Discrete Laplacians, Ribbon Graphs, and Link Polynomials

Neal Stoltzfus (LSU)

Abstract: With the resurgence of the graph Laplacian in the study of classical link theoy (e.g. Silver-Williams arXiv:1809.06492), I will discuss an extension to ribbon graph involving three laplacians! These laplacians interact to contruct a three-parameter homology theory on the state space of a ribbon graph which gives an Euler characteristic formalism for the topological rank (Bollobas-Riordan-Tutte) polynomial. This can be specialzed to the Jones' polynomial of links, the chromatic polynomical of graphs and the the Termperley-Lieb algebra invariant.