Link diagrams in Seifert manifolds and applications to skein modules

TL;DR

This survey explores link diagrams in Seifert manifolds using arrow diagrams, detailing Reidemeister moves, diagram transitions, and results on skein modules, including new bases for certain manifolds.

Contribution

It introduces new bases for the Kauffman bracket and HOMFLYPT skein modules of solid tori and lens spaces, advancing the understanding of link invariants in Seifert manifolds.

Findings

Reidemeister moves for arrow diagrams enable link invariant studies

Transitions between arrow diagrams and alternative diagrams are established

New bases for skein modules of solid torus and lens spaces are presented

Abstract

In this survey paper we present results about link diagrams in Seifert manifolds using arrow diagrams, starting with link diagrams in $F\times S^1$ and $N\hat{\times}S^1$, where $F$ is an orientable and $N$ an unorientable surface. Reidemeister moves for such arrow diagrams make the study of link invariants possible. Transitions between arrow diagrams and alternative diagrams are presented. We recall results about %the knot group presentation for lens spaces and the Kauffman bracket and HOMFLYPT skein modules of some Seifert manifolds using arrow diagrams, namely lens spaces, a product of a disk with two holes times $S^1$, $\mathbb{R}P^3 \# \mathbb{R}P^3$, and prism manifolds. We also present new bases of the Kauffman bracket and HOMFLYPT skein modules of the solid torus and lens spaces.