# Periodic projections of alternating knots

## TL;DR

This paper proves that prime alternating q-periodic knots have q-periodic alternating projections using the Flyping theorem, and derives consequences for crossing numbers and knot periodicity without heavy computations.

## Contribution

It establishes the existence of q-periodic alternating projections for prime alternating q-periodic knots, extending understanding of their symmetry and structure.

## Key findings

- q-periodic alternating knots have q-periodic projections
- Crossing number of q-periodic alternating knots is divisible by q
- Elementary proof that knot 12_{a634} is not 3-periodic

## 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]).