The Poincaré-Hopf theorem for relative braid classes

This paper studies the existence and multiplicity of closed integral curves of smooth vector fields on the closed 2-disc D² in the complement of a suitable geometric braid y—called a skeleton. The strands in a skeleton y must be trajectories of the vector field and the 1-periodic orbits to be counted must also be representatives of a suitable relative braid class x in the complement of y. From Leray–Schauder degree theory a signed count of the closed trajectories representing x in the complement of y is independent of the vector field. It therefore suffices to compute, or estimate the signed count of periodic orbits for a particularly convenient vector field. We show that the signed count equals the Euler-Floer characteristic of Braid Floer homology, (cf. van den Berg et al. in J Differ Equ 259(5):1663–1721, 2015). The latter can be computed via a finite cube complex which serves as a model for the given braid class.