# Plumbing is a natural operation in Khovanov homology

## TL;DR

This paper explores the natural operation of plumbing in Khovanov homology, establishing isomorphisms for chain groups and demonstrating the existence of specific enhancements for homogeneously adequate states, with implications over different coefficient rings.

## Contribution

It introduces a chain-level plumbing isomorphism in Khovanov homology and constructs explicit enhancements for homogeneously adequate states, advancing understanding of their homological properties.

## Key findings

- Chain groups decompose via plumbing for link diagrams.
- Homogeneously adequate states have specific enhancements with nonzero homology classes.
- Results hold over both  and  coefficients under certain conditions.

## 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.