Climbing a Legendrian mountain range without Stabilization

TL;DR

This paper develops a braid-theoretic framework to classify Legendrian and transversal knots without stabilization, enhancing understanding of knot invariants and providing new visualization tools.

Contribution

It introduces a Legendrian Markov Theorem without Stabilization, enabling classification without stabilization and extending the Legendrian mountain range for specific knot types.

Findings

Existence of a nontrivial knot-type specific Legendrian and transversal MTWS.

Elementary negative flypes facilitate movement toward maximal tb without stabilization.

New visualization methods for convex tori and Legendrian divides using tilings and braided diagrams.

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.