sub EM NUL 2 a .. r .. period j a-c NUL s-b j = to the power of two-i slash period P .. r .. N-o .. o .. ft .. j e-a NUL b t-j OE-h to the power of two-one semicolon a-two c141 .. t 2 EM NUL 2 a .. r .. period j c-a NUL b j = 2 slash n DC4 5 EM 3 = .. A .. s p .. r .. Vdash .. v i .. Vdash CR-.notdef a .. b .. Vdash p t .. r Vdash Vdash h .. f r comma t .. l Vdash t .. t Vdash a .. n a t n i a d Vdash Vdash a r-n .. c d f r a a .. p Wq Vdash .. h b a CR-two a v .notdef-a n grave sub a d NAK q EM .. s .. o w .. m .. Vdash grave sub q less 2 EM-s 3 = m r e .. t .. h .. Vdash t .. r .. i .. v .. i S .... i CR sub q-n .... i c Vdash Vdash e Vdash Vdash Vdash grave sub q-n NAK p EM a Vdash sub greater-q comma .... b .notdef-grave b-.notdef 2 CR sub q .... p .... I f-b B-two period p slash .... h .... e .... t h .... e CR w-a q t a .. y .. p .. B-u period p slash period i L-d ellipsis .. te .. o .. t e .. t h .. e .. r Vdash at B period p slash .. s .. F-p i r to the power of i-n o g .. period u F-c r a .. ellipsis a p ellipsis sub ellipsis sub q to the power of b .. b .. q i .. g .. Vdash .. r .. Vdash 1 1 Vdash .. L I .. e n .. m t .. comma m-h pa .. r p 8 o .. CR j-a two-.notdef .notdef-e sub Vdash c-t sub h .notdef-t f u i n n i a .. r Vdash .. p i d n .. c .. r Vdash .. a .. s .. Vdash .. s .. n .. i .. t .. h .. Vdash r .. i .. t .. s .. l h .. Vdash .. p .. r Vdash Vdash Vdash .. c .. t .. Vdash .. d .. c Vdash .. r ellipsis a a i ellipsis n-q sub d cw lh Vdash i dc .. h l .. e .. n .. g .. t .. h L .. e .. m .. comma m .. t a h 3 .. i period .. s 3 p .. r .. o .. j e .. c .. t .. i o .. j ellipsis a NUL ellipsis o-b j e .. a-DC4 s b-j j n e .. Vdash 2 EM NUL 2 a .. r .. period j ellipsis a s ellipsis-i b j n-equal 2 slash NAK 2 EM NUL 2 a .. r .. period j c-a NUL b-s j = to the power of two-i slash period n .. I t .. t .. h .. e .. r c .. o .. s .. d CR a .. SUB at-t sub B period p e-slash period .. c F .. o .. e .. Vdash .. n y .. x-t sub semicolon w-y two-o at p-B period p-o slash .. n .. t .. s .. s d .. i .. s .. t .. a .. n c .. two-e a .. b r .. period j-e c-x NUL y j equal-w 2 slash period n T-e grave sub a b sub D grave sub a C-q grave sub q NAK b EM C 2 a .. r .. period j q-c NUL b j = to the power of i-two slash D EM C 2 a .. r period j-c a NUL b j = to the power of two-o slash s D 2 EM NUL 2 a .. r .. period j c-a NUL b-s j = to the power of i-two slash e .. n .. ow .. Vdash .. j b NUL p j less-m 1 e .. t L h-CR sub q Vdash b t-t Vdash t .. h Vdash Vdash h .. Vdash r .. t Vdash Vdash Vdash Vdash Vdash .. n .. q i a .. n .. d j q NUL y j .. two-t sub NAK Vdash S .. i-grave sub q n y EM greater sub q grave Vdash h a .. b-v .notdef-two CR sub q .. y L Vdash h W .. D-p NUL j-y j .. grave-DC4 sub y G e o .. m .. T-e sub o t p r sub o-comma .. yl V .. ampersand-o-o .. g l y u Ropelength boundsfrom quadrisecants .... 19 a .. n .. d .. n .. h o-two OE to the power of t-zero semicolon 1 c141-e .. i t .. b-h 2 a-B-e period t-p slash period open parenthesis F-S .. i e g .. period .. u closing parenthesis .. rS j q-e NUL y-one j n 2 c-two sub NAK e-w Vdash j p NUL q j .... two-NAK NUL h .... grave sub a Vdash NAK q EM 2 C-equal f 2 period h slash .... C .... o .... r .... o period l .... Ol .... n-a t-r y-h .... e 4 period-o grave sub b DC4 p EM 2 = .... b .... L .... y e .... m .... closing parenthesis m a a-grave sub q n 3 y period-EM 1 w e .... h grave sub q-.notdef NAK b EM 2 C-equal grave sub y NAK p EM 2 equal-C h h .. grave sub a NAK b EM C f 2 period h slash C h .. A .. n .. Vdash .. l .. Vdash .. m .. Vdash .. n .. t .. a .. r .. y .. c .. a s .... i .... d .... Vdash .... i .... s .... a .... n .... i .... h-n 2 c-OE 0 semicolon 1 r-c141 Vdash m .... a i s i i .... i-n-h z D 0 d-f w Vdash Vdash h n Vdash cr t Vdash i .. EM-s C f 2 slash D 7 EM 6 C-equal p-line 3 greater 5 EM 3 equal-period .. T .. h .. a .. t .. i .. s .. comma grave sub a NAK b 5 EM 3 NAK = 2 EM NUL 2 a .. r .. period j c-a NUL b-s j = to the power of i-two slash line-line a .. h from y b to p .. x .. q i .. g .. Vdash .. r Vdash 1 2 Vdash .. I .. n .. t .. h .. Vdash .. m .. Vdash L s e t .. m i comma m-n .. w t-a e-CR-r a-eight b .. t .. Vdash Vdash .. r Vdash t .. Vdash Vdash .. nt a-i .. a .notdef-g a-i r v c i fr n Vdash g .. a .. n a .. p sq Vdash Vdash t l .. CR sub q t b .. t .. Vdash Vdash .. r Vdash t .. Vdash Vdash .. n t i-q .. a g a i r v c i f n-r Vdash m g .. a .. n .. q Vdash x s S .. i j x-n NUL c-y j Vdash one-NAK a .. j-n x NUL q j .. 1 NAK .. s .. Vdash h-t D .. p j y j-n g-e Vdash .. j v-p NUL q j .. two-NAK sub NUL h I .. f .. w .. e .. d .. e .. fi .. n .. e .. t g period r slash W .. Case 1 2 EM NUL 2 a r period r-c = 2 s-slash i-i 0 DC4 n r DC4 2 comma Case 2 EM i r NAK 2 period t .. h .. e .. n .. w .. e .. c .. a .. n .. c .. o .. l .. l .. e .. c .. t .. t C .. o .. r .. o .. I .. CR-l-f a a i .. r s .. y a .. 8 n .. e .. 3 s .. s e .. K o .. tfi u a nl ia tr grave sub a NAK b g period j a NUL b j slash 9 .. M .. a .. i W .. e .. n .. o .. w .. p .. r .. o .. v .. e .. r .. o .. p .. e .. l .. e f .. o .. l .. l .. o .. w .. i .. n .. g .. l .. e G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l y u 20 .... Elizabeth Denne comma Yuanan Diao and John M Sullivan Lemma 9 period 1 .. Recall hat f period r slash W D line-NUL sub r 2 1 C arcsin period 1 = r slash semicolon g period r slash W D Case 1 2 EM NUL 2 arcsin period r = 2 slash for r DC4 2 semicolon Case 2 EM for r NAK 2 : Then comma for r NAK 1 comma open parenthesis 1 closing parenthesis .. the minimum of f period r slash is EM = 2 and occurs at r D 1 comma open parenthesis 2 closing parenthesis .. the minimum of f period r slash C g period r slash is 7 EM = 6 C p-line 3 greater 5 : 3 97 and occurs at r D 2 comma open parenthesis 3 closing parenthesis .. the minimum of g period r slash C r is EM C 2 greater 5 : 1 4 1 and also occurs at r D 2 comma and open parenthesis 4 closing parenthesis .. t .. h .. e .. m .. two-i f r-n slash C-i sub g period m-r slash C r-u i .. s-m .. o Vdash nine-f sub : to the power of s-three t-seven a Vdash 7 v-n 4 d r .. Vdash .. r EM c 1 sub c-colon 0 0 r P .... r .... N-o .... o o .... ft f e .... s t i h n .... a .... r t e g i .... i-s .... n c Vdash Vdash comma n-r-.notdef NAK s 2 .... t Vdash a .... h n .... ts ft Vdash h r c .. c Vdash .. r .. i r 2 OE 1 semicolon two-t c141 .. h w h Vdash e-f to the power of 0 D r 1 divided by r to the power of r-line-n-NUL-g 2 1 a Vdash g-n to the power of 0 D d .. 2 NUL-equal to the power of NUL-line 4 r to the power of 2 .. E Vdash Vdash m Vdash n c .... a .... Vdash .... c .... u .... a .... i .... n Vdash .... t h .... n .... gi .... v Vdash t .... r EM 1 : to the power of zero-.notdef u-zero .... i three-t s 0 w a-.notdef 5 wa p n Vdash t .... l e .... x .... p .... r .... e .... s .... s i .... b .... F l .... ie .... g i period S-line-line F .. i .. g .. u .. r .. e .. 