Lattices, graphs, and Conway mutation (Q1955773)

For a link \(L\), let \(\Sigma(L)\) denote its double-cover of \(S^2\) branched along \(L\), and let \(\widehat{HF}(\Sigma(L))\) denote the Heegaard Floer homology of \(\Sigma(L)\). The main result of this paper is that for alternating links, the Heegaard Floer homology of the branched double cover provides a complete invariant of mutation type: if \(L\) and \(L'\) are alternating links, then \(L\) and \(L'\) are mutants if and only if \(\widehat{HF}(\Sigma(L))\cong \widehat{HF}(\Sigma(L'))\) as absolutely graded, relatively \(\mathrm{spin}^c\)-graded groups. In particular, this means that alternating links with homeomorphic branched double covers are homeomorphic. Note that this is not true for non-alternating links as non-mutant alternating links can posses homeomorphic branched double covers. The approach in the paper is combinatorial in nature. \textit{P. Ozsváth} and \textit{Z. Szabó} [Adv. Math. 194, No. 1, 1--33 (2005; Zbl 1076.57013)] proved that when \(L\) is a non-split alternating link, the isomorphism type of \(\widehat{HF}(\Sigma(L))\) is completely determined by the \(d\)-invariant of the lattice of integral flows, \(\mathcal{F}(G)\), where \(G\) is a Tait graph of \(L\) (the Tait graph should be constructed from a reduced alternating diagram of \(L\)). The author proves that two 2-edge-connected graphs \(G\) and \(G'\) are 2-isomorphic if and only if \(\mathcal{F}(G)\) and \(\mathcal{F}(G')\) have isomorphic \(d\)-invariants. The result then follows from the well-known correspondence between 2-isomorphism of Tait graphs and mutation of link diagrams.