On a generalization of Jones polynomial and its categorification for Legendrian Knots

TL;DR

This paper introduces a new polynomial invariant for Legendrian knots, extending the Jones polynomial, and provides a categorification akin to Khovanov homology, incorporating the Thurston-Bennequin invariant.

Contribution

It presents a novel skein relation for Legendrian knots and develops a categorification that extends Khovanov homology to Legendrian knot theory.

Findings

Defined a new polynomial invariant for Legendrian knots.

Constructed a categorification extending Khovanov homology.

Revealed the role of Thurston-Bennequin invariant in the homology.

Abstract

In this article, we explore a polynomial invariant for Legendrian knots which is a natural extension of Jones polynomial for (topological) knots. To this end, a new type of skein relation is introduced for the front projections of Legendrian knots. Further, we give a categorification of the polynomial invariant for Legendrian knots which is a natural extension of Khovanov homology for knots. The Thurston-Bennequin invariant of Legendrian knot appears naturally in the construction of the homology as the grade-shift. The constructions of the polynomial invariant and its categorification are natural in the sense that if we treat Legendrian knots as only knots (that is, we forget the geometry on the knots), then we recover the Jones polynomial and Khovanov homology respectively. In the end, we discuss strengths and limitations of these invariants.