# Legendrian and transverse twist knots

John B. Etnyre; Lenhard L. Ng; Vera Vértesi

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 3, page 969-995
- ISSN: 1435-9855

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topEtnyre, 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 -

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