Legendrian and transverse twist knots

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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 n²2 Legendrian representatives with maximal Thurston--Bennequin number, and /2 transverse representatives with maximal self-linking number. Our techniques include convex surface theory, Legendrian ruling invariants, and Heegaard Floer homology.