4-dimensional Skein modules, Handle attachments, and Tangles

TL;DR

This paper develops general formulas for handle attachments in skein modules of 4-manifolds, extending previous work and providing a detailed description of gluing homomorphisms in functorial link theories.

Contribution

It introduces comprehensive formulas for handle attachments in skein modules, generalizing prior results and describing the gluing homomorphism with a new boundary-structured skein module variation.

Findings

Formulas for 1-, 2-, and 3-handle attachments in skein modules.

A complete description of the gluing homomorphism on skein modules.

Introduction of a skein module variation with distinguished boundary 3-manifolds.

Abstract

Skein lasagna modules are a recent tool developed for the study of 4-manifolds. We provide general formula for 1-, 2-, and 3-handle attachments for skein modules defined with any functorial link theory in $S^3 \times I$ generalizing existing formula of Chen, Manolescu-Neithalath, Manolescu-Walker-Wedrich, and Ren-Willis. These formula are derived from a complete description of the gluing homomorphism on skein modules. For this description, we introduce a variation of these skein modules in the presence of distinguished 3-manifolds in the boundary. A similar construction was recently introduced independently by Blackwell-Krushkal-Luo.