The ${\rm SL}(2,\mathbb{C})$-character variety of an arborescent knot

Haimiao Chen

The ${\rm SL}(2,\mathbb{C})$-character variety of an arborescent knot

Geometric Topology 2024-03-05 v2

Authors: Haimiao Chen

Abstract

We clarify steps for determining the ${\rm SL}(2,\mathbb{C})$ -character variety of any arborescent knot. Interestingly, we show that the `excellent parts' of arborescent knots $K_1,K_2$ are isomorphic if $K_1$ can be related to $K_2$ through certain moves on projection diagrams. Furthermore, we give a sufficient condition in terms of diagram for the existence of components of dimension larger than $1$ . A generalization to arborescent links is sketched.

Keywords

sl(2,c)-character variety arborescent knot excellent part projection diagram arborescent link

Cite

@article{arxiv.2304.10345,
  title  = {The ${\rm SL}(2,\mathbb{C})$-character variety of an arborescent knot},
  author = {Haimiao Chen},
  journal= {arXiv preprint arXiv:2304.10345},
  year   = {2024}
}

Comments

19 pages, 8 figures. This version is better than the 1st one, but is still not satisfactory enough to be submitted to a journal

The ${\rm SL}(2,\mathbb{C})$-character variety of an arborescent knot

Abstract

Keywords

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