Graph Complexity and Link Colorings

Abstract

The (torsion) complexity of a finite signed graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When G is d-periodic (i.e., G has a free Z d -action by graph automorphisms with finite quotient) the Mahler measure of its Laplacian polynomial is the growth rate of the complexity of finite quotients of G. Any 1-periodic plane graph G determines a link ∪ C with unknotted component C. In this case the Laplacian polynomial of G is related to the Alexander polynomial of the link. Lehmer's question, an open question about the roots of monic integral polynomials, is equivalent to a question about the complexity growth of signed 1-periodic graphs that are not necessarily embedded. MSC: 05C10, 37B10, 57M25, 82B20 MotivationConsider a locally finite graph without multiple edges or isolated vertices. Let p be an odd prime. We can attempt to form a "p-coloring" of the graph, assigning elements ("colors") of Z/p to the vertices in such a way that the color of any vertex is the average of the colors of the adjacent vertices. Obviously, the graph can be colored trivially or "monochromatically," that is, with any single color. But nontrivial p-colorings might exist as well. It is clear that the set of all p-colorings of the graph is a vector space under vertex-wise addition and scalar multiplication. Now consider a plane diagram of a knot or link. (As a knot is a link with only one component, henceforth we will use the more general term.) We can try to form a "p-coloring," assigning colors to the arcs in such a way that the color of any overcrossing arc is the average of the colors of its two undercrossing arcs (see Section 4). Again we can color trivially, but nontrivial p-colorings can exist too. The set of the all p-colorings of the diagram is a vector space under arc-wise addition and scalar multiplication.There is a strong relationship between p-colorings of plane graphs and p-colorings of link diagrams [27,33]. We review the relationship in Section 3. By substituting the continuous palette T = R/Z, which contains Z/p as a subgroup, and replacing the plane with a compact orientable surface S, we enter the world of algebraic dynamical systems [28]. When S is an annulus, we obtain a version of Lehmer's unanswered question about roots of monic integral polynomials, which we formulate in terms of signed graphs. * Both authors are grateful for the support of the Simons Foundation.