Cabling and transverse simplicity

Details Cabling and transverse simplicity Cabling and transverse simplicity

2003-06-23

arxiv.org/abs/math/0306330v2

Ann. of Math. (2) 162 (2005), no. 3, 1305--1333

Mathematics

Symplectic Geometry

29 pages, published version

Scientific paper

We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type K, we analyze the Legendrian knots in knot types obtained from K by cabling, in terms of Legendrian knots in the knot type K. As a corollary of this analysis, we show that the (2,3)-cable of the (2,3)-torus knot is not transversely simple and moreover classify the transverse knots in this knot type. This is the first classification of transverse knots in a non-transversely-simple knot type. We also classify Legendrian knots in this knot type and exhibit the first example of a Legendrian knot that does not destabilize, yet its Thurston-Bennequin invariant is not maximal among Legendrian representatives in its knot type.

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