A recursive relation in the $(2p+1,2)$ torus knot

TL;DR

This paper explores recursive relations for colored Jones polynomials of knots in 3-manifolds, specifically analyzing a $(2p+1,2)$ torus knot complement, and extends the concept to knots in such manifolds, revealing new recursive structures.

Contribution

It introduces a novel approach by extending colored Jones polynomials to knots in 3-manifolds and identifies recursive relations and polynomials for a specific knot in the $(2p+1,2)$ torus knot complement.

Findings

Existence of recursive relations for the studied knot

Identification of associated recurrence polynomials

Development of skein basis representations in genus two handlebody

Abstract

It is known that the colored Jones polynomials of a knot in the 3-dimensional sphere satisfy recursive relations, it is also known that these recursive relations come from recurrence polynomials which have been related, by the AJ conjecture, to the geometry of the knot complement. In this paper we propose a new line of thought, by extending the concept of colored Jones polynomials to knots in the 3-dimensional manifold such as a knot complement, and then examining the case of one particular knot in the complement of the $(2p+1,2)$ torus knot for which an analogous recursive relation exists, and moreover, this relation has an associated recurrence polynomial. Part of our study consists of the writing in the standard basis of the genus two handlebody of two families of skeins in this handlebody.