Topological quantum field theory and polynomial identities for graphs on the torus

We establish a relation between the trace evaluation in SO(3) topological quantum field theory and evaluations of a topological Tutte polynomial. As an application, a generalization of the Tutte golden identity is proved for graphs on the torus.

[1]  Sammie Bae,et al.  Graphs , 2020, Algorithms.

[2]  Calvin McPhail-Snyder,et al.  Planar diagrams for local invariants of graphs in surfaces , 2018, 1805.00575.

[3]  I. Agol,et al.  Structure of the flow and Yamada polynomials of cubic graphs , 2018, Proceedings of Symposia in Pure Mathematics.

[4]  Ramanujan Santharoubane,et al.  Asymptotics of quantum representations of surface groups , 2016, Annales scientifiques de l'École Normale Supérieure.

[5]  R. Mong,et al.  Topological defects on the lattice: I. The Ising model , 2016, 1601.07185.

[6]  I. Agol,et al.  Tutte relations, TQFT, and planarity of cubic graphs , 2015, 1512.07339.

[7]  Noah Snyder,et al.  Knot polynomial identities and quantum group coincidences , 2010, 1003.0022.

[8]  Vyacheslav Krushkal,et al.  Graphs, Links, and Duality on Surfaces , 2009, Combinatorics, Probability and Computing.

[9]  V. Rivasseau,et al.  Topological Graph Polynomials and Quantum Field Theory, Part I: Heat Kernel Theories , 2008, 0811.0186.

[10]  P. Fendley,et al.  Link invariants, the chromatic polynomial and the Potts model , 2008, 0806.3484.

[11]  K. Walker,et al.  On Picture (2+1)-TQFTs , 2008, 0806.1926.

[12]  P. Fendley,et al.  Tutte chromatic identities from the Temperley-Lieb algebra , 2007, 0711.0016.

[13]  N. Reshetikhin,et al.  Dimers on Surface Graphs and Spin Structures. II , 2007, 0704.0273.

[14]  Nicolai Reshetikhin,et al.  Dimers on Surface Graphs and Spin Structures. I , 2006, math-ph/0608070.

[15]  Oliver T. Dasbach,et al.  The Jones polynomial and graphs on surfaces , 2006, J. Comb. Theory B.

[16]  Béla Bollobás,et al.  A polynomial of graphs on surfaces , 2002 .

[17]  Seiya Negami,et al.  Diagonal flips in pseudo-triangulations on closed surfaces , 2001, Discret. Math..

[18]  N. Habegger,et al.  Topological Auantum Field Theories derived from the Kauffman bracket , 1995 .

[19]  J. Barrett,et al.  Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds , 1994 .

[20]  V. Jones.  Subfactors and Knots , 1991 .

[21]  V. Pasquier,et al.  Lattice derivation of modular invariant partition functions on the torus , 1987 .

[22]  J. Zuber,et al.  Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models , 1987 .

[23]  A. K. Dewdney,et al.  Wagner's theorem for Torus graphs , 1973, Discret. Math..

[24]  W. T. Tutte.  On chromatic polynomials and the golden ratio , 1970 .

[25]  V. Rivasseau,et al.  Topological graph polynomials and quantum field theory , 2010 .