Knot primality: knot Floer homology, metacyclic representations and twisted homology

TL;DR

This paper introduces algebraic methods combining Heegaard Floer homology and classical covering space techniques to efficiently determine knot primality, achieving high accuracy across large knot families.

Contribution

It presents a novel unified approach using twisted homology and covering space methods to test knot primality, covering a broad class of knots.

Findings

Proved primality for over 99.6% of knots in large families.

Successfully applied methods to all prime knots with 15 or fewer crossings.

Unified algebraic framework enhances primality testing efficiency.

Abstract

We develop purely algebraic methods for proving that a knot is prime. Our approach uses the Heegaard Floer polynomial in conjunction with classical knot-theoretic methods: cyclic, dihedral, and metacyclic covering spaces. The theory of twisted homology allows us to view these approaches from a unified perspective. Collectively, the primality tests developed here have proved primality for over 99.6% of knots in large families of prime knots, including all prime knots with 15 or fewer crossings.