A proof of Tait’s Conjecture on prime alternating - achiral knots

Nicola Ermotti[1]; Cam Van Quach Hongler[1]; Claude Weber[1]

  • [1] Section de mathématiques, Université de Genève, CP 64, CH-1211 GENEVE 4, Switerland

Annales de la faculté des sciences de Toulouse Mathématiques (2012)

  • Volume: 21, Issue: 1, page 25-55
  • ISSN: 0240-2963

Abstract

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In this paper we are interested in symmetries of alternating knots, more precisely in those related to achirality. We call the following statement Tait’s Conjecture on alternating - achiral knots:Let K be a prime alternating - achiral knot. Then there exists a minimal projection Π of K in S 2 ⊂ S 3 and an involution ϕ : S 3 → S 3 such that:1) ϕ reverses the orientation of S 3 ;2) ϕ ( S 2 ) = S 2 ;3) ϕ ( Π ) = Π ;4) ϕ has two fixed points on Π and hence reverses the orientation of K .The purpose of this paper is to prove this statement.For the historical background of the conjecture in Peter Tait’s and Mary Haseman’s papers see [16].

How to cite

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Ermotti, Nicola, Cam Van Quach Hongler, and Weber, Claude. "A proof of Tait’s Conjecture on prime alternating $-$achiral knots." Annales de la faculté des sciences de Toulouse Mathématiques 21.1 (2012): 25-55. <http://eudml.org/doc/251018>.

@article{Ermotti2012,
abstract = {In this paper we are interested in symmetries of alternating knots, more precisely in those related to achirality. We call the following statement Tait’s Conjecture on alternating $-$achiral knots:Let $K$ be a prime alternating $-$achiral knot. Then there exists a minimal projection $\Pi $ of $K$ in $S^2 \subset S^3$ and an involution $\varphi : S^3 \rightarrow S^3$ such that:1) $\varphi $ reverses the orientation of $S^3$;2) $\varphi (S^2) = S^2$;3) $\varphi (\Pi ) = \Pi $;4) $\varphi $ has two fixed points on $\Pi $ and hence reverses the orientation of $K$.The purpose of this paper is to prove this statement.For the historical background of the conjecture in Peter Tait’s and Mary Haseman’s papers see [16].},
affiliation = {Section de mathématiques, Université de Genève, CP 64, CH-1211 GENEVE 4, Switerland; Section de mathématiques, Université de Genève, CP 64, CH-1211 GENEVE 4, Switerland; Section de mathématiques, Université de Genève, CP 64, CH-1211 GENEVE 4, Switerland},
author = {Ermotti, Nicola, Cam Van Quach Hongler, Weber, Claude},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {alternating knot; achiral knot; Tait conjectures},
language = {eng},
month = {1},
number = {1},
pages = {25-55},
publisher = {Université Paul Sabatier, Toulouse},
title = {A proof of Tait’s Conjecture on prime alternating $-$achiral knots},
url = {http://eudml.org/doc/251018},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Ermotti, Nicola
AU - Cam Van Quach Hongler
AU - Weber, Claude
TI - A proof of Tait’s Conjecture on prime alternating $-$achiral knots
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2012/1//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - 1
SP - 25
EP - 55
AB - In this paper we are interested in symmetries of alternating knots, more precisely in those related to achirality. We call the following statement Tait’s Conjecture on alternating $-$achiral knots:Let $K$ be a prime alternating $-$achiral knot. Then there exists a minimal projection $\Pi $ of $K$ in $S^2 \subset S^3$ and an involution $\varphi : S^3 \rightarrow S^3$ such that:1) $\varphi $ reverses the orientation of $S^3$;2) $\varphi (S^2) = S^2$;3) $\varphi (\Pi ) = \Pi $;4) $\varphi $ has two fixed points on $\Pi $ and hence reverses the orientation of $K$.The purpose of this paper is to prove this statement.For the historical background of the conjecture in Peter Tait’s and Mary Haseman’s papers see [16].
LA - eng
KW - alternating knot; achiral knot; Tait conjectures
UR - http://eudml.org/doc/251018
ER -

References

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