# Invariants of Welded Virtual Knots Via Crossed Module Invariants of Knotted Surfaces

## TL;DR

This paper introduces a new invariant for welded virtual knots derived from finite crossed modules, using invariants of associated ribbon knotted surfaces, and demonstrates their non-triviality through explicit examples.

## Contribution

It develops a novel method to define invariants of welded virtual knots and graphs via crossed modules and knotted surfaces, expanding the tools for knot theory analysis.

## Key findings

- Invariants are non-trivial, confirmed by explicit calculations.
- Extended invariants to welded virtual graphs.
- Established a connection between crossed modules and knotted surface invariants.

## Abstract

We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non trivial by calculating explicit examples. We define welded virtual graphs and consider invariants of them defined in a similar way.