Periodic knots and Heegaard Floer correction terms

TL;DR

This paper introduces new obstructions to knot periodicity using Heegaard Floer correction terms, successfully classifying many alternating knots with prime periods and providing comprehensive listings for knots with higher crossings.

Contribution

It develops novel obstructions based on Heegaard Floer theory and applies them to classify alternating knots with prime periods, surpassing previous methods.

Findings

Successfully obstructs periodicity in many cases where previous methods failed

Nearly classifies all alternating 12-crossing knots with odd prime periods

Provides complete listings for knots with 13-15 crossings and odd prime periods > 3

Abstract

We derive new obstructions to periodicity of classical knots by employing the Heegaard Floer correction terms of the finite cyclic branched covers of the knots. Applying our results to two fold covers, we demonstrate through numerous examples that our obstructions are successful where many existing periodicity obstructions fail. A combination of previously known periodicity obstructions and the results presented here, leads to a nearly complete (with the exception of a single knot) classification of alternating, periodic, 12-crossing knots with odd prime periods. For the case of alternating knots with 13, 14 and 15 crossings, we give a complete listing of all periodic knots with odd prime periods greater than 3.