On virtual link invariants

TL;DR

This paper characterizes virtual link invariants that are partition functions of vertex models, demonstrating their structure as affine varieties using invariant theory and algebraic geometry techniques.

Contribution

It provides a complete characterization of such invariants, linking virtual link invariants to algebraic varieties via invariant theory.

Findings

Virtual link invariants form an affine variety for fixed states.

The characterization uses the first and second fundamental theorems of invariant theory.

Applicable to both real and complex cases.

Abstract

Virtual links were introduced by Kauffman in 1999. We characterize the virtual link invariants that are partition functions of vertex models (as considered by de la Harpe and Jones), both in the real and in the complex case. We show that for any fixed number of states, these invariants form an affine variety. Basic techniques are the first and second fundamental theorem of invariant theory for the orthogonal group (in the sense of Weyl) and some related methods from algebraic geometry.