Non-conjugate braids with the same closure link from density of representations

Abstract

We use some Lie group theory and the unitarizations of the Burau and Lawrence–Krammer representation, to prove that for generic parameters of definite form the image of these representations (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, all links have infinitely many conjugacy classes of braid representations on any non-minimal number of (and at least 4) strands.