On a generalization of Jones polynomial and its categorification for Legendrian Knots
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909823938854912 |
|---|---|
| author | Kulkarni, Dheeraj Yadav, Monika |
| author_facet | Kulkarni, Dheeraj Yadav, Monika |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_00777 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On a generalization of Jones polynomial and its categorification for Legendrian Knots Kulkarni, Dheeraj Yadav, Monika Geometric Topology Symplectic Geometry 57K10, 57K14, 57K33 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. |
| title | On a generalization of Jones polynomial and its categorification for Legendrian Knots |
| topic | Geometric Topology Symplectic Geometry 57K10, 57K14, 57K33 |
| url | https://arxiv.org/abs/2207.00777 |