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In this note we provide a higher-dimensional analogue of Tutte's celebratedtheorem on colorings and flows of graphs, by showing that the theory ofarithmetic Tutte polynomials and quasi-polynomials encompasses invariantsdefined for CW complexes by Beck-Breuer-Godkin-Martin and Duval-Klivans-Martin.Furthermore, we answer a question by Bajo-Burdick-Chmutov, concerning themodified Tutte-Krushkal-Renhardy polynomials defined by these authors: to thisend, we prove that the product of two arithmetic multiplicity functions on amatroid is again an arithmetic multiplicity function.

Colorings and flows on CW complexes, Tutte quasi-polynomials and arithmetic matroids