1 .. f 3 r slash-period : n d-L-g period r slash f .. r two-r OE 1 semicolon 3 c141 .. a Vdash .. n p d .. l .. i-o Vdash t-h 2 f r slash-period C g period r slash C r f .. r-o 2 OE 1 semicolon 1 : 0 c141-zero sub period T .... h .... e .... o .... r-A .... ek .... mn 9 o .... period t 2 .... w .... i .... t .... h .... a .... n .... e .... s .... s .... e .... n .... t .... i 1 EM 0 3 equal-C 2 line-p 3 C 2 greater 1 : 9 5 period-three P .. r .. R-o .. oe .. s-f .. c .. a K t .. e-.notdef .. t h .. h a .. v e Vdash .. k .. n n .. a i-o b-t-t bc .. Vdash h t .. ih .. c Vdash .. kq .. .notdef-n r .. K i-.notdef n-n .. t t .. h Vdash Vdash s .. Vdash .. a .. l .. K a-i grave sub y-a-s C-period grave sub b Vdash C grave sub c Vdash C grave-n sub d .. t A h .. b-.notdef .. l l .. r-e D .. a j b j comma s D .. b j c j a .. t-n D d .. c j d j period C .. o .. r .. o .. bl .. o-l CR-u a r n .. CR y-b sub d .. a eight-s n .. CR-period d-c 3 .. T .. q u a .. d .. r .. i Vdash Vdash .. c .. an .. ti L .. e .. m .. wm .. e-a eight-r-h sub semicolon s semicolon a-t-period NAK one-one .. e a L .. n .. e d .. m .. m-m a-a .. y 4 b-period five-e .. a .. grave sub d .. p T l h G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l .. y-u Ropelength boundsfrom quadrisecants .... 21 the length of K is at least g period r slash C g period s slash C g period t slash C NUL f period r slash C s C f period t slash SOH D .. NUL g period r slash C f period r slash SOH C NUL g period s slash C s SOH C NUL g period t slash C f period t slash SOH : i n .. c .. t .. h .. s .. i .. s .. Vdash .. u .. m .. o .. f .. f .. u n .. ct .. i .. o n s .. i .. n t .... e .... r .... m .... se .... p .... a .... r .... a .... t .... e .... l .... y .... period .... L T e .... h m .... e-comma ms s a o 9 at period h 1 a .. c .. h .. Row 1 e Row 2 D . v D t e D 2 d .. A a .. d-t .. d .. i .. n .. g .. t .. h .. Vdash .. t .. h .. r Vdash Vdash .. v o .... K-f i .... s .... a .... 1 t EM 0 to the power of l-three equal-C 2 three-line-a-p C 2 greater 1 : 95 period-three line-line T .... h .... e .... o .... A-r k-e .... mn .... o 9 t-period 3 w .... i .... t .... h .... a .... n .... e .... s .... s .... e .... n .... t .... i 1 EM 0 3 equal-C 2 line-p 3 greater 1 : 9 3 period-three P .... r .... R-o .... oe .... K t-c Vdash a h .... e-a .... v .... Vdash Vdash .... n .... i-a t-b b t-c Vdash h-d tihc Vdash kq .... Vdash Vdash a s .... d s .. Vdash .. a .. l .. r .. i .. Vdash .. n K t-i grave sub a-s-a C-t grave sub i-b C-.notdef grave sub n-d C-comma grave sub c t .. S h i Vdash n l .. .notdef-.notdef t n .. g q .. t Vdash .. n .. t i L a .. Vdash e comma .. m r am m nLa d e 8 .. m period b w-m-one d sub ea h-r sub semicolon s semicolon a-t NAK one-v-one .. e .. Vdash .. a .. C p .. o .. p r l o t .. CR-o a .. a .. CR d .. a .. nC .. d .. o .. r .. o tl o-CR b l aa .. CR-r c .. period-d T .. h .. u .. s .. t .. K-h ie .. s .. l a g period r slash C NUL f r slash C f s slash period-SOH C g period t slash C NUL f s slash C f t slash SOH-period D .. NUL g period r slash C f r slash period-SOH C 2 f s slash C NUL g period t slash C f t slash SOH-period sub : Vdash Vdash i .. n w e .. i .. ni .. m i .. z .. t .. h .. L .. t e e .. r m .. period m-s s-a-W .. e 9 p .. fi-period r to the power of n-one Vdash d-t o .... K-f i .... s .... a .... 1 t EM 0 to the power of l-three equal-C 2 three-line-a-p greater 1 : to the power of t-nine 3 three-period line-line T .. h .. e .. o .. A-r .. ek .. mn .. nine-o t-period 4 w .. i .. t .. h .. a .. n .. e .. s s .. e .. n .. t l .. e .. 