# Up-down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II

## TL;DR

This paper introduces an up-down coloring method for virtual-link diagrams to establish lower bounds on the number of Reidemeister type II moves needed for diagram transformations, using quandle cocycle invariants.

## Contribution

It presents a novel coloring technique and applies quandle cocycle invariants to analyze the necessity of Reidemeister moves of type II in virtual-link diagram transformations.

## Key findings

- Up-down colorings provide lower bounds on Reidemeister II moves.
- Quandle cocycle invariants determine move necessity for trivial virtual-knots.
- Any virtual-knot diagram can be transformed with at least one Reidemeister II move.

## Abstract

We introduce an up-down coloring of a virtual-link diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two 2-component virtual-link diagrams. By using the notion of a quandle cocycle invariant, we determine the necessity of Reidemeister moves of type II for a pair of diagrams of the trivial virtual-knot. This implies that for any virtual-knot diagram $D$, there exists a diagram $D'$ representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.