The unknotting numbers of \(10_{139}\) and \(10_{152}\) are 4
In the 1960s, Milnor conjectured that the unknotting number of any algebraic knot would be equal to the genus of the Milnor fiber. In the 1990s, Kronheimer and Mrowka proved this conjecture using gauge theory. In particular, the unknotting number of \((p,q)\)-torus knot is \(((p-1) (q-1))/2\). In 1995, \textit{D. Auckly} [Surgery, knots, and the Seiberg-Witten equations, Lectures for the 1995 TGRCI-W, Preprint] gave an alternative proof of this conjecture for certain torus knots using Seiberg-Witten theory. In this paper we shall compute unknotting numbers of certain knots using Seiberg-Witten theory. By extending the argument in [D. Auckly, loc. cit.], the author obtains the following Theorem. Let \(K\) be an oriented knot in \(S^3\times \{1\}\) and \(L_{m,n}\) the link in \(S^3\times \{0\}\). If there is a compact connected oriented surface \(\widehat F\) in \(S^3\times [0,1]\) such that \(\partial \widehat F=L_{m,n} \sqcup K\), then \(u(K)\geq (m-1)(n-1)- g(\widehat F)\) where \(f(\widehat F)\) denotes the genus of \(\widehat F\). By means of that Theorem the author determines the unknotting numbers of certain knots. In particular, in the present paper, he shows the following result: Theorem. (1) The unknotting number of the knot \(10_{139}\) is 4. (2) The unknotting number of the knot \(10_{152}\) is 4.
scientific article; zbMATH DE number 1216669