1 a : 65 sub s-period P .... r .... R-o .... oe .... K t c Vdash a h l .... a e v .... Vdash Vdash .... n .... ai .... t-b bc t Vdash d h-t .... ih c Vdash .... kq .... n .... Vdash a .... sd s-K i .. n .. t .. h .. Vdash .. s .. Vdash .. a .. l .. w .. a .. K-y i .. grave sub a-s C-c-h grave sub c Vdash C grave sub b .. C h-grave sub d .. A r .. a-.notdef .. i p r D .. a NUL b j comma s D .. b NUL c j a .. n-t d D .. c NUL d j period T .... h .... e .... q .... u .... a .... d .... r .... i .... s .... eL .... c e a m .... amt nLa di e 8 s .... m period e w-one m .... e s-a r semicolon four-s-e sub semicolon t period n e-one 2 h .. L Vdash .. s e .. m .. m-m a-a .. y 4 .. b period e-five CR d .. a W p .. a p p lC p i o Vdash e y r d o tt o-CR a-o l a a n CR d-b-r .. d y a C .. o .. r .. o tl CR-o c l period W .. e .. fi .. n .. d .. t .. h K a-i t-s t-a NUL f r slash C f s slash period-SOH C NUL f s slash C f t slash SOH-period C g period s slash C NUL f r slash C s C f t slash period-SOH D .. 2 f r slash C NUL 2 f s slash C g period s slash C s SOH C 2 f t slash period Vdash Vdash .. i .. comma .. w .. e .. c .. m i .. n i .. i z .. e .. n .. L .. c e .. m .. period-v .notdef-m H-r a i e 9 r .... o .... p .... e .... K-l i .... es .... n ag 2 t-EM C nine-h sub : 3 eo a-greater-seven one-f sub 7 sub colon-s 65 sub t-period line-line G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l .. y-u 22 .... Elizabeth Denne comma Yuanan Diao and John M Sullivan Theorem 9 period 5 .. Any nontrivial knot has ropelength at leas t 1 3 : 9 36 period Proof .... A .... n .... y .... k .... n .... o .... t .... o .... C-f to the power of 1 semicolon 1 fi Vdash n .... i-C-b .... y t o e r .... o-r .... i .... l-o sub tl p h a e ar l sy e q .. u .. a .. d .. r .. i .. s .. e .. c .. a .. n .. t .. period .. T .. h .. i .. s .. m .. u .. s .. t .. b a .... b .... o .... v .... e .... a .... p .... p .... l .... i .... e .... s .... semicolon .... w .... e .... line-line I .. n .. h .. e .. r .. d .. o .. open square bracket sub c-five closing square bracket comma t Do .. r-e a-n T .. h .. e .. o .. A-r e-n .. y m .. n 9 C to the power of 1 semicolon 1 n-k-six n-t r-o .. i t v hi a a .. sl .. a n .. e .. s .. s C .. o .. m .. b .. i .. T .. i h n .. e .. g o gtrih C .. o .. r .. o .. A-l l n a y r n y o 9 .. n period .. t 7 r .. i .. v .. i .. 1 : to the power of six-l 5 period e .. n .. o .. t .. e .. t .. h .. a .. t .. t .. h .. i .. s .. b .. o .. u .. n .. d .. i .. s .. b open square bracket 4 comma .. C .. o .. n .. closing square bracket sub period-j .. e W ce t .. a u .. lr .. se o 2 .. n 6 o .. t .. e .. t .. h .. a .. t .. o .. u i .... C-s to the power of 1 a .... t-b c .... a .... n .... n .... Vdash t .... s .... i .... m .... Vdash l .... t .... a grave sub c n a .notdef-n grave sub b d d to the power of s Vdash s-n-l EM y 1 : 0 .... a 0 r .. Vdash .. b .. a .. b .. l .. y .. a .. c .. a .. r .. Vdash f .. Vdash .. l .. a .. n b aa .. n-c-l c d-y Vdash .. s l-i .. d s .. l .. i .. g .. h .. t .. l y .. b .. Vdash .. t .. t .. Vdash .. r .. b .. Vdash Vdash .. n .. d .. period .. H .. Vdash w .. Vdash .. v t .. r .. e .. f .. o .. i .. l .. k .. n .. o .. t .. s .. w .. i .. t .. h .. r .. o .. p .. e .. l .. e .. n m .. u .. c .. h .. f .. u .. r .. t .. h .. e 1 .. 0 .. Q .. u .. a .. d .. r Q .. u .. a .. d .. r .. i .. s .. e .. c .. a .. n .. t .. s .. m .. a .. y .. b o .. f .. a .. n .. o .. n .. n .. t s r .. c i .. a v n-n .. i ts l a .. ci l an .. l n .. s t-i a-s n n a k .. r-d c .. a .. n .. b .. c .. l .. a .. s .. s .. i Vdash Vdash .. d .. i .. n .. t .. r .. m .. s .. Vdash .. f o .. f .. t .. h .. e .. l .. i .. n .. k .. period .. W .. e .. b .. e .. g .. i .. n .. o .. u .. r .. c .. o a .. n .. y .. n .. o .. n .. s .. p .. l L .... e .... m .... m .... S au .... 