Genera, band sum of knots and Vassiliev invariants

Recently Stoimenow showed that for every knot K and any n∈N and u0⩾u(K) there is a prime knot Kn,uo which is n-equivalent to the knot K and has unknotting number u(Kn,uo) equal to u0. The similar result has been obtained for the 4-ball genus gs of a knot. Stoimenow also proved that any admissible value of the Tristram–Levine signature σξ can be realized by a knot with the given Vassiliev invariants of bounded order. In this paper, we show that for every knot K with genus g(K) and any n∈N and m⩾g(K) there exists a prime knot L which is n-equivalent to K and has genus g(L) equal to m.