Periodic projections of alternating knots

2019/05/31 by Costa, Antonio F., Hongler, Cam Van Quach
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1905.13718

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≥ 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≥ 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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