1 L zero-i sub p .... s o 1 a s .... e A i-i .... n ka comma c .... a Vdash L-n-m ndp Vdash .notdef-t .... ns .... pn .... l t t .. h .. Vdash .. l .. i n .. k .. a 2 A-.notdef .. h t Vdash h .. n .. r Vdash Vdash r i-a a to the power of 0 .. n a-a y at 2 A-r b-i sub i s-b l n-t Vdash o .. n .. s .. o .. m .. e .. o .. L h-h e-a .. s r .. u-c n-o .. im .. A-p t-h sub o .. ahnse cl .. nke t two-n sub EM-n period P .... r .... F-o o-o f-r .... t a .... h t e h .... g .... i Vdash n-v .... e i .... n-a n to the power of 0 a .... Vdash a f .... a s l-k .... s l-p a-A s n Vdash B-i c Vdash c-A-a i .... s .... n .... Vdash .... t L X A p .... l-t .... ih .... i fs .... r d Vdash .... im s-L X k-A .... m a .... t-.notdef .... ss t .... b b m Vdash Vdash hcpi Vdash t .... h .... Vdash .... d .... Vdash .... s .... r-L .notdef-h d-a .... s .... r .... n-i .... s i .... Vdash t c-A sa ht ia tcy period s-B-k period-I b slash-n Vdash Vdash s t-o .... s C .... o .... r .... o comma l .... wl e grave sub a h to the power of 0 NAK a a-EM a three-v n-grave-e sub a-period 0 sub a-d NAK 4 EM period line-line G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l .. y-u Ropelength boundsfrom quadrisecants .... 23 T .. h .. i .. s .. c .. o .. n .. s .. t .. r .. u .. c .. t i .. o .. n .. o .. f .. a .. t .. r i s .. e l .. i .. n .. k .. p .. r .. two-o period parenright-b .. l T eh m e period l .. e S ne g et .. h .. b .. o .. u .. n .. d .. c open square bracket 4 comma .. T .. h .. e closing square bracket comma o a r .. n-e .. dm .. i 1 m .. 0 m .. e .. d .. i .. a .. t .. e C .. o .. r .. o .. A l n a .. o r .. n-y .. s 1 k-p cl period o-i 2 t-m p-l i o n n ke .. nw .. i-t s-two-t sub EM h-k h-period T .. h i .. s .. b .. o .. u .. n .. d .. i .. s .. s .. h .. a .. r .. p .. i .. n .. t .. h .. e .. c e .. x .. a-two sub EM period W .. e k .. n .. o .. w .. t .. h .. a .. t .. n .. o .. n .. t .. r i .. v .. i .. a .. l .. l .. i h .. a .. v .. e .. q .. u .. a .. d open square bracket 1 r closing square bracket six-w-i a-s e-s .. c t .. ah n .. t fis .. r period s .. Pt .. at .. no .. ns w f .. o .. r .. c .. e P .... r .... o .... p .... o .... Is A a .... t n-B ad o r .... .notdef-n d-one .... i 0 s period Vdash 3 .... in .... t .... g .... Vdash .... n .... Vdash .... r t .... h .... a .... t .... n .... Vdash .... i .... t .... h .... Vdash .... r .... n .... i .... s .... h .... Vdash .... m .... Vdash Vdash A-p open square bracket B-i h .. a .. s .. a .. q .. u .. a .. d .. r .. i s .. e .. c .. a .. A n B t period N .. o .. t .. e .. A h .. r-B-a i .. t s comma Vdash .. w n .. h k e n .. n .. t .. t Vdash .. d .. P .. a .. n .. n .. w n .. Vdash .. n .. Vdash Vdash .. r Vdash .. l .. i .. n .. k .. i .. n .. g .. n .. Vdash .. m .. b .. Vdash r .. period Vdash .. h .. Vdash l .. i K .... u .... p .... open square bracket to the power of one-e closing square bracket r-e-three x-b e-t e-r .... ng .... d .... e .... d .... t .... h .... i .... s .... r .... e .... s .... u .... l .... t .... t n .. o .. i .. n .. f .. o .. r .. m .. a .. t .. i .. o .. n .. a .. b .. o .. u .. t .. t .. h .. e .. o e .. s .. s .. e .. n .. t P .. r .. o .. p .. o .. A-s g-i .. t e .. i n .. o e .. n r .. 1 c-zero .. n .. 