Climbing a Legendrian mountain range without stabilization

William Menasco, University at Buffalo

Joint work with Doug LaFountain.

Etnyre and Honda provide a classification of the Legendrian isotopy
classes for a (2,3)-cable of a (2,3)-torus knot as it is embedded in
3-sphere with the standard contact structure.  To do this, they use
the theory of convex surfaces in a tight contact structure.  Their
classification takes the visual form of a mountain range formed from
points having values of (r,tb), where r is the rotation number and tb
is the Thurston-Bennequin number.  In this talk we show how the two
Legendrian classes at (r,tb)=(2,5) can be realized as rectangular
braided diagrams, and are seen to be related by an elementary negative
flype.  We will talk about how we can obtain for this particular cable
knot type a Legendrian Markov Theorem without Stabilization and a
transverse Markov Theorem without Stabilization.