Warping labeling for twisted knots and twisted virtual braids

TL;DR

This paper introduces the warping degree and labeling scheme for twisted knots and virtual braids, creating invariants that help distinguish these complex structures in knot theory.

Contribution

It develops a new invariant based on warping labeling for twisted knots and virtual braids, extending existing concepts in knot theory.

Findings

Warping labeling can be extended to twisted virtual braids.

A function invariant under all R-moves except R2 is constructed.

Invariants for twisted virtual braids are developed using $\\mathbb{Z}_2$ labeling.

Abstract

In this paper, we introduce the concept of the warping degree for twisted knots, construct an invariant for them, and utilize it to establish a labeling scheme for these knots, known as ``warping labeling". We have identified that a warping labeling can be extended to twisted virtual braids, enabling the creation of a function that remains invariant under all R-moves except the R2 move. By limiting the labeling set to $\mathbb{Z}_2$, we can develop invariants for twisted virtual braids.