Topological Symmetry Groups of K_{4r+3}

We present the concept of the topological symmetry group as a way to analyze the symmetries of non-rigid molecules. Then we characterize all of the groups which can occur as the topological symmetry group of an embedding of the complete graph K_{4r+3} in S^3

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

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

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

Geometry of Schreieder's varieties and some elliptic and K3 moduli curves

We study the geometry of a class of $n$-dimensional smooth projective varieties constructed by Schreieder for their noteworthy Hodge-theoretic properties. In particular, we realize Schreieder's surfaces as elliptic modular surfaces and Schreieder's threefolds as one-dimensional families of Picard rank $19$ $K3$ surfaces.Comment: 28 pages. Contains arXiv:1603.0561