Link diagrams in Seifert manifolds and applications to skein modules

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.