Plumbing is a natural operation in Khovanov homology

Abstract

Given a connect sum of link diagrams, there is an isomorphism which decomposes unnormalized Khovanov chain groups for the product in terms of normalized chain groups for the factors; this isomorphism is straightforward to see on the level of chains. Similarly, any plumbing \(x*y\) of Kauffman states carries an isomorphism of the chain subgroups generated by the enhancements of \(x*y\), \(x\), \(y\): \[ \mathcal{C}_R(x*y)\to \left(\mathcal{C}_{R,p\to1}(x)\otimes \mathcal{C}_{R,p\to1}(y)\right)\oplus\left(\mathcal{C}_{R,p\to0}(x)\otimes \mathcal{C}_{R,p\to0}(y)\right). \] We apply this plumbing of chains to to prove that every homogeneously adequate state has enhancements \(X^\pm\) in distinct \(j\)-gradings whose \(A\)-traces (which we define) represent nonzero Khovanov homology classes over \(\mathbb{F}_2\), and that this is also true over \(\mathbb{Z}\) when all \(A\)-blocks' state surfaces are two-sided. We construct \(X^\pm\) explicitly.