Coding of Knots by T-Graphs

Knots and Graphs

A classical matrix-tree theorem expresses the determinant of some matrix constructed from a graph (principal minor of the Laplacian) as a sum over all spanning trees of the graph. There are generalizations of this theorem to hypergraphs or simplicial complexes [MV, DKM]. Some version of this theorem provides a formula for the first non-zero coefficient of the Conway polynomial of a (virtual) li...

Mutant knots and intersection graphs

We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. Conversely, if we have a weight system depending only on the intersection graphs of chord diagrams, then the composition of such a weight system with the Kontsevich invariant determines a kn...

Knots in Graphs in Subsets

The probability that an embedding of a graph in Z 3 is knotted is investigated. For any given graph (embeddable in Z 3) without cut edges, it is shown that this probability approaches 1 at an exponential rate as the number of edges in the embedding goes to innnity. Furthermore, at least for a subset of these graphs, the rate at which the probability approaches 1 does not depend on the particula...

Spatial graphs with local knots

It is shown that for any locally knotted edge of a 3-connected graph in S^3, there is a ball that contains all of the local knots of that edge and it is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in S^3. Response to Reviewers: See attachment. SPATIAL GRAPHS WITH LOCAL KNOTS ERICA FLAPAN, BLAKE MELLOR, A...

Graphs on Surfaces - Dualities, Polynomials, and Knots