Knot lattice homology in \(L\)-spaces (Q2788556)

Lattice homology was introduced by \textit{A. Némethi} [Publ. Res. Inst. Math. Sci. 44, No. 2, 507--543 (2008; Zbl 1149.14029)] for negative definite plumbing trees. For such a graph \(G\), the lattice homology \(\mathbb{HF}^-(G)\) is a finitely generated \(\mathbb{Z}_2[U]\)-module. As usual, the graph \(G\) gives rise to a surgery presentation of the plumbed \(3\)-manifold \(Y_G\), and moreover a simply connected \(4\)-manifold \(X_G\) whose boundary is \(Y_G\).NEWLINENEWLINEFor all negative definite plumbing trees, it is expected that lattice homology is isomorphic to Heegaard Floer homology HF\(^-(Y_G)\). In fact, this is verified for certain plumbings.NEWLINENEWLINEIn the present paper under review, the authors show that the filtered chain complexes in the two homology theories are filtered chain homotopic for plumbing trees/forests under a certain condition. (Indeed, the corresponding \(3\)-manifolds are \(L\)-spaces.) This extends the class of plumbing graphs for which two homology theories coincide.NEWLINENEWLINEAs the simplest case, it is shown that the knot lattice homology of a knot in the \(3\)-sphere is equal to the knot Floer homology for the same knot. (In this situation, the involved knots are only iterated torus knots or their connected sums.)