The lattice cohomology of \(S_{-d}^3(K)\) (Q2918499)

Consider \(\{K_{i}\}_{i=1}^{\nu}\subset S^{3}\) a finite collection of algebraic knots and define their connected sum \(K=K_{1}\#\dots\# K_{\nu}\subset S^{3}\). The articles computes the Lattice Cohomology \(\mathbb{H}^{\ast}(M)\) of the 3-manifold \(M=S^{3}_{-d}(K)\) obtained by performing a \(-d\)-surgery on \(K\subset S^{3}\). The basics of Lattice Cohomology were introduced by the first author in [Sel. Math., New Ser. 5, No.1, 161--179 (1999; Zbl 0936.32015), Publ. Res. Inst. Math. Sci. 44, No. 2, 507--543 (2008; Zbl 1149.14029)] where it was defined for negative plumbed 3-manifolds.NEWLINENEWLINEThe present article focuses on the combinatorial aspects of \(\mathbb{H}^{\ast}(M)\) and also on the connections with Seiberg-Witten and Heegard-Floer theories. The author conjectures the Heegard-Floer Homology of \(M\) to be contained in \(\mathbb{H}^{\ast}(M)\). This is based on the following cases;NEWLINENEWLINE (i) \(\nu=1\), \(\mathbb{H}^{0}(M)=HF^{+}_{even}(M)\), \(\mathbb{H}^{q}(M)=HF^{+}_{odd}(M)\) (\(q\geq 1\)).NEWLINENEWLINE\noindent (ii) \(\nu=2\), \(\mathbb{H}^{0}(M)=HF^{+}_{even}(-M)\), \(\mathbb{H}^{1}(M)=HF^{+}_{odd}(-M)\) and \(\mathbb{H}^{q}(M)=0\) (\(q\geq 1\)).NEWLINENEWLINE\noindent In what concerns the Seiberg-Witten theory, the article proves that the Euler charcateristic of the Lattice Cohomology is equal to the normalized Seiberg-Witten invariant of \(M\). In order to compute the groups \(\mathbb{H}^{\ast}(M)\), a negative definite graph is defined to be rational if it is the resolution graph of a rational singularity. A rational graph is endowed with a set of bad vertices. Since the Lattice Cohomology of rational graphs is trivial, the strategy is to reduce any non-rational graph into a rational graph by surgery along some of its bad vertices.NEWLINENEWLINEFor the entire collection see [Zbl 1242.11004].