Writhe polynomials and shell moves for virtual knots and links

TL;DR

This paper introduces shell moves as a new local move for virtual knots and proves they preserve the writhe polynomial, providing a classification method for 2-component virtual links using invariants.

Contribution

It establishes shell moves as an equivalence relation preserving the writhe polynomial and classifies 2-component virtual links via invariants.

Findings

Shell moves preserve the writhe polynomial.

Two virtual knots are related by shell moves iff they have the same writhe polynomial.

Classification of 2-component virtual links using invariants.

Abstract

The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a kind of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of shell moves. The second aim of this paper is to classify oriented $2$-component virtual links up to shell moves by using several invariants of virtual links.