Higher-Dimensional Algebra IV: 2-Tangles

TL;DR

This paper extends the algebraic description of knots and links to 4-dimensional surfaces using 2-tangles, establishing a universal property that leads to invariants of 2-tangles in four dimensions.

Contribution

It proves that the 2-category of 2-tangles in 4D is the free semistrict braided monoidal 2-category with duals on one unframed self-dual object, enabling new invariants.

Findings

Established the universal property of the 2-category of 2-tangles in 4D.

Connected the algebraic structure to invariants of 2-tangles.

Extended knot theory concepts to higher dimensions.

Abstract

Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R^4 can be described as certain 2-morphisms in the 2-category of `2-tangles in 4 dimensions'. Using the work of Carter, Rieger and Saito, we prove that this 2-category is the `free semistrict braided monoidal 2-category with duals on one unframed self-dual object'. By this universal property, any unframed self-dual object in a braided monoidal 2-category with duals determines an invariant of 2-tangles in 4 dimensions.