A recursive relation in the $(2p+1,2)$ torus knot (2409.18960v2)

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.