Realizing the Tutte polynomial as a cut-and-paste K-theoretic invariant

TL;DR

This paper demonstrates how cut-and-paste K-theory can be used to realize the Tutte polynomial as a K-theoretic invariant for matroids, connecting combinatorial invariants with algebraic K-theory.

Contribution

It introduces a novel application of cut-and-paste K-theory to matroid theory, realizing the Tutte polynomial as a K-theoretic invariant via K-theory spectra.

Findings

Tutte polynomial can be realized as a K-theoretic invariant for matroids.

K-theory spectra induce the Tutte-Grothendieck invariant.

Connects combinatorial invariants with algebraic K-theory.

Abstract

Cut-and-paste $K$-theory is a new variant of higher algebraic $K$-theory that has proven to be useful in problems involving decompositions of combinatorial and geometric objects, e.g., scissors congruence of polyhedra and reconstruction problems in graph theory. In this paper, we show that this novel machinery can also be used in the study of matroids. Specifically, via the $K$-theory of categories with covering families developed by Bohmann-Gerhardt-Malkiewich-Merling-Zakharevich, we realize the Tutte polynomial map of Brylawski (also known as the universal Tutte-Grothendieck invariant for matroids) as the $K_0$-homomorphism induced by a map of $K$-theory spectra.