Splitting numbers and signatures (Q2827391)

Let \(L\) be any link in \(S^3\). The splitting number of \(L\), denoted \(sp(L)\), is the minimal number of crossing changes between different components of \(L\) required to convert it into a split link. Given a diagram of \(L\), it is easy to find upper bounds for \(sp(L)\), but finding lower bounds may be tricky. An elementary lower bound for \(sp(L)\) is given by the total linking number. \textit{J. Batson} and \textit{C. Seed} [Duke Math. J. 164, No. 5, 801--841 (2015; Zbl 1332.57011)] gave a lower bound for \(sp(L)\) based on a spectral sequence coming from Khovanov homology. \textit{J.C. Cha} et al. [Proc. Edinb. Math. Soc., II. Ser. 60, No. 3, 587--614 (2017; Zbl 1377.57014)] introduced two techniques for computing \(sp(L)\), one based on covering calculus and the other one based on the multivariable Alexander polynomial. This last one was strengthened by \textit{M. Borodzik} et al. [J. Math. Soc. Japan 68, No. 3, 1047--1080 (2016; Zbl 1359.57003)]. \textit{M. Borodzik} and \textit{E. Gorsky} [``Immersed concordances of links and Heegaard Floer homology'', Preprint, \url{arXiv:1601.07507}] gave a lower bound for \(sp(L)\) based on Heegaard Floer homology, and used it to show that the 2-bridge link with Conway normal form \(C(2a,1,2a)\) has splitting number \(2a\), although the linking number of the two components is zero.NEWLINENEWLINEIn the paper under review a new method for computing \(sp(L)\) is given. In fact, a lower bound for \(sp(L)\) is given in terms of the multivariable signature and nullity of \(L\). These two invariants are generalizations of the one variable Levine-Tristam signature and nullity, which were developed by \textit{D. Cimasoni} and \textit{V. Florens} [Trans. Am. Math. Soc. 360, No. 3, 1223--1264 (2008; Zbl 1132.57004)]. This lower bound is somewhat easier to compute than previous lower bounds. The authors say that it gives sharp lower bounds for 127 out of the 130 prime links with up to 9 crossings. This lower bound is used to show that the splitting number of the 2-bridge link with Conway normal form \(C(2a_1,b_1,2a_2,b_2,\ldots ,2a_{n-1},b_{n-1},2a_n)\) is equal to \(a_1 + \dots+a_n\). The results obtained also hold for colored links, where in this case for a given partition of \(L\) into sublinks, \(sp(L)\) is defined to be the minimal number of crossing changes between the components of different sublinks required to obtain the split union of these links.