On the existence of real R-matrices for virtual link invariants

TL;DR

This paper characterizes which virtual link invariants can be represented by real R-matrices, using weak reflection positivity and advanced algebraic techniques, thereby linking topological invariants with algebraic structures.

Contribution

It provides a complete characterization of virtual link invariants realizable via real R-matrices through the concept of weak reflection positivity.

Findings

Virtual link invariants correspond to weakly reflection positive functions.

The characterization uses invariant theory, Brauer algebra decomposition, and orbit inequalities.

Establishes a connection between topological invariants and algebraic conditions.

Abstract

We characterize the virtual link invariants that can be described as partition function of a real-valued R-matrix, by being weakly reflection positive. Weak reflection positivity is defined in terms of joining virtual link diagrams, which is a specialization of joining virtual link diagram tangles. Basic techniques are the first fundamental theorem of invariant theory, the Hanlon-Wales theorem on the decomposition of Brauer algebras, and the Procesi-Schwarz theorem on inequalities for closed orbits.