Categorifying the magnitude of a graph

@article{Hepworth2015CategorifyingTM,
  title={Categorifying the magnitude of a graph},
  author={Richard A. Hepworth and Simon Willerton},
  journal={arXiv: Combinatorics},
  year={2015}
}
The magnitude of a graph can be thought of as an integer power series associated to a graph; Leinster introduced it using his idea of magnitude of a metric space. Here we introduce a bigraded homology theory for graphs which has the magnitude as its graded Euler characteristic. This is a categorification of the magnitude in the same spirit as Khovanov homology is a categorification of the Jones polynomial. We show how properties of magnitude proved by Leinster categorify to properties such as a… 

Figures and Tables from this paper

Torsion in the magnitude homology of graphs
Magnitude homology is a bigraded homology theory for finite graphs defined by Hepworth and Willerton, categorifying the power series invariant known as magnitude which was introduced by Leinster. We
The magnitude of a graph
  • T. Leinster
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2017
Abstract The magnitude of a graph is one of a family of cardinality-like invariants extending across mathematics; it is a cousin to Euler characteristic and geometric measure. Among its
A categorification of the Vandermonde determinant
  • Alex Chandler
  • Mathematics
    Journal of Knot Theory and Its Ramifications
  • 2022
In the spirit of Bar-Natan’s construction of Khovanov homology, we give a categorification of the Vandermonde determinant. Given a sequence of positive integers [Formula: see text], we construct a
Graph magnitude homology via algebraic Morse theory
We compute magnitude homology of various graphs using algebraic Morse theory. Specifically, we (1) give an alternative proof that trees are diagonal, (2) identify a new class of diagonal graphs, (3)
Magnitude homology of graphs and discrete Morse theory on Asao-Izumihara complexes
Recently, Asao and Izumihara introduced CW-complexes whose homology groups are isomorphic to direct summands of the graph magnitude homology group. In this paper, we study the homotopy type of the
Magnitude meets persistence. Homology theories for filtered simplicial sets
The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely
Girth, magnitude homology, and phase transition of diagonality
This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. Magnitude and magnitude homology are formulations of the Euler
Magnitude homology of enriched categories and metric spaces
Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,\infty)$-enriched categories. We show that in many cases magnitude can be categorified to a
Magnitude cohomology
Magnitude homology was introduced by Hepworth and Willerton in the case of graphs, and was later extended by Leinster and Shulman to metric spaces and enriched categories. Here we introduce the dual
Geometric approach to graph magnitude homology
In this paper, we introduce a new method to compute magnitude homology of general graphs. To each direct sum component of magnitude chain complexes, we assign a pair of simplicial complexes whose
...
1
2
3
...

References

SHOWING 1-10 OF 26 REFERENCES
The magnitude of a graph
  • T. Leinster
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2017
Abstract The magnitude of a graph is one of a family of cardinality-like invariants extending across mathematics; it is a cousin to Euler characteristic and geometric measure. Among its
A categorification for the chromatic polynomial
For each graph we construct graded cohomology groups whose graded Euler characteristic is the chromatic polynomial of the graph. We show the cohomology groups satisfy a long exact sequence which
Categorification of the Dichromatic Polynomial for Graphs
For each graph and each positive integer n, we define a chain complex whose graded Euler characteristic is equal to an appropriate n-specialization of the dichromatic polynomial. This also gives a
The magnitude of metric spaces
Magnitude is a real-valued invariant of metric spaces, analogous to the Euler characteristic of topological spaces and the cardinality of sets. The definition of magnitude is a special case of a
The Euler characteristic of a category
The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality
A categorification for the Tutte polynomial
For each graph, we construct a bigraded chain complex whose graded Euler characteristic is a version of the Tutte polynomial. This work is motivated by earlier work of Khovanov, Helme-Guizon and
Positive definite metric spaces
Magnitude is a numerical invariant of finite metric spaces, recently introduced by Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic of
A categorification of the Jones polynomial
Author(s): Khovanov, Mikhail | Abstract: We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.
An introduction to Heegaard Floer homology
Contents 1. Introduction 1 2. Heegaard decompositions and diagrams 2 3. Morse functions and Heegaard diagrams 7 4. Symmetric products and totally real tori 8 5. Disks in symmetric products 10 6. Spin
Magnitude, Diversity, Capacities, and Dimensions of Metric Spaces
Magnitude is a numerical invariant of metric spaces introduced by Leinster, motivated by considerations from category theory. This paper extends the original definition for finite spaces to compact
...
1
2
3
...