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

Haimiao Chen

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arXiv · 2304.10345

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

Haimiao Chen

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Abstract

We describe a procedure for computing the ${\rm SL}(2,\mathbb{C})$-character variety of an arborescent knot. Along the way, we clarify several facts about representations of arborescent tangles. Then we study a family of hyperbolic knots whose exteriors contain closed essential surfaces, showing that each of these knots has $1$-dimensional character variety. This provides infinitely many positive answers to a question of Boyer and Zhang posed in 1998.

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Haimiao Chen. 2023-04-20. The ${\rm SL}(2,\mathbb{C})$-character variety of an arborescent knot. https://arxiv.org/abs/2304.10345

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