Extending Tutte and Bollobás-Riordan Polynomials to Rank 3 Weakly-Colored Stranded Graphs

Remi C. Avohou, Joseph Ben Geloun, Mahouton N. Hounkonnou

Published 2013-01-09, updated 2017-08-24Version 3

The Bollob\'as-Riordan polynomial [Math. Ann. 323, 81 (2002)] is a universal polynomial invariant for ribbon graphs. We find an extension of this polynomial for a particular family of combinatorial objects called rank 3 weakly-colored stranded graphs. Stranded graphs arise in the study of tensor models for quantum gravity in physics, and generalize graphs and ribbon graphs. We present a seven-variable polynomial invariant of these graphs which obeys a contraction/deletion recursion relation similar to that of the Tutte and Bollob\'as-Riordan polynomials. However, it is defined on a much broader class of objects, and furthermore captures properties that are not encoded by the Tutte or Bollob\'as-Riordan polynomials of the underlying graphs.