Legendrian and transverse twist knots

Etnyre, John B., Ng, Lenhard L., and Vértesi, Vera. "Legendrian and transverse twist knots." Journal of the European Mathematical Society 015.3 (2013): 969-995. <http://eudml.org/doc/277687>.

@article{Etnyre2013,
abstract = {In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the $m(5_2)$ knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least $n$ different Legendrian representatives with maximal Thurston-Bennequin number of the twist knot $K_\{-2n\}$ with crossing number $2n+1$. In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that $K_\{-2n\}$ has exactly $\left\lceil \frac\{n^2\}\{2\}\right\rceil $ Legendrian representatives with maximal Thurston–Bennequin number, and $\left\lceil \frac\{n\}\{2\}\right\rceil $ transverse representatives with maximal self-linking number. Our techniques include convex surface theory, Legendrian ruling invariants, and Heegaard–Floer homology.},
author = {Etnyre, John B., Ng, Lenhard L., Vértesi, Vera},
journal = {Journal of the European Mathematical Society},
keywords = {Legendrian knot; transverse knot; twist knots; Legendrian knot; transverse knot; twist knots},
language = {eng},
number = {3},
pages = {969-995},
publisher = {European Mathematical Society Publishing House},
title = {Legendrian and transverse twist knots},
url = {http://eudml.org/doc/277687},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Etnyre, John B.
AU - Ng, Lenhard L.
AU - Vértesi, Vera
TI - Legendrian and transverse twist knots
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 3
SP - 969
EP - 995
AB - In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the $m(5_2)$ knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least $n$ different Legendrian representatives with maximal Thurston-Bennequin number of the twist knot $K_{-2n}$ with crossing number $2n+1$. In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that $K_{-2n}$ has exactly $\left\lceil \frac{n^2}{2}\right\rceil $ Legendrian representatives with maximal Thurston–Bennequin number, and $\left\lceil \frac{n}{2}\right\rceil $ transverse representatives with maximal self-linking number. Our techniques include convex surface theory, Legendrian ruling invariants, and Heegaard–Floer homology.
LA - eng
KW - Legendrian knot; transverse knot; twist knots; Legendrian knot; transverse knot; twist knots
UR - http://eudml.org/doc/277687
ER -