Intrinsic knotting and linking of complete graphs

We show that for every m in N, there exists an n in 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 in N such that every embedding of K_r in R^3 contains a knot Q with |a_2(Q)| > m-1, where a_2(Q) denotes the second coefficient of the Conway polynomial of Q.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-17.abs.htm

Predicting Knot or Catenane Type of Site-Specific Recombination Products

Site-specific recombination on supercoiled circular DNA yields a variety of knotted or catenated products. We develop a model of this process, and give extensive experimental evidence that the assumptions of our model are reasonable. We then characterize all possible knot or catenane products that arise from the most common substrates. We apply our model to tightly prescribe the knot or catenane type of previously uncharacterized data.Comment: 17 pages, 4 figures. Revised to include link to the companion paper, arXiv:0707.3896v1, that provides topological proofs underpinning the conclusions of the current paper. References update

A Prime Strongly Positive Amphicheiral Knot Which Is Not Slice

We begin by giving several definitions. A knot K in S3 is said to be amphicheiral if there is an orientation-reversing diffeomorphism h of S3 which leaves K setwise invariant. Suppose, in addition, that K is given an orientation. ThenK is said to be positive amphicheiral if h preserves the orientation of K. If, in addition, the diffeomorphism h is an involution then K is strongly positive amphicheiral. Finally, we say a knot is slice if it bounds a smooth disc in B4. In this note we shall give a smooth example of a prime strongly positive amphicheiral knot which is not slice

Rigid and Nonrigid Achirality

In order to completely characterize a molecule it is useful to understand the symmetries of its molecular bond graph in 3-space. For many purposes the most important type of symmetry that a molecule can exhibit is mirror image symmetry. However, the question of whether a molecular graph is equivalent to its mirror image has different interpretations depending on what assumptions are made about the rigidity of the molecular structure. If there is a deformation of 3-space taking a molecular bond graph to its mirror image then the molecule is said to be topologically achiral. If a molecular graph can be embedded in 3-space in such a way that it can be rotated to its mirror image, then the molecule is said to be rigidly achiral. We use knot theory in R^3 to produce hypothetical knotted molecular graphs which are topologically achiral but not rigidly achiral, this answers a question which was originally raised by a chemist

Complete graphs whose topological symmetry groups are polyhedral

We determine for which $m$, the complete graph $K_m$ has an embedding in $S^3$ whose topological symmetry group is isomorphic to one of the polyhedral groups: $A_4$, $A_5$, or $S_4$.Comment: 27 pages, 12 figures; v.2 and v.3 include minor revision

A Topological Approach to Molecular Chirality

Topology is the study of deformations of geometric figures. Chemistry is the study of molecular structures. At first glance these fields seem to have nothing in common. But let’s take a closer look to see how these fields come together in the study of molecular symmetries