A state sum invariant of tangles in surfaces

TL;DR

This paper introduces a new state sum invariant for tangles in surfaces, extending knot theory tools to surfaces with holes, with potential applications to framed links and categorification.

Contribution

It defines the $u$-invariant based on regions in tangles on surfaces, providing a novel regular isotopy invariant with algebraic and categorical properties.

Findings

The $u$-invariant is valued in $\\mathbb{Z}[u]$ with $u$ a primitive fifth root of unity.

The invariant is compatible with tangle composition via matrix multiplication.

It can be specialized to invariants of framed links in 3D space.

Abstract

In this paper we define a new state sum based on the regions defined by tangles on a surface which is an oriented closed surface with a finite number of open holes drilled. From this state sum we obtain an invariant of regular isotopy for the tangles named $u$-invariant. The values of the $u$-invariant are in $\mathbb{Z}[u]$, where $u$ is a primitive fifth root of the unity. Various basic properties of the invariant are proved and discussed. It can be specialized to invariants of framed links in $\mathbb{R}^3$. Its categoric aspect is emphasized: composition of tangles in surfaces correspond to matrix multiplication. The values of the $u$-invariant conjugate under taking the mirror of the tangle.