Knot Graphs

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph which is reducible by some finite sequence of these moves, to a graph with no edges is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes

Intrinsic Linking and Knotting in Virtual Spatial Graphs

We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and non-terminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the {\it virtual unknotting number} of a knot, and show that any knot with non-trivial Jones polynomial has virtual unknotting number at least 2.Comment: 13 pages, 13 figure

Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs

We define 2-decompositions of ribbon graphs, which generalise 2-sums and tensor products of graphs. We give formulae for the Bollobas-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case. This study was initially motivated from knot theory, and we include an application of our formulae to mutation in knot diagrams.Comment: Version 2 has minor changes. To appear in Annals of Combinatoric

Knot graphs and Gromov hyperbolicity

We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that overwhelmingly, the knot graphs are not Gromov hyperbolic, with the exception of a particular family of quotient knot graphs. We also investigate the property of homogeneity, and prove that the concordance knot graph is homogeneous. Finally, we prove that that for any $n$, there exists a knot $K$ such that the ball of radius $n$ in the Gordian graph centered at $K$ contains no connected sum of torus knots