Homogeneous braids are visually prime

If a knot decomposes as connected sum, is this decomposition visible in a diagram of the knot? We shall see that the answer is "yes", for diagrams coming from homogeneous braids. This partially resolves a conjecture made by Cromwell in 1991. The proof relies on a new criterion for the existence of fixed arcs in open books. The talk will not presume previous knowledge of braids and open books. It is based on joint work with Peter Feller and Miguel Orbegozo Rodriguez.