Intersection graph and writhe polynomial

TL;DR

This paper establishes a fundamental equivalence between virtual knots' intersection graphs and their writhe polynomials, revealing a new way to classify and distinguish virtual knots.

Contribution

It proves that two virtual knots are equivalent if and only if their intersection graphs and writhe polynomials match, linking graph theory and knot invariants.

Findings

Intersection graphs characterize virtual knots.

Writhe polynomial is a complete invariant for intersection graphs.

Equivalence of virtual knots corresponds to identical intersection graphs and writhe polynomials.

Abstract

We prove that two virtual knots have equivalent intersection graphs if and only if they have the same writhe polynomial.