The Alexander polynomial of links in lens spaces

Abstract

We show how the Alexander polynomial of links in lens spaces is related to the classical Alexander polynomial of a link in the 3-sphere, obtained by cutting out the exceptional lens space fibre. It follows from these relationship that a certain normalization of the Alexander polynomial satisfies a skein relation in lens spaces.

The Alexander polynomial for closed braids in lens spaces

“…It is also shown in [10] that a normalized version of the Alexander polynomial in lens spaces, denoted by ∇ L (t), respects a skein relation…”

“…Example 4.4. It has been calculated in [10] that the Alexander polynomial for the knot in Figure 1(a) is equal to t 2p − t p + 1. The braid representative of this knot is represented in Figure 1 det I − ρ(tσ 3 1 )(t q , t p ) = −t 2p+q + t 3p+q − t 4p+q + t 2p − t p + 1.…”

Knot Invariants in Lens Spaces

“…In this section we describe a Torres-type formula (see [31]), constructed in [16] for the Alexander polynomial of links in lens spaces defined by Fox's free differential calculus [9,22,33].…”

“…[1,12]). We briefly recall the construction of the Alexander polynomial using Fox calculus [33,16]. Suppose P = x1, .…”

“…For a link in L(p, q), the abelianization of its link group may also contain torsion, see [16,Corollary 2.10]. In this case, we need the notion of a twisted Alexander polynomial.…”