Ribbon concordance and fibered predecessors (John Baldwin)

PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)

Abstract: A concordance from a knot $J$ to a knot $K$ is a smoothly embedded annulus in $S^3 \times [0,1]$ with boundary $K \times \{1\} \cup - J \times \{0\}$. Concordance defines an equivalence relation on knots, and concordance classes of knots form a much-studied group under connected sum. A ribbon concordance from $J$ to $K$ is a concordance for which projection to the interval $[0,1]$ defines a Morse function on the concordance with no maxima. Ribbon concordance is not a symmetric relation. In fact, Ian Agol recently proved that it defines a partial order on knots, affirming an old conjecture of Cameron Gordon. I'll focus on another conjecture stemming from Gordon's work: for each knot $K$ there are only finitely many knots $J$ that are ribbon concordant to $K$. I'll discuss work with Sivek and Hanselman in which we prove this finiteness for fibered $J$. Our proof makes use of the behavior of Floer homology under ribbon homology cobordism, as well as a relationship between the knot Floer homology of a fibered knot and the number of fixed points of its monodromy. 

In the morning talk (10am in Park 338), I'll motivate the study of concordance and ribbon concordance. I'll give concrete examples, and I will survey what's known about the behaviors of classical invariants like the Alexander polynomial under ribbon concordance. I'll additionally review some facts about fibered knots that will be relevant for the afternoon talk.