Fixed point-free pseudo-Anosovs and the cinquefoil

I will discuss joint work with Braeden Reinoso and Ethan Farber on showing that the pseudo-Anosov representative of any genus-two, hyperbolic, fibered knot in the 3-sphere with nonzero fractional Dehn twist coefficient has an interior fixed point. Combined with work of Baldwin-Hu-Sivek where they use the number of fixed points of appropriate area-preserving diffeomorphism to constrain ranks of Floer homology, this shows that knot Floer homology detects the cinquefoil T(2,5). In particular, T(2,5) is the only genus-two L-space knot. I will discuss some tools from train tracks and train track maps, and outline the main ideas of our proof.