Skein lasagna modules and handle decompositions

TL;DR

This paper introduces a method to compute the skein lasagna module, an invariant extending Khovanov-Rozansky homology to four-manifolds, and explores its properties through examples and handle decompositions.

Contribution

It provides a general procedure to express the skein lasagna module via handle decompositions, enabling calculations and analysis of its dimensional properties.

Findings

The skein lasagna module can be expressed using handle decompositions.

Examples show the module can be locally infinite dimensional.

The method facilitates computations of the invariant for specific four-manifolds.

Abstract

The skein lasagna module is an extension of Khovanov-Rozansky homology to the setting of a four-manifold and a link in its boundary. This invariant plays the role of the Hilbert space of an associated fully extended (4+epsilon)-dimensional TQFT. We give a general procedure for expressing the skein lasagna module in terms of a handle decomposition for the four-manifold. We use this to calculate a few examples, and show that the skein lasagna module can sometimes be locally infinite dimensional.