Periodic projections of alternating knots

Antonio F. Costa; Cam Van Quach Hongler

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arXiv · 1905.13718

Periodic projections of alternating knots

Antonio F. Costa·Cam Van Quach Hongler

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Abstract

This paper is devoted to prove the existence of $q$-periodic alternating projections of prime alternating $q$-periodic knots. The main tool is the Menasco-Thistlethwaite's Flyping theorem. Let $K$ be an oriented prime alternating knot that is $q$-periodic with $q\geq 3$, i.e. $K$ admits a symmetry that is a rotation of order $q$. Then $K$ has an alternating $q$-periodic projection. As applications, we obtain the crossing number of a $q$ -periodic alternating knot with $q\geq 3$ is a multiple of $q$ and we give an elementary proof that the knot $12_{a634} $ is not 3-periodic; this proof does not depend on computer computations as in "Periodic knots and Heegaard Floer correction terms" by Stanilav Jabuka and Swatee Naik (arXiv:1307.5116 [math.GT]).

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BibTeX RIS

Antonio F. Costa, Cam Van Quach Hongler. 2021-03-04. Periodic projections of alternating knots. https://arxiv.org/abs/1905.13718

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