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.
We show that for every m ∈ N, there exists an n ∈ N such that every embedding of the complete graph K n 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 K r in R 3 contains a knot Q with |a 2 (Q)| ≥ m, where a 2 (Q) denotes the second coefficient of the Conway… (More)
We prove that a graph is intrinsically linked in an arbitrary 3-manifold M if and only if it is intrinsically linked in S 3. Also, assuming the Poincaré Conjecture, a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S 3. The study of intrinsic linking and knotting began in 1983 when Conway and Gordon [CG] showed that every… (More)
Over the past twenty years, knot theory has rekindled its historic ties with biology, chemistry, and physics. While the original motiviation for understanding and classifying knots—Lord Kelvin's correlation of chemical elements with particular knotted configurations in the " ether " — proved erroneous, mathematicians continued to develop the theory of… (More)
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 homeomor-phism of the ambient space. We prove strong restrictions on the groups that can occur as the topological symmetry group of some embedded graph. In addition, we characterize the orientation… (More)
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… (More)
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… (More)
Site-specific recombination on supercoiled circular DNA yields a variety of knotted or catenated products. Here, we present a topological model of this process and characterize all possible products of the most common substrates: unknots, unlinks, and torus knots and catenanes. This model tightly prescribes the knot or catenane type of previously… (More)
We answer the question " Does the Y-triangle move preserve intrinsic knottedness? " in the negative by giving an example of a graph that is obtained from the intrinsically knotted graph K 7 by triangle-Y and Y-triangle moves but is not intrinsically knotted.
We present a model of how DNA knots and links are formed as a result of a single recombination event, or multiple rounds of (processive) recombination events, starting with an unknotted, unlinked, or a (2, m)-torus knot or link substrate. Given these substrates, according to our model all DNA products of a single recombination event or processive… (More)