Locality and the Uniqueness of Quantum Invariants

TL;DR

This paper introduces a new framework for analyzing quantum invariants of tangles using state functions, classifies all such functions for key invariants, and compares these classifications to existing literature.

Contribution

It defines the concept of state functions for framed tangles, classifies all such functions for the Kauffman bracket and quantum SU(3) invariants, and relates findings to prior work.

Findings

Classification of all state functions for key quantum invariants.

Comparison of new classifications with existing literature.

Insights into the locality and uniqueness of quantum invariants.

Abstract

We introduce the notion of a "state function" for framed tangles in a disk. After choosing a finite set of states for each marked disk, a state function is a projection from the vector space spanned by all tangles to the vector space spanned by the states, that is local, and topologically invariant. Given the states for the Kauffman bracket, and the quantum $SU(3)$-invariant we classify all state functions, and then compare our results to the literature.