# The Wriggle polynomial for virtual tangles

## TL;DR

This paper extends the Wriggle polynomial to virtual tangles, demonstrating it as a Vassiliev invariant of order one for virtual knots and exploring properties related to tangle connected sums.

## Contribution

The paper introduces a generalized Wriggle polynomial for virtual tangles, establishing it as a Vassiliev invariant of order one and analyzing its properties.

## Key findings

- The generalized polynomial is a Vassiliev invariant of order one.
- It applies to virtual tangles considering self-crossings.
- Properties related to tangle connected sums are studied.

## Abstract

We generalize the Wriggle polynomial, first introduced by L. Folwaczny and L. Kauffman, to the case of virtual tangles. This generalization naturally arises when considering the self-crossings of the tangle. We prove that the generalization (and, by corollary, the original polynomial) are Vassiliev invariants of order one for virtual knots, and study some simple properties related to the connected sum of tangles.