Equivariant crossing numbers for two-bridge knots
Abstract
Symmetries of knots have been studied extensively, and strongly invertible knots are one of them. Lamm defined the equivariant crossing number ct(K), the minimum crossing number among all symmetric diagrams for a strongly invertible knot K. In this paper, we define c2(K) for two-bridge knots by restricting diagrams to two types. This gives an upper bound for ct(K). We give an algorithm to determine c2(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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