Genera, band sum of knots and Vassiliev invariants |
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Authors: | Leonid Plachta |
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Affiliation: | Institute of Mathematics, University of Gdańsk, 80 952 Wita Stwosza, 57, Gdańsk, Poland Institute of Applied Problems of Mechanics and Mathematics of NAS of Ukraine, Naukova 3b, 79 060 Lviv, Ukraine |
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Abstract: | 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. |
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Keywords: | 57M25 |
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