On Positive Braid Knot Monodromies

A fibered knot in the 3-sphere is one whose complement is a surface bundle over the circle. The topology of such knots is tied to their monodromies. Positive braid knots are fibered by a theorem of Stallings from the 1970s. Using knot Floer homology, we verify a rank-one condition which, by a theorem of Ni, forces the monodromy to be fixed-point-free. Consequently, every prime positive braid knot admits a fixed-point-free monodromy. This is joint work with Matthew Hedden.