Erica Flapan

INTRINSICALLY n-LINKED GRAPHS

For every natural number n, we exhibit a graph with the property that every embedding of it in ℝ3 contains a non-split n-component link. Furthermore, we prove that our graph is minor minimal in the sense that every minor of it has an embedding in ℝ3 that contains no non-split n-component link.

Intrinsically Triple Linked Complete Graphs

Abstract We prove that every embedding of K 10 in R 3 contains a non-split link of three components. We also exhibit an embedding of K 9 with no such link of three components.

Rigidity of Graph Symmetries in the 3-Sphere

In this paper we show that, for any 3-connected graph G, if an automorphism of G is realizable by a homeomorphism of (S3, G) then there is an embedding of G such that that automorphism is induced by a finite order homeomorphism of (S3, G). We then use this result to characterize which automorphisms of an arbitrary complete graph can be induced by a homeomorphism of S3 for some embedding of the graph.

Topological symmetry groups of graphs embedded in the 3-sphere

The topological symmetry group of a graph embedded in the 3-sphere is the group consisting of those automorphisms of the graph which are induced by some homeomorphism of the ambient space. We prove strong restrictions on the groups that can occur as the topo- logical symmetry group of some embedded graph. In addition, we characterize the orientation preserving topological symmetry groups of embedded 3-connected graphs in the 3-sphere.

A topological characterization of knots and links arising from site-specific recombination

We develop a topological model of knots and links arising from a single (or multiple processive) round(s) of recombination starting with an unknot, unlink, or (2, m)-torus knot or link substrate. We show that all knotted or linked products fall into a single family, and prove that the size of this family grows linearly with the cube of the minimum number of crossings. Additionally, we prove that the only possible nontrivial products of an unknot substrate are (2, m)-torus knots and links and those knots and links which consist of two non-adjacent rows of crossings. (In the special case where one row contains only two crossings, these are the well-known twist knots and links.) In the (common) case of (2, m)-torus knot or link substrates whose products have minimal crossing number m + 1, we prove that the types of products are tightly prescribed, and use this to examine previously uncharacterized experimental data. Finally, we illustrate how the model can help determine the sequence of products in multiple rounds of processive recombination.

Topological Symmetry Groups of Complete Graphs in the 3-Sphere

The orientation preserving topological symmetry group of a graph embedded in the 3-sphere is the subgroup of the automorphism group of the graph consisting of those automorphisms which can be induced by an orientation preserving homeomorphism of the ambient space. We characterize all possible orientation preserving topological symmetry groups of embeddings of complete graphs in the 3-sphere.

Intrinsic Knotting and Linking of Complete Graphs

We show that for every m ∈ N, there exists an n ∈ N such that every embedding of the complete graph Kn in R 3 contains a link of two components whose linking number is at least m. Furthermore, there exists an r ∈ N such that every embedding of Kr in R 3 contains a knot Q with |a2(Q)| ≥ m, where a2(Q) denotes the second coefficient of the Conway polynomial of Q. AMS Classification 57M25; 05C10

Complete Graphs whose Topological Symmetry Groups are Polyhedral

We determine for which

On π-Hyperbolic Knots which are Determined by their 2-Fold and 4-Fold Cyclic Branched Coverings

m

Intrinsic chirality of complete graphs

, the complete graph