Realizing the Tutte polynomial as a cut-and-paste K-theoretic invariant
Mauricio Gomez Lopez
TL;DR
The work builds a bridge between matroid theory and cut-and-paste K-theory by realizing the Tutte polynomial as a K0-map between categories of matroids endowed with Tutte-compatible coverings. It constructs two covering-family structures on Mat_+, identifies K0(Mat_cong) with the free abelian group on matroids and K0(Mat_tc) with the Tutte-Grothendieck ring, and shows the Tutte polynomial arises from the induced map on K-theory spectra. Moreover, it endows these groups with ring structures and connects the categorical construction to Brylawski's universal Tutte-Grothendieck invariant, suggesting a spectrum-level lift and potential E-infinity structure. The results provide a conceptual and computational framework for understanding matroid invariants through higher algebraic K-theory, with explicit universal properties and decomposition-theoretic interpretations. The approach promises further spectrum-level lifts and monoidal enhancements, linking combinatorial decompositions to modern K-theory machinery.
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.