4 o .. n .. t .. r .. i H .. e .. r .. e .. comma .. a .. L i e .. s c .. a-a Vdash n .. t .. o m .. f .. a .. a t .. i-l .. i c .. n a .. k-l .. l .. y .. Vdash c .... Vdash m .... p .... L-.notdef .... n .... I Vdash .... n it ts .... s .... Vdash .... n .... d .... p .... Vdash .... i .... K-n .... tw s Vdash a-a .... r p Vdash p Vdash .... l d .... Vdash Vdash .... n .... i t .... i .... Vdash .... n .... Vdash .... f .... i .... n .... s .... s .... Vdash .... n .... t .... i .... a L .... l .... n s .... Vdash Vdash t .... c c .. o .. m .. K-p sub period C .. o .. m .. b .. i .. n .. i .. n .. g .. K-P r-u p-o .. p .. parenleft-o b-s we h .. triig c quoteright-o h n-s .. e r 6 xe period i .. m .. m .. e .. d .. i T .. h .. e .. o .. r-E .. e v .. e-m .. r 1 y zero-C to the power of 1 semicolon 1 l five-o i .. nn .. t kr .. hi .. a v s-i .. aa .. n-l D .. e .. p .. e .. n .. d .. i .. n .. g .. o .. n .. t .. h .. e .. o .. r .. d .. e .. r .. i .. n .. w t .. h .. e .. l .. i .. n .. k .. comma .. w .. e .. c .. a .. n .. h .. o .. p .. e .. t .. o .. g-A .. e B q .... Vdash .... a .... d .... r .... i .... s .... c .... a .... n .... t .... s .... comma .... C .... Vdash o .... c-r o-a period n H-l n .... .notdef-e-a t-r sub r i-y sub e .... m 1 e .. x .. a .. m .. p .. l e .. o .. f .. t .. h .. e .. t .. A g .. B-h q A tu B Ha od .. r-p fi .. s-l t .. h .. i .. s .. b .. o .. u .. n .. d .. h a N .. o .. t .. Le .. e t m h ema x a-t .. t 8 e period .. n 1 d .. s .. i .. m .. m .. a a .. d b-n a-i d r-a t-.notdef .notdef-e .. n l c .. o .. m .. p .. o .. n .. e .. n .. t .. s .. o .. f .. a .. u .. n .. j a-i NUL b j hyphen NAK-one period-t G e o .. m .. T-e sub o t p r comma .. yl V .. o-o-ampersand .. g l .. y-u 24 .... Elizabeth Denne comma Yuanan Diao and John M Sullivan Theorem 10 period 6 .. L .. L-e b t Vdash .. a .. l i .. n .. k .. Vdash .. f .. Vdash .. n .. i .. t .. t .. h .. i .. c-Q I .... Q-f h .... a s AtAy .... A-A p B-A Vdash B-A .... BA .... B r-A AB .... C t .... Ah .... n .... t .... h .... Vdash .... l .... .notdef-A i .. s .. a .. 7 EM-t 3 C-equal two-e line-p 3 a-comma 8 s-EM 3 equal-C 1 C line-p 3 comma 2 EM C 2 o .. 2 r-EM C 2 comma .. r .. e .. s P .. r .. S-o o-u .. p a sub 1 to the power of a-p sub 2 a o-three b is .. s e a .. n .. e .. s .. s .. e .. n .. A i A a comma A la .. B-q n r W .. D-a sub 2 NUL a sub 1 j semicolon .. s W .. a-D sub 3 NUL a sub 2 j : h .. e .. q .. u .. Vdash .. i s .. e .. c .. Vdash .. n t L .. i e s .. m .. w-s m .. a-e sub e r-h sub semicolon-eight s-t NAK i 1 a-one-v l n-comma s-d o w f-.notdef C .. o .. r .. o tl CR-o a one-l a sub 3 .. a n r-d .. C y .. o 4 .. r .. period o t-four sub l CR-o a one-l a sub 2 .. a n d-r CR a two-y a sub 3 .. Th period-u s .. 3 h e len .. t .. o-A i .... s .... a .... f tr slash C-l g period r a-slash C f-s s t-slash C g period s slash period .... s .... b .... e .... f .... r .... e comma .... w e .... c .... Vdash n .... m s .. e .. p .. a .. r La .. e-t e-m period m H-y a .. comma e 9 n-u period s-c 1 e-i A i-t .. g h-s e-a 7 l-t-EM 3 equal-C e-two p-line 3 period-g-a N o .. w .. s .. u a sub 1 to the power of a-p sub 2 b p sub 3 i-a o .. s s ae .. n t .. e-h a-s s-t .. e .. n .. A i A a comma B la .. q-A n r W .. D-a sub 1 NUL a sub 2 j semicolon .. s W .. a-D sub 2 NUL b j semicolon .. t W .. D-b NUL j-a sub 3 j : i .. c e .. t .. e .. q .. u .. Vdash .. d .. r .. i s .. e r semicolon s semicolon t n NAK 1 a i n-s e-d s w s Vdash e C sub m-n .. t o a i .. Case 1 y Case 2 l o t .... CR-o a sub 1 a sub 2 .... C .... o .... r .... o tl o-CR a two-l a sub 3 a nd .... L y .... e .... 4 m period tm o-CR-four a one-a a sub 3 W-four period 5 .... ha .... t .... el .... en gt A i .. s .. a g period r slash C NUL f s slash C f t slash period-SOH C NUL f r slash C s C f t slash SOH-period D .. NUL f r slash C g period r slash SOH C NUL f s slash C s SOH C 2 f t slash period Vdash Vdash .. i n comma .. m .. i .. i z .. n .. Vdash .. i .. e .. L .. h e .. m-v comma r-m w-i sub a Vdash b-nine fi period e n 1 s o .. A-f i .. s .. a .. 