The multivariate signed Bollobas-Riordan polynomial

TL;DR

This paper introduces a multivariate signed Bollobás-Riordan polynomial, extending previous work, and demonstrates its invariance under partial duality, linking it to the multivariate Tutte polynomial.

Contribution

It generalizes the signed Bollobás-Riordan polynomial to a multivariate form and proves its invariance under partial duality, connecting it to the multivariate Tutte polynomial.

Findings

Proves invariance of the multivariate signed polynomial under partial duality.

Shows the duality transformation of the multivariate Tutte polynomial follows from this invariance.

Extends previous polynomial invariants to a multivariate setting.

Abstract

We generalise the signed Bollobas-Riordan polynomial of S. Chmutov and I. Pak [Moscow Math. J. 7 (2007), no. 3, 409-418] to a multivariate signed polynomial Z and study its properties. We prove the invariance of Z under the recently defined partial duality of S. Chmutov [J. Combinatorial Theory, Ser. B, 99 (3): 617-638, 2009] and show that the duality transformation of the multivariate Tutte polynomial is a direct consequence of it.