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

[摘要] Polynomials on stranded graphs are higher dimensional generalization of Tutte and Bollobás-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.

[关键词] Tutte polynomial;Bollob´as–Riordan polynomial;graph polynomial invariant;colored graph;Ribbon graph;Euler characteristic