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4-dimensional Skein modules, Handle attachments, and Tangles

Gage Martin, Mary Stelow, Mira Wattal

TL;DR

We address the problem of describing how skein lasagna modules $S^\star(X; L)$ behave under fundamental 4-manifold handle attachments by establishing a general gluing formula and then deducing explicit $1$-, $2$-, and $3$-handle attachment rules for arbitrary functorial link theories $H^\star$. The core method is a gluing theorem expressing $S^\star(X; T_1\cup_P T_2) \cong S^\star(X_1; Y; T_1) \otimes_{S^\star(Y; P)} S^\star(X_2; Y; T_2)$, with a constructive map built from fillings and an inverse obtained by cutting along the gluing interface. The main contributions are: (i) a unified framework for $1$-, $2$-, $3$-handle formulae applicable to any $H^\star$ satisfying minimal monoidal and cobordism-extension properties; (ii) explicit reductions to known formulas recovering Manolescu-Neithalath, Ren-Willis, Chen, and related results in $H^\star$, KhR$_n$, Kh, and Lee theories; (iii) explicit algebra/module constructions controlling gluing and actions. This approach advances 4-manifold studies by providing a versatile toolkit for cutting-and-pasting skein-theoretic invariants and clarifies the structural reasons behind previously observed similarities among different 2-handle formulae.

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.

4-dimensional Skein modules, Handle attachments, and Tangles

TL;DR

We address the problem of describing how skein lasagna modules behave under fundamental 4-manifold handle attachments by establishing a general gluing formula and then deducing explicit -, -, and -handle attachment rules for arbitrary functorial link theories . The core method is a gluing theorem expressing , with a constructive map built from fillings and an inverse obtained by cutting along the gluing interface. The main contributions are: (i) a unified framework for -, -, -handle formulae applicable to any satisfying minimal monoidal and cobordism-extension properties; (ii) explicit reductions to known formulas recovering Manolescu-Neithalath, Ren-Willis, Chen, and related results in , KhR, Kh, and Lee theories; (iii) explicit algebra/module constructions controlling gluing and actions. This approach advances 4-manifold studies by providing a versatile toolkit for cutting-and-pasting skein-theoretic invariants and clarifies the structural reasons behind previously observed similarities among different 2-handle formulae.

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 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.
Paper Structure (6 sections, 7 theorems, 31 equations, 2 figures)

This paper contains 6 sections, 7 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. The skein lasagna module of a triple
  4. The gluing homomorphism
  5. Topological Picture
  6. Handle attachment formulae

Key Result

Theorem 1.1

If $X_1$ is obtained from $X$ by the addition of a 1-handle along $S^0 \times B^3$ and $T$ is the tangle obtained from $L$ by cutting along the cocore of the 1-handle, then $S^\star(X_1; L)$ is the 0-th Hochschild homology of a bimodule associated to the triple: $(X; S^0 \times B^3; T)$. The bimodul

Figures (2)

  • Figure 1: Multiplication in $\text{KhR}_\star$.
  • Figure 2: A pushout diagram of gluing homomorphisms.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 3.1: The skein lasagna module of a triple
  • Definition 3.3: The algebra associated to a 3-manifold
  • Remark 3.4
  • Example 3.5: Test case
  • Lemma 3.6
  • proof
  • ...and 11 more