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

Samantha Allen; Charles Livingston

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arXiv · 2508.08102

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

Samantha Allen·Charles Livingston

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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 99.67% of knots in a large family of prime knots that includes all prime knots with 15 or fewer crossings. There are additional ways in which our approach highlights the power of Heegaard Floer methods. For one, a single computation can prove the primality of an infinite family of knots. We also illustrate the application of our approach to the setting of general three-manifolds by proving the primality of a knot in a nontrivial homology sphere.

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Samantha Allen, Charles Livingston. 2025-08-11. Knot primality: knot Floer homology, metacyclic representations and twisted homology. https://arxiv.org/abs/2508.08102

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