Unknotting number and genus of 3-braid knots (Q2850716)

Let \(u(K)\), \(g(K)\) and \(g^*(K)\) denote the unknotting number, genus and 4-ball genus of a knot \(K\) respectively. While \(g^*(K)\leq u(K)\) and \(g^*(K) \leq g(K)\) for all knots \(K\), there is no such general relationship between \(u(K)\) and \(g(K)\). In this paper the authors show that for 3-braid knots, \(g^*(K)\leq u(K) \leq g(K)\). This implies that for strongly quasipositive 3-braid knots, \(g^*(K)=u(K)=g(K)\). The authors also show that if the equality \(u(K)=g(K)\) holds for a 3-braid knot \(K\), \(K\) is either a 2-braid knot, a connected sum of 2-braid knots, the figure-eight knot, a strongly quasipositive knot, or the mirror image of a strongly quasipositive knot. That is, for `generic' 3-braid knots, the strict inequality \(u(K)<g(K)\) holds. The main strategy of proof in the paper involves analyzing the shortest representation (as a word in band generators of the braid group) in the conjugacy class for any non-trivial 3-braid knot.