A structural approach to tree decompositions of knots and spatial graphs

Lunel, Corentin; de Mesmay, Arnaud

doi:10.2140/agt.2026.26.2015

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ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

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A structural approach to tree decompositions of knots and spatial graphs

Corentin Lunel and Arnaud de Mesmay

Algebraic & Geometric Topology 26 (2026) 2015–2047

DOI: 10.2140/agt.2026.26.2015

Abstract

Knots are commonly represented and manipulated via diagrams, which are decorated planar graphs. The treewidth of a graph is a parameter quantifying how close a graph is to a tree. When a knot diagram has low treewidth, parameterized graph algorithms can be leveraged to ensure the fast computation of many invariants and properties of the knot. It was recently proved that there exist knots which do not admit any diagram of low treewidth, and the proof relied on intricate low-dimensional topology techniques. In this work, we initiate a thorough investigation of tree decompositions of knot diagrams (or more generally, diagrams of spatial graphs) using ideas from structural graph theory. We define an obstruction on spatial embeddings that forbids low treewidth diagrams, and we prove that it is optimal with respect to a related width invariant. We then show the existence of this obstruction for knots of high representativity, which include, for example, torus knots, providing a new and self-contained proof that those do not admit diagrams of low treewidth. This last step is inspired by a result of Pardon on knot distortion.

Keywords

knots, spatial graphs, tree decompositions, tangles, representativity

Mathematical Subject Classification

Primary: 05C10

Secondary: 57K10, 57M15

References

Publication

Received: 21 December 2023

Revised: 18 December 2024

Accepted: 26 May 2025

Published: 22 June 2026

Authors

Corentin Lunel
Inria Université Côte d’Azur Sophia Antipolis France

Arnaud de Mesmay
LIGM, CNRS Université Gustave Eiffel Marne la Vallée France

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