Topology and Geometry Seminar

Speaker : Dr. Jun Ge, Sichuan Normal University

Time:   3:30-5:30pm Wednesday, Oct 19, 2016

Place:  X2511, Xipu campus SWJTU

Title: Kenyon's conjecture

        Abstract:  In graph theory, $\tau(G)$, the number of spanning trees of a graph $G$ (with $v(G)$ vertices and $e(G)$ edges), usually plays a central role. The tree entropy, $\frac{\log \tau(G)}{v(G)}$ has been extensively studied, especially for lattices. We call $\frac{\log \tau(G)}{e(G)}$ the edge spanning tree entropy of graph $G$. In 1996, Kenyon raised a conjecture on the upper bound of the edge-spanning tree density of a planar graph when studying tiling a rectangle with fewest squares. Interestingly, it can be translated into the language of knot theory, which is called the determinant density conjecture for knots and links. Both the graph theory version and the knot theory version of the conjecture are now named after Kenyon. In this talk, we will first sketch the combinatorial and geometrical backgrounds of Kenyon's conjecture. Then we will point out that a Stoimenow's result can give the best known upper bound for edge spanning tree entropy of planar graphs and determinant density of links. We will verify the Konyon's conjecture for links with at most 47 crossings, and equivalently, for planar graphs with at most 47 edges. Special classes such as torus links, Montesinos links, generalized theta graphs,

outerplanar graphs, 2D Archimedean lattices and Lave lattices, 3-regular graphs will also be verified. The forbidden structures in minimal counter-examples will also be mentioned.