The Alexander polynomials of immersed concordant links
In this low-dimensional topology article, the author relates the 4-dimensional clasp distance of two links to their beta-distance and their torsion Alexander polynomials. The main result unifies results of Kanenobu and the author. Let \(L\) and \(L'\) be two links in the 3-sphere \(S^3\) with \(r\geq2\) components. The 4-dimensional clasp distance \(c^4(L,L')\) is given by minimizing the number of transversal double points of \(r\) immersed annuli in \(S^3\times[0,1]\) that fill \(L\times\{0\}\cup L'\times\{1\}\). Let \(H_L\) be the link \(\Lambda\)-module \(H_L\), where \(\Lambda\) denotes the ring of Laurent polynomials with integer coefficients in \(r\) indeterminates. The 4-dimensional clasp distance is bounded from above by the Gordian distance \(u(L,L')\). The beta-rank \(\beta(L)\) is defined to be the \(\Lambda\)-rank of \(H_L\), the (torsion) Alexander polynomial \(\Delta_L\) (\(\Delta_L^T\)) is the characteristic polynomial of (the torsion of) \(H_L\), and the beta-distance \(\beta(L,L')\) is defined to be \(|\beta(L')-\beta(L)|\). Note that \(\Delta_L=0\) if \(\beta(L)>0\) and \(\Delta_L=\Delta_L^T\) if \(\beta(L)=0\). The main result of this article establishes that \[ \beta(L,L')\leq c^4(L,L')\leq u(L,L'), \] for all links \(L,L'\) with the same number of components \(r\geq 2\). Furthermore, if \(\beta(L,L')=c^4(L,L')\), then there exists \(f\in\Lambda\) such that \(\Delta_L^T\Delta_{L'}^T\) equals \(g(L,L')ff^*\) up to multiplication by units, where \(f^*\in\Lambda\) is the element obtained from \(f\) by replacing the indeterminates by their inverses. Here, \(g(L,L')\in\Lambda\) is the so called gap factor and can be described explicitly. This generalizes the \textit{Concordance Theorem} by the author [Osaka J. Math. 15, 151--159 (1978; Zbl 0401.57013)]. As a corollary, the author obtains a \textit{4-dimensional Clasp Number One Theorem}, which in turn recovers the \textit{Unlinking Number One Theorem} of \textit{Kanenobu} (Annonced in an abstract of the fall term meeting of the Math. Soc. Japan (1993)). For the proof of the main result, the author nicely blends surgery techniques and homological algebra over \(\Lambda\). The paper ends with applying the main result to explicit examples of links.
scientific article; zbMATH DE number 6384878