Magnitude, homology, and the Whitney twist

TL;DR

This paper explores the relationship between magnitude, a graph invariant, and magnitude homology, providing a homological perspective on the invariance of magnitude under Whitney twists, and extending known results.

Contribution

It introduces a homological approach to understanding magnitude invariance under Whitney twists, broadening the class of gluings where this invariance holds.

Findings

Homological account of magnitude invariance under Whitney twists

Extension of invariance results to wider class of gluings

First theorem about magnitude proved using magnitude homology

Abstract

Magnitude is a numerical invariant of metric spaces and graphs, analogous, in a precise sense, to Euler characteristic. Magnitude homology is an algebraic invariant constructed to categorify magnitude. Among the important features of the magnitude of graphs is its behaviour with respect to an operation known as the Whitney twist. We give a homological account of magnitude's invariance under Whitney twists, extending the previously known result to encompass a substantially wider class of gluings. As well as providing a new tool for the computation of magnitudes, this is the first new theorem about magnitude to be proved using magnitude homology.