Knot invariants in lens spaces

TL;DR

This survey reviews various knot invariants in lens spaces, including skein modules and polynomials, highlighting their ability to distinguish complex knots with symmetries.

Contribution

It compiles and compares multiple knot invariants in lens spaces, emphasizing their effectiveness in differentiating complex symmetric knots.

Findings

Knot invariants in lens spaces generalize classical polynomials.

Some invariants can distinguish knots with symmetries.

Comparison shows varying effectiveness of invariants.

Abstract

In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT, Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial of links in lens spaces, which we represent as mixed link diagrams. These invariants generalize the corresponding knot polynomials in the classical case. We compare the invariants by means of the ability to distinguish between some difficult cases of knots with certain symmetries.