Using Heegaard Floer homology and dihedral covers to detect prime knots

In this talk, I will present knot primality tests that are built from knot Floer homology.  The most basic of these is an elementary consequence of Heegaard Floer theory: if the knot Floer polynomial is irreducible, then the knot is prime.  I will describe improvements on this test which yield a primality condition that is over 92% effective in identifying prime knots up to 15 crossings.  This work was inspired by the barcode complex associated to a knot and, with that perspective, we give a short proof of the Krcatovich’s result that L-space knots are prime.  
In addition, I will describe ongoing work which further addresses the case where the knot Floer polynomial factors nontrivially.  Here we pursue an unexpected coupling of Heegaard Floer theory with the use of dihedral covers of knots, incorporating a method of Reidemeister.
This talk is based on joint work with Charles Livingston, Misha Temken, and C.-M. Michael Wong and ongoing work-in-progress with Charles Livingston.