Genera, band sum of knots and Vassiliev invariants

Recently Stoimenow showed that for every knot K and any n ∈ N and u 0 ⩾ u ( K ) there is a prime knot K n , u o which is n-equivalent to the knot K and has unknotting number u ( K n , u o ) equal to u 0 . The similar result has been obtained for the 4-ball genus g s of a knot. Stoimenow also proved that any admissible value of the Tristram–Levine signature σ ξ 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.