Topological steps toward the Homflypt skein module of the lens spaces L(p,1) via braids

Abstract

In this paper, we work toward the Homflypt skein module of the lens spaces [Formula: see text], [Formula: see text] using braids. In particular, we establish the connection between [Formula: see text], the Homflypt skein module of the solid torus ST, and [Formula: see text] and arrive at an infinite system, whose solution corresponds to the computation of [Formula: see text]. We start from the Lambropoulou invariant [Formula: see text] for knots and links in ST, the universal analog of the Homflypt polynomial in ST, and a new basis, [Formula: see text], of [Formula: see text] presented in [I. Diamantis and S. Lambropoulou, A new basis for the Homflypt skein module of the solid torus, J. Pure Appl. Algebra 220(2) (2016) 577–605, http://dx.doi.org/10.1016/j.jpaa.2015.06.014 , arXiv:1412.3642 [math.GT]]. We show that [Formula: see text] is obtained from [Formula: see text] by considering relations coming from the performance of braid band move(s) [bbm] on elements in the basis [Formula: see text], where the bbm are performed on any moving strand of each element in [Formula: see text]. We do that by proving that the system of equations obtained from diagrams in ST by performing bbm on any moving strand is equivalent to the system obtained if we only consider elements in the basic set [Formula: see text]. The importance of our approach is that it can shed light on the problem of computing skein modules of arbitrary c.c.o. [Formula: see text]-manifolds, since any [Formula: see text]-manifold can be obtained by surgery on [Formula: see text] along unknotted closed curves. The main difficulty of the problem lies in selecting from the infinitum of band moves some basic ones and solving the infinite system of equations.