Trefoils and hexafoils in 3-braids

Abstract

We determine the cobordism distance between all positive 3-braid links and connected sums of trefoil knots, up to a constant error. This may be seen as a reinterpretation of Coxeter's finiteness result on the braid group on three strands modulo third powers of the generators. We go on to explore the limits of Coxeter's result, and find the cobordism distance between three strand torus links and almost all connected sums of two strand torus links, up to a constant error.