Cornered skein lasagna theory

TL;DR

This work extends the skein lasagna invariant to 4-manifolds with corners, establishing a TQFT-like framework with gluing formulas that compose cornered invariants into global ones, and it applies these methods to trisected 4-manifolds. It introduces cornered skein lasagna bimodules and proves tensor-product gluing theorems, including a self-gluing Hochschild interpretation that connects cornered data to closed invariants. The authors further develop a dimension-two extension using bicategories of closed surfaces and 3-manifolds, constructing modules over a surface bicategory and proving gluing results for 3-manifolds. Collectively, the paper provides computational tools and conceptual structure for calculating and understanding skein lasagna invariants across dimensions via cut-and-paste techniques.

Abstract

We extend the skein lasagna theory of Morrison-Walker-Wedrich to 4-manifolds with corners and formulate gluing formulas for 4-manifolds with boundary and, more generally, with corners. As an application, we develop a categorical framework for a presentation of the skein lasagna module of trisected closed 4-manifolds. Further, we extend the theory to dimension two by introducing bicategories for closed oriented surfaces and proving a gluing formula for the categories associated with 3-manifolds with boundary.

Figures (14)

• Figure 1: The $4$-manifold $Y \times I$ with the link $L = (-T' \times \{-1\}) \cup (P \times I) \cup (T \times \{1\})$ on its boundary.
• Figure 2: The $4$-manifold $X$ with the link $-T_1 \cup T_2$ on its boundary.
• Figure 3: Two schematics for gluing two cornered $4$-manifolds along part of their boundary. The left hand side is useful for a big picture view of what pieces are being glued, while the right hand side (in which all pieces have increased in dimension) is useful for seeing where the points, tangles, and surfaces (which will be relevant in \ref{['def:cornered-skein-lasagna-bimodule']}) live. Here, the gray schematically represents lasagna fillings of $(X_1, T' \cup T")$ and $(X_2,T" \cup T"')$.
• Figure 4: A schematic representing the equivalence relation $(a \cdot \beta) \otimes c \sim_1 a \otimes (\beta \cdot c)$. In words, this relation captures the idea that acting by a morphism on the "left" side of $X_1$ or "right" side of $X_2$ gives the same result once the two pieces are glued along their common boundary. (Orientations have been omitted from this schematic for the sake of clarity.)
• Figure 5: An isotopy of an input ball.

Theorems & Definitions (37)

• Theorem
• Remark 1.1
• Definition 2.1: MWW2
• Definition 2.2: MWW2
• Definition 2.3
• Remark 2.1
• Remark 2.2
• Definition 3.1
• Remark 3.1
• Remark 3.2