Equivariant crossing numbers for two-bridge knots

Jundai Nanasawa

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arXiv · 2304.00540

Equivariant crossing numbers for two-bridge knots

Jundai Nanasawa

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Abstract

Symmetries of knots have been studied extensively, and strongly invertible knots are one of them. Lamm defined the equivariant crossing number $c_t(K)$, the minimum crossing number among all symmetric diagrams for a strongly invertible knot $K$. In this paper, we define $c_2(K)$ for two-bridge knots by restricting diagrams to two types. This gives an upper bound for $c_t(K)$. We give an algorithm to determine $c_2(K)$ for any two-bridge knot. The results of calculation by a computer up to 14 crossings are shown. As a corollary, we show 20 examples of knots up to 10 crossings in Rolfsen's knot table whose symmetry can be improved without increasing the number of crossings.

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Jundai Nanasawa. 2023-04-02. Equivariant crossing numbers for two-bridge knots. https://arxiv.org/abs/2304.00540

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