Geodesic
Legendrian and transverse twist knots
Journal of the European Mathematical Society, Tome 15 (2013) no. 3, pp. 969-995

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In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the m(52​) 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 ⌈2n2​⌉ Legendrian representatives with maximal Thurston–Bennequin number, and ⌈2n​⌉ transverse representatives with maximal self-linking number. Our techniques include convex surface theory, Legendrian ruling invariants, and Heegaard–Floer homology.
DOI : 10.4171/jems/383
Classification : 57-XX, 53-XX, 00-XX
Keywords: Legendrian knot, transverse knot, twist knots 
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John B. Etnyre; Lenhard L. Ng; Vera Vértesi. Legendrian and transverse twist knots. Journal of the European Mathematical Society, Tome 15 (2013) no. 3, pp. 969-995. doi: 10.4171/jems/383

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