On knots and links in lens spaces

Abstract

In this paper we study some aspects of knots and links in lens spaces. Namely, if we consider lens spaces as quotient of the unit ball B 3 with suitable identification of boundary points, then we can project the links on the equatorial disk of B 3 , obtaining a regular diagram for them. In this contest, we obtain a complete finite set of Reidemeister type moves establishing equivalence, up to ambient isotopy, a Wirtinger type presentation for the fundamental group of the complement of the link and a diagrammatic method giving the first homology group. We also compute Alexander polynomial and twisted Alexander polynomials of this class of links, showing their correlation with Reidemeister torsion.Mathematics Subject Classification 2010: Primary 57M25, 57M27; Secondary 57M05.

Knot Invariants in Lens Spaces

“…x i3 (b) negative crossing Given a mixed link diagram of U −p/q ∪ L the following proposition allows us to describe the fundamental group of L(p, q) \ L (cf. [1,12]). We briefly recall the construction of the Alexander polynomial using Fox calculus [33,16].…”

“…In this case, we need the notion of a twisted Alexander polynomial. We recall the following from [1].…”

Equivalence of two diagram representations of links in lens spaces and essential invariants

“…In this section we recall the notion of the disk diagram for links in lens spaces developed in [8], and the corresponding equivalence moves.…”

“…We denote with F : The construction of the disk diagram. We briefly recall the construction of the disk diagram for a link in a lens space developed in [8]. Throughout the section we assume p > 1.…”

“…In [8] it is shown that two disk diagrams represent the same link if and only if they are connected by a finite sequence of the seven Reidemeister-type moves depicted in Fig. 3.…”