Categorification of Chromatic, Dichromatic and Penrose Polynomials
Louis H Kauffman
TL;DR
This work develops a unified framework to categorify chromatic, dichromatic, and Penrose polynomials by interpreting their state-sum definitions as graded Euler characteristics of carefully constructed chain complexes, inspired by Whitney deletion-contraction. Using enhanced states and algebras like $M=\mathbb{Z}[x]/(x^2)$, it constructs chromatic and (bi)graded dichromatic homologies that recover $C_G(n)$ and $Z_G(v,\lambda)$ as $\chi(H^{*})$, while introducing impropriety polynomials $C_G^{i}(\lambda)$ tied to graded Euler characteristics. It further links to statistical mechanics by expressing the Potts partition function as a graded Euler characteristic of the dichromatic homology with $p=-e^{-E/(kT)}$ and $\lambda=n$, thereby associating energy levels with homological degrees. An alternative color-algebra approach $A(n)$ shows that $C_G(n)=\chi(H^{*}(G,A(n)))$ with potential vanishing higher homology in favorable models, illustrating multiple categorifications of chromatic evaluations. The results provide a foundation for calculations, physics interpretations, and broader connections to coloring problems via homological invariants, with future work aimed at detailed computations and deeper links to statistical physics.
Abstract
This paper discusses ways to categorify chromatic, dichromatic and Penrose polynomials, including categorifications of integer evaluations of chromatic polynomials. We show that with an appropriate choice of variables the coefficients of the Potts partition function at different energy levels are given by Euler characteristics of appropriate parts of a bigraded homology theory associated with the model. In the case of the dichromatic polynomial for graphs, we show that the two variable polynomial can be seen as a sum of powers of one variable multiplied by coefficients that are "impropriety" coloring polynomials for the underlying graph. An impropriety polynomial $C_{G}^{i}(n)$ counts the number of colorings in $n$ colors of the graph that are not proper at a given number $i$ of edges in the graph. The last section of the paper categorifies coloring evaluations rather than coloring polynomials. We then obtain a range of possible chain complexes and homology theories such that the chromatic evaluation is equal to the Euler characteristic of the homology. The freedom of choice in making such chain complexes is related to possible associative algebra structures on the set of colors.