Cornered skein lasagna theory

TL;DR

This paper extends skein lasagna theory to 4-manifolds with corners, develops a categorical framework for skein modules of trisected 4-manifolds, and introduces bicategories for surfaces, with gluing formulas for manifolds with boundary.

Contribution

It introduces new extensions of skein lasagna theory to manifolds with corners and develops a categorical framework for skein modules of 4-manifolds and surfaces.

Findings

Formulated gluing formulas for 4-manifolds with corners.

Developed a categorical presentation of skein lasagna modules.

Extended the theory to 2-dimensional bicategories for surfaces.

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.