8 t-EM 3 equal-C 1 C line-a-p 3 period F .... i .... n .... a .... l a sub 1 b .... y two-c i .... a .... u-a .... n p Vdash p-.notdef Vdash .notdef-o n s t i-e .... a Vdash .... q .... A a d-B .... r B r-i Vdash A Ay B-a p n B .... Vdash .... c .... a .... Vdash .... s .... Vdash .... t .... h .... Vdash .... q .... Vdash .... a .... d .... r .... j a sub 1 NUL b j c one-NAK j-a b NUL c j t one-NAK .... a .... s j c NUL a sub 2 j .... one-NAK Vdash .... s .... t .... a .... s .... i-L .... n e .... m-t comma mw a .... e-p 4 r fi grave sub a one-n to the power of .notdef-a-five sub 2 NAK f EM C-t 1 .... a .... a t .... h CR a sub 2 to the power of a-a sub 1 t NAK EM C 1 comma .... s .... h .... o .... w .... i .... n .... g A i t .... s h .... aa 2 t-t-EM C two-t a h-e .... sae .... d-s sub l s-line-line U .. n .. f .. o .. r .. t .. u .. n a .. t .. e .. l .. y .. comma .. w .. e .. d .. o .. n .. o .. t .. k o .. n .. e .. o .. f .. t .. h .. e .. s .. e .. t .. y p .. e s .. o .. f .. q .. u .. a A .. c .. k .. n .. o W .. e .. e x .. t .. e .. n .. d .. o .. u .. r .. t .. h .. a .. n .. k .. s .. t .. o .. S K .. u .. s .. n .. e .. r .. a .. n .. d .. N .. a .. n .. c .. y .. W .. r .. i .. n .. k p .. a .. r .. t .. i .. a .. l .. l .. y .. s .. u .. p .. p .. o .. r .. t .. e .. d .. b .. y D .. i .. a .. o .. closing parenthesis .. a .. n .. d .. D .. M .. S .. endash .. 0 .. 0 .. endash .. 7 .. 1 .. 5 .. 2 .. 0 B .. o G e o .. m .. T-e sub o t p r sub o-comma .. yl V .. ampersand-o-o .. g l y u Ropelength boundsfrom quadrisecants .... 25 References open square bracket 1 closing square bracket .. R Budney J .. C .. o K n Pa nS .. tc D a Sn i N-n n .. e e h w-l .. a l .. p .. e .. r .. s .. p .. e .. cA t M .. a .. t .. h .. period .. 1 .. 9 .. M 1 open parenthesis .. R 2 .. 2 0 open square bracket .. J closing square bracket C .. a .. n J t .. F a-R r-u sub K e ul J to the power of s-l n-M sub a e-S ru N l l-C .. iW v r-C a irnni t-k .. il .. c-e .. a .. l l .. i .. n .. k .. p-a .. r r .. o X .. b i .. l v .. e : m .. m .. a open square bracket .. J closing square bracket C .. a .. n G .. aK ru ep R-l e r-B sub a bK e J-u r Mg .. nSe .. u r-T l h .. l .. e i .. sv .. e-a c n o .. n c .. u comma r A v .. m e .. e .. period .. J period .. M .. a .. t .. hM .. period R .. 1 2 .. 2 open square bracket .. J closing square bracket C .. a .. n Rt B-a .. rK e u-J l M-l .. n aSe .. u r-O l .. n l .. it .. v-h e-a .. nm .. i .. n .. i .. m .. u l .. i comma .. n I k n s v .. e .. n .. t period .. M .. a .. tM h R period .. 1 1 open square bracket .. E closing square bracket D .. e A n l nt ee .. r .. n a .. t .. i .. n .. g .. Pq uh .. aD .. dt r h i e ss e i cs .. comma U .. r .. b .. a .. n .. a .. endash .. C .. a .. h .. r a .. X m .. i .. p .. v a .. colon-i .. m g .. a open square bracket .. E closing square bracket D .. e J n Mn .. Se .. u Tl h l ei .. d-v i-a .. sn .. t .. o r .. t a .. i r .. o X .. n i .. o v .. f : .. a m .. k-a open square bracket .. E closing square bracket D .. e J n Mn .. S-e .. u C lol .. i-n v-v .. ea rn .. g .. e .. n .. c .. e .. a .. n .. d .. i s .. o open parenthesis 2 open square bracket .. Y closing square bracket D .. T-i .. a h .. e-o .. l .. o .. w .. e .. r .. b .. o .. u .. n .. d J-s period oKf .. n .. oh te .. Tl h 1 .. 2 .. open parenthesis .. 2 0 .. M .. 0 .. 