### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 030, 23 pages      arXiv:1504.07165      https://doi.org/10.3842/SIGMA.2016.030

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

Remi Cocou Avohou
International Chair in Mathematical Physics and Applications, ICMPA-UNESCO Chair, 072BP50, Cotonou, Republic of Benin

Received June 26, 2015, in final form March 14, 2016; Published online March 22, 2016

Abstract
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.

Key words: Tutte polynomial; Bollobás-Riordan polynomial; graph polynomial invariant; colored graph; Ribbon graph; Euler characteristic.

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