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

Dheeraj Kulkarni, Monika Yadav

TL;DR

The paper extends the Jones polynomial and Khovanov homology to Legendrian knots by developing a front-projection based bracket polynomial P_K(A,r) that is invariant under Legendrian Reidemeister moves and a categorification in the form of a trigraded Legendrian Khovanov homology H_{i,j,k}(K_F) whose Euler characteristic equals P_K(A,r). The Thurston-Bennequin invariant tb(K) appears as a natural grading shift in the new k-grading, linking Legendrian geometry to the classical invariants. The construction recovers the classical Jones polynomial and Khovanov homology when the Legendrian structure is ignored and is proven invariant under LR moves, with integer coefficients and a discussion of strengths and limitations, including cases like Chekanov knots where distinctions remain challenging. Overall, the work provides a natural and computable Legendrian refinement of quantum invariants and a categorification that encodes contact-geometric data.

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.

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

TL;DR

The paper extends the Jones polynomial and Khovanov homology to Legendrian knots by developing a front-projection based bracket polynomial P_K(A,r) that is invariant under Legendrian Reidemeister moves and a categorification in the form of a trigraded Legendrian Khovanov homology H_{i,j,k}(K_F) whose Euler characteristic equals P_K(A,r). The Thurston-Bennequin invariant tb(K) appears as a natural grading shift in the new k-grading, linking Legendrian geometry to the classical invariants. The construction recovers the classical Jones polynomial and Khovanov homology when the Legendrian structure is ignored and is proven invariant under LR moves, with integer coefficients and a discussion of strengths and limitations, including cases like Chekanov knots where distinctions remain challenging. Overall, the work provides a natural and computable Legendrian refinement of quantum invariants and a categorification that encodes contact-geometric data.

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.
Paper Structure (11 sections, 13 theorems, 51 equations, 10 figures, 7 tables)

This paper contains 11 sections, 13 theorems, 51 equations, 10 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of Jones Polynomial for Legendrian Knots
  4. Extension of Khovanov Homology for Legendrian knots
  5. The Legendrian State Sum
  6. Construction of the Legendrian Khovanov Chain Complex
  7. The boundary map
  8. Invariance of the Legendrian Khovanov homology under LR moves
  9. The Legendrian Khovanov homology with integer coefficients
  10. Strengths and limitations of Legendrian Khovanov homology
  11. Appendix

Key Result

Theorem 1.1

Let $K$ be a Legendrian knot in $(\mathbb{R}^3,\xi).$ Let $K_F$ be its front projection, then there is a polynomial in two variables, denoted by $P_K(A,r)$, (see Section Legend_Jones_poly for detailed definition) that is an in invariant of Legendrian knot upto Legendrian isotopy. Moreover, the subst

Figures (10)

  • Figure 1: Examples of front projections
  • Figure 2: LR moves for front projection
  • Figure 3: Resolutions of a crossing
  • Figure 4: List of all the incident states
  • Figure 5: Case 1. $x$ and $y$ are part of one loop in $S$ and three loops in $U$
  • ...and 5 more figures

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 34 more