# Knot Diagrams of Treewidth Two

## TL;DR

This paper investigates knot diagrams with underlying graphs of treewidth two, providing linear time algorithms to determine if they represent the unknot or unlink, and showing such diagrams can be simplified efficiently.

## Contribution

It introduces linear time algorithms for unknot and unlink recognition in treewidth two knot diagrams and proves their diagrams can be simplified with a bounded number of Reidemeister moves.

## Key findings

- Linear time algorithm for unknot recognition in treewidth two diagrams
- Linear time algorithm for unlink recognition in treewidth two diagrams
- Bound on the number of Reidemeister moves needed for simplification

## Abstract

In this paper, we study knot diagrams for which the underlying graph has treewidth two. We give a linear time algorithm for the following problem: given a knot diagram of treewidth two, does it represent the unknot? We also show that for a link diagram of treewidth two we can test in linear time if it represents the unlink. From the algorithm, it follows that a diagram of the unknot of treewidth 2 can always be reduced to the trivial diagram with at most $n$ (un)twist and (un)poke Reidemeister moves.