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

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.