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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)
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)
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)
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 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)