# Mutant knots and intersection graphs

## TL;DR

This paper investigates the relationship between mutant knots, intersection graphs, and weight systems, showing that invariants insensitive to mutation depend only on intersection graphs, with implications for Lie algebra weight systems.

## Contribution

It establishes a connection between knot invariants, mutant knots, and intersection graphs, highlighting when invariants depend solely on intersection graphs.

## Key findings

- Invariants not distinguishing mutants depend on intersection graphs.
- The converse that such invariants depend on diagrams is well known.
- Discusses links to Lie algebra weight systems.

## Abstract

We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. The converse statement is easy and well known. We discuss relationship between our results and certain Lie algebra weight systems.