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We present knot primality tests that are built from knot Floer homology. Themost basic of these is a simply stated and elementary consequence of HeegaardFloer theory: if the two-variable knot Floer polynomial of a knot K isirreducible, then K is prime. Improvements in this test yield a primalitycondition that has been over 90 percent effective in identifying prime knots ofup 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; thetests 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 Khas a unique minimal-dimension representative that is the direct sum of aone-dimensional complex and two-dimensional complexes, each of which can beassigned a parity. Let delta(K), b_e(K), and b_o(K) denote the dimension ofthis minimal representative and the number of even and odd two-dimensionalsummands, respectively. For a composite knot K, we observe that there is anon-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 simpleproof of Krcatovich's result that L-space knots are prime.

Using knot Floer invariants to detect prime knots