Polynomial Invariants for Arbitrary Rank $D$ Weakly-Colored Stranded Graphs

TL;DR

This paper extends polynomial invariants for stranded graphs, generalizing previous work to higher dimensions and arbitrary rank D, introducing a modified Euler characteristic with multiple parameters.

Contribution

It introduces a modified Euler characteristic for rank D weakly-colored stranded graphs and extends polynomial invariants to arbitrary dimensions.

Findings

Defined a modified Euler characteristic with D-2 parameters for dimension D

Extended the polynomial invariant from rank 3 to arbitrary rank D

Preserved key properties of the polynomial invariant across dimensions

Abstract

Polynomials on stranded graphs are higher dimensional generalization of Tutte and Bollob\'as-Riordan polynomials [Math. Ann. 323 (2002), 81-96]. Here, we deepen the analysis of the polynomial invariant defined on rank 3 weakly-colored stranded graphs introduced in arXiv:1301.1987. We successfully find in dimension $D\geq3$ a modified Euler characteristic with $D-2$ parameters. Using this modified invariant, we extend the rank 3 weakly-colored graph polynomial, and its main properties, on rank 4 and then on arbitrary rank $D$ weakly-colored stranded graphs.