Geometry and Topology Seminar: "Fixed-point-free pseudo-Anosovs and the cinquefoil"

Abstract: In a recent and marvelous paper, Baldwin-Hu-Sivek (BHS) show that if there exists a knot K in the 3-sphere that is distinct from the torus knot T(2,5) while having the same knot Floer homology, then there exists a certain pseudo-Anosov (pA) homeomorphism. This pA must have many special properties: among other things, it is defined on a genus-2 Seifert surface for K, and has no fixed points in the interior of the surface. As it happens, there is an excellent tool ready-made for studying fixed points of pseudo-Anosovs: a train track. In this talk, I will introduce train tracks and discuss how to work with them. In particular I will sketch the proof that knot Floer homology detects T(2,5), by using train tracks to show that there is no pA having the requisite properties articulated by BHS. This is joint work with Braeden Reinoso and Luya Wang.

Host: Michael Landry