Distinguishing Legendrian knots of topological type $7_4$, $9_{48}$ and $10_{136}$

TL;DR

This paper applies a new method to distinguish Legendrian knots with symmetries, confirming conjectures and completing the classification of certain knot types up to complexity 9.

Contribution

It introduces a novel comparison method for Legendrian knots with symmetries and verifies conjectures for specific topological knot types.

Findings

Confirmed conjectures of Ng and Chongchitmate for specific knots.

Completed classification of Legendrian types for knots up to complexity 9.

Validated the effectiveness of the new comparison method.

Abstract

In a recent work of I. Dynnikov and M. Prasolov a new method of comparing Legendrian knots with nontrivial symmetry group is proposed. Using this method we confirm conjectures of Ng and Chongchitmate about Legendrian knots in topological types $7_4$, $9_{48}$ and $10_{136}$. This completes the classification of Legendrian types of rectangular diagrams of knots of complexity up to 9.