Using knot Floer invariants to detect prime knots

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 two-variable knot Floer polynomial of a knot K is irreducible, then K is prime. Improvements in this test yield a primality condition that has been over 90 percent effective in identifying prime knots of up to 30 crossings. As another illustration of the strength of these tools, there are 1,315 non-hyperbolic prime knots with crossing number 20 or less; the tests we develop prove the primality of over 96 percent of them. The filtered chain homotopy class of the knot Floer hat complex of a knot 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(K), and b_o(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 non-trivial factorization delta(K) = mn satisfying (m-1)(n-1) \le 4 min(b_e(K), b_o(K)). This yields another knot primality test. One corollary is a simple proof of Krcatovich's result that L-space knots are prime.