Climbing a Legendrian mountain range without stabilization
Volume 100 / 2014
Abstract
We introduce a new braid-theoretic framework with which to understand the Legendrian and transversal classification of knots, namely a Legendrian Markov Theorem without Stabilization which induces an associated transversal Markov Theorem without Stabilization. We establish the existence of a nontrivial knot-type specific Legendrian and transversal MTWS by enhancing the Legendrian mountain range for the $(2,3)$-cable of a $(2,3)$-torus knot provided by Etnyre and Honda, and showing that elementary negative flypes allow us to move toward maximal $tb$ value without having to use Legendrian stabilization. In doing so we obtain new ways to visualize convex tori and Legendrian divides and rulings, using tilings and braided rectangular diagrams.
