Fixed-point-free pseudo-Anosovs, and genus-two L-space knots in the Poincare sphere

We classify genus-two L-space knots in the Poincar\'e homology sphere. This leads to the second knot Floer homology detection result for a knot of genus at least two, and the first such result outside of $S^3$. The argument uses the theory of train tracks and folding automata, and builds off of recent work connecting knot Floer homology of fibered knots to fixed points of mapping classes. As a consequence of our proof technique, we almost completely classify genus-two, hyperbolic, fibered knots with knot Floer homology of rank 1 in their next-to-top grading in any 3-manifold.