Using knot Floer invariants to detect prime knots

Samantha Allen, Charles Livingston, Misha Temkin and C.-M. Michael Wong

Vol. 345 (2026), No. 2, 225–237

DOI: 10.2140/pjm.2026.345.225

Abstract

We present knot primality tests that are built from knot Floer homology. The most basic of these is a simply stated and elementary consequence of Heegaard Floer theory: if the knot Floer polynomial ${\mathrm{\Omega }}_K(s,t)$ is irreducible, then $K$ is prime. Building off this basic observation, we develop a primality condition that is over 92% effective in identifying prime knots of up to 15 crossings. As another illustration of the strength of these tools, there are 1,315 nonhyperbolic prime knots with crossing number 20 or less; the tests we develop prove the primality of over 97% of them.

This basic approach has implications for the knot Floer botany problem. As a simple example, there are infinitely many knots $K$ which have $\widehat{\mathrm{HFK}<!--FUNCTION\; APPLICATION-->}(K)$ isomorphic to that of the Kinoshita–Terasaka knot $11n_{42}$. However, the basic irreducibility criterion immediately implies that any such $K$ is prime.

The filtered chain homotopy class of $\widehat{\mathrm{CFK}<!--FUNCTION\; APPLICATION-->}(K)$ has a unique minimal-dimension representative that is the direct sum of a one-dimensional complex and two-dimensional complexes, each of which can be assigned a parity. Let $\delta (K)$, $b_{e<!--FUNCTION\; APPLICATION-->}(K)$, and $b_{o<!--FUNCTION\; APPLICATION-->}(K)$ denote the dimension of this minimal representative and the number of even and odd two-dimensional summands, respectively. For a composite knot $K$, we observe that there is a nontrivial factorization $\delta (K)=d_1{d}_2$ satisfying

$$(d_1-1)(d_2-1)\le 4\min <!--FUNCTION\; APPLICATION-->(b_{e<!--FUNCTION\; APPLICATION-->}(K),b_{o<!--FUNCTION\; APPLICATION-->}(K)).$$

This yields another knot primality test. One corollary is a simple proof of Krcatovich’s result that $L$-space knots are prime.

Keywords

Knot Floer homology, prime knots, knot primality tests, bar-complex, knot Floer botany problem, L-space knots

Mathematical Subject Classification

Primary: 57K10

Secondary: 57K18

Milestones

Received: 19 June 2025

Revised: 17 February 2026

Accepted: 21 May 2026

Published: 26 October 2026