Writhes and $2k$-moves for virtual knots

TL;DR

This paper investigates how $2k$-moves affect virtual knots, establishing congruence relations between their writhes and providing conditions for equivalence based on odd writhes modulo $2k$.

Contribution

It introduces new invariants related to $2k$-moves and characterizes when virtual knots share the same odd writhes modulo $2k$, advancing understanding of virtual knot transformations.

Findings

$n$-writhes are congruent modulo $k$ after $2k$-moves

Odd writhes are congruent modulo $2k$ after $2k$-moves

Provides necessary and sufficient conditions for odd writhes congruence

Abstract

A $2k$-move is a local deformation adding or removing $2k$ half-twists. We show that if two virtual knots are related by a finite sequence of $2k$-moves, then their $n$-writhes are congruent modulo $k$ for any nonzero integer $n$, and their odd writhes are congruent modulo $2k$. Moreover, we give a necessary and sufficient condition for two virtual knots to have the same congruence class of odd writhes modulo $2k$.