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

Abstract

The Bollobás-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 Bollobás-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 Bollobás-Riordan polynomials.

41 pages, 32 figures; improved version (published), some statements corrected

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• Random tensor models in the large N limit: Uncoloring the colored tensor models
• The 1/N expansion of colored tensor models
• Lost in Translation: Topological Singularities in Group Field Theory
• Combinatorial Hopf algebra for the Ben Geloun-Rivasseau tensor field theory
• Renormalization and Hopf Algebraic Structure of the 5-Dimensional Quartic Tensor Field Theory
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Cited by in corpus (4)

• The Tensor Track, III
• Parametric Representation of Rank d Tensorial Group Field Theory: Abelian Models with Kinetic Term ${\sum }_s|p_s|+\mu$
• Polynomial Invariants for Arbitrary Rank $D$ Weakly-Colored Stranded Graphs
• Universality for polynomial invariants on ribbon graphs with half-ribbons