New results and conjectures on slice knots and knot homologies

New results and conjectures on slice knots and knot homologies

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Marco Marengon, Alfred Renyi Institute of Mathematics
Fine Hall 314

In-Person and Online Talk

Inspired by a classical result of Fox-Milnor for the Alexander polynomial, in joint work with Hockenhull and Willis we prove that for a family of knots (a subfamily of symmetric unions) the ranks of knot Floer homology is an odd square integer.

In joint work with Dunfield, Gong, Hockenhull, and Willis, we conjecture that the weaker statement that the rank of knot Floer homology is congruent to 1 modulo 8 should hold for all ribbon knots: this would imply that the modulo 8 residue of the rank defines a homomorphism from the concordance group to . I will show that the knot Floer side of the conjecture holds true for all ribbon knots of fusion number 1, and I will explain why a simple skein proof of such a fact cannot exist. I will also discuss the Khovanov counterparts of the above.