3 R .. one-parenright open square bracket .. A closing square bracket .. D .. u .. m A .. tE rb eb .. secr u-A hyphen G .. Br R udieresis u-K n .. m-G-e eaRin n-o-O .. n t-n .. et h .. e d i .. l .. a .. t i .. o .. n .. o f .. c .. l .. o .. s .. a e .. r d X c i u .. v r comma : v m e .. a s open square bracket 1 A-zero closing square bracket E .. b .. b .. e .. r A .. hyphen G .. Br R udieresis-a K-u n e-m-G e e-a ionn .. n-m .. e .. t .. r i .. c .. d .. il a .. n .. e .. w .. l comma o fwr .. e-o r m b : o .. u-P r-n .. o d c .. e .. e d .. i .. n .. g s .. o t .. i .. o .. n .. a .. l G .. e o .. m .. e .. t open square bracket 1 O-one closing square bracket G .. o .. J-n z Ha .. lM ea .. G-z d .. ld .. o o bc .. a k .. l s .. c .. u .. r .. v .. a .. t .. u .. r .. e k .. n comma o P t r s .. o .. c .. N .. a t .. l .. period .. A .. c a .. dM period .. R .. S .. 1 c open square bracket 1 L-two .. H .. K .. a .. I u nf .. v f .. a m .. ra i n .. a .. n .. t .. s .. o .. f-T .. r g a-r .. n asp .. h .. A .. i-m open parenthesis .. 1 .. 9 .. 8 .. 9 .. closing parenthesis M .. 6 R 9 9 open square bracket 1 G-three closing square bracket K .. u .. p .. Q e ur .. ab .. d e .. r r ig .. s e .. c .. a Jn period t Ks n o f-o t k .. Tn .. h o et o 4 .. 1 .. endash .. M 5 R 0 1 2 open square bracket 1 R-four B-bracketright .. K .. J-u .. M .. nS eu r-O l .. nl .. di .. v .. s-a tn .. o .. r .. t i .. o fnr .. ao n m d .. quotedblleft-t g .. e o .. m .. e .. t .. r .. y .. i n .. p .. o l .. y .. m .. e .. r .. s .. c i e .. n .. c S .. p .. r in .. g .. e .. r .. comma .. N .. M-e .. w R .. 1 Y open square bracket 1 H-five .. R .. M .. o-D r M tQo .. n-M-C o-l .. o n s d .. e .. d .. c .. u .. r .. v .. eT s o w p .. o t lh 2 3 5 M-endash 2 R .. 46 open square bracket 1 E-six closing square bracket P .. a .. n E .. n .. wn e-i .. te zl .. e .. m .. e .. n .. t .. a .. r .. g .. e .. o .. m .. e .. t .. r M .. a .. t .. h .. period .. A .. n .. n period .. 1 M .. 0 R .. 8 1 .. open parenthesis open square bracket 1 P-seven closing square bracket P i-acute-n .. s e Ir kn to the power of a-i .. s .. e .. a r .. c .. hf .. or .. f-o m-i d .. quotedblleft .. I a dl ek .. an l o kt .. s-n W .. o .. r .. l d .. S .. c .. i .. period .. P .. u .. b .. l .. is .. M-h .. i R .. n 1 .. g G e o .. m .. e-T sub ot p r comma-o yl V ampersand-o o .. l-g y-u .. m 26 .... Elizabeth Denne comma Yuanan Diao and John M Sullivan open square bracket 1 8 closing square bracket .. E J Rawdon .. C .. a .. n .. c .. o .. m .. p .. u .. t .. e .. rEs x dpi .. sr c .. o mv ee 2 8 7 M-endash 3 R 2 .. 2 open square bracket .. 1 .. J 9 M-bracketright .. S .. u A lp l i-p .. rv .. oa .. ni .. m .. a .. t .. i .. n .. g .. r for .. poe m l : e quotedblleft n k .. n .. o .. t .. t i .. n .. g .. l .. i .. n .. k .. i .. n-H11938 to the power of 3 g L comma-a Vdash a-V Vdash n .. a d .. comma f .. comma-o two-l 0 d 0 i M .. a .. t .. h .. period .. 3 .. 0 .. 4 .. comma .. A .. m .. e .. r .. period .. M .. a .. t .. M-h period R .. S 1 .. o D .. e .. p .. a .. r t m .. e .. n .. t .. o f .. M C a .. m .. b r .. i d .. g .. e .. comma .. M D .. e .. p .. a .. r t m .. e .. n .. t .. o .. f .. M .. a .. t .. h C h .. a r .. l .. o .. t .. t .. e .. comma .. N I .. n .. s .. t .. i .. t .. u .. t .. f .. udieresis .. r .. M .. a .. t .. h .. e .. m .. a D .. endash .. 1 .. 0 .. 6 .. 2 .. 3 d .. e .. n .. n .. e .. at .. m .. a .. comma .. y h .. d period .. i h .. a a .. or .. comma .. j u a mn r .. scd .. at-c period i period P r .. o .. p .. o .. s .. e .. a .. n S .. e .. c .. n .. d .. e .. d .. : .. D .. a .. i .. comma .. W .. a l t .. e .. r G e o .. m .. T ot p ro comma yl V ampersand-o o .. l-g .. y u .. m