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arXiv · 1802.03645

Link diagrams in Seifert manifolds and applications to skein modules

Boštjan Gabrovšek·Maciej Mroczkowski
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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.

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Boštjan Gabrovšek, Maciej Mroczkowski. 2018-02-13. Link diagrams in Seifert manifolds and applications to skein modules. https://doi.org/10.1007/978-3-319-68103-0_6

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