Braids, fibered knots, and concordance questions
Abstract
Given a knot in S3, one can associate to it a surface diffeomorphism in two different ways. First, an arbitrary knot in S3 can be represented by braids, which can be thought of as diffeomorphisms of punctured disks. Second, if the knot is fibered -- that is, if its complement fibers over S1 -- one can consider the monodromy of the fibration. One can ask to what extent properties of these surface diffeomorphisms dictate topological properties of the corresponding knot. In this article we collect observations, conjectures, and questions addressing this, from both the braid perspective and the fibered knot perspective. We particularly focus on exploring whether properties of the surface diffeomorphisms relate to four-dimensional topological properties of knots such as the slice genus.