Kauffman bracket skein module of the connected sum of handlebodies and non-injectivity

For the handlebody [Formula: see text] of genus [Formula: see text], Przytycki studied the (Kauffman bracket) skein module [Formula: see text] of the connected sum [Formula: see text] at [Formula: see text]. One of his results is that, in the case when [Formula: see text] is invertible for any [Formula: see text], a homomorphism [Formula: see text] is an isomorphism, which is induced by a natural way. In this paper, in the case when [Formula: see text], the ground ring is [Formula: see text], and [Formula: see text] is a [Formula: see text]-th root of unity ([Formula: see text]), we show that [Formula: see text] is not injective.

The Kauffman Skein Module at 1st Order

For a three-manifold $M$ with boundary, we study the Kauffman module with indeterminate equal to $-1+\epsilon $ where $\epsilon ^2=0$. We conjecture an explicit relation between this module and the Reidemeister torsion of $M$, which we prove in particular cases. As a maybe-useful tool, we then introduce a notion of twisted self-linking and prove that it satisfies the Kauffman relations at 1st order. These questions come from considerations on asymptotics of quantum invariants.

The Kauffman bracket skein module of the handlebody of genus 2 via braids

In this paper we present two new bases, [Formula: see text] and [Formula: see text], for the Kauffman bracket skein module of the handlebody of genus 2 [Formula: see text], KBSM[Formula: see text]. We start from the well-known Przytycki-basis of KBSM[Formula: see text], [Formula: see text], and using the technique of parting we present elements in [Formula: see text] in open braid form. We define an ordering relation on an augmented set [Formula: see text] consisting of monomials of all different “loopings” in [Formula: see text], that contains the sets [Formula: see text], [Formula: see text] and [Formula: see text] as proper subsets. Using the Kauffman bracket skein relation we relate [Formula: see text] to the sets [Formula: see text] and [Formula: see text] via a lower triangular infinite matrix with invertible elements in the diagonal. The basis [Formula: see text] is an intermediate step in order to reach at elements in [Formula: see text] that have no crossings on the level of braids, and in that sense, [Formula: see text] is a more natural basis of KBSM[Formula: see text]. Moreover, this basis is appropriate in order to compute Kauffman bracket skein modules of closed–connected–oriented (c.c.o.) 3-manifolds [Formula: see text] that are obtained from [Formula: see text] by surgery, since isotopy moves in [Formula: see text] are naturally described by elements in [Formula: see text].

An important step for the computation of the HOMFLYPT skein module of the lens spaces L(p,1) via braids

We prove that, in order to derive the HOMFLYPT skein module of the lens spaces [Formula: see text] from the HOMFLYPT skein module of the solid torus, [Formula: see text], it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant [Formula: see text] for knots and links in the solid torus, braid band moves that are performed only on the first moving strand of elements in a set [Formula: see text], augmenting the basis [Formula: see text] of [Formula: see text].

The Dubrovnik and Kauffman skein modules of the lens spaces Lp,1

We compute the Dubrovnik skein module of the lens spaces [Formula: see text], [Formula: see text], as well as the Kauffman two variables skein module when [Formula: see text] is odd. We also show that there is torsion in the Kauffman skein module when [Formula: see text] is even.