Extending the Tutte and Bollob\'as-Riordan Polynomials to Rank 3 Weakly-Colored Stranded Graphs

Abstract

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.