Classification of Flat Virtual Pure Tangles

TL;DR

This paper provides a complete classification of flat virtual pure tangles, enhancing the understanding of virtual knot theory and offering a new invariant for virtual pure tangles and braids.

Contribution

It introduces a complete classification of flat virtual pure tangles, which was not previously established, and proposes their use as invariants in virtual knot theory.

Findings

Complete classification of flat virtual pure tangles.

Introduction of a new invariant for virtual pure tangles and braids.

Potential applications in topological and quantum algebra contexts.

Abstract

Virtual knot theory, introduced by Kauffman, is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation of Etingof and Kazhdan's theory of quantization of Lie bi-algebras. Classical knots inject into virtual knots, and flat virtual knots is the quotient of virtual knots which equates the real positive and negative crossings, and in this sense is complementary to classical knot theory within virtual knot theory. We completely classify flat virtual tangles with no closed components (pure tangles). This classification can be used as an invariant on virtual pure tangles and virtual braids.