The following is a short introduction to an invited lecture to be presented at the upcoming 2018 SIAM Annual Meeting (AN18) in Portland, Ore., from July 9-13.
Mariel Vazquez, University of California, Davis.
Knots have appeared in art and architecture for centuries. Like many other areas of mathematics, the field of knot theory is rooted in questions related to the physical world. In 1833, Carl Friedrich Gauss reported on the magnitude of a magnetic field produced by a current traveling through a circular wire. He formulated the Gauss linking number while computing the magnetic field induced in a second loop interlinked with the first one but not carrying current. In the 1860s, Lord Kelvin postulated that matter is composed of “vortex atoms,” some of which are knotted. This idea led Peter Guthrie Tait to first tabulate knots in 1867. Though Lord Kelvin’s table of elements was incorrect, interest in knots continued — albeit with reduced enthusiasm. At the turn of the 20th century, notable mathematicians—including Max Dehn, Kurt Reidemeister, and James Waddell Alexander II—began a rigorous mathematical study of knots. For many decades, knot theory and low-dimensional topology epitomized the beauty and power of pure mathematics.
Knots in the computer are represented as polygons. We utilize Monte Carlo sampling in the cubic lattice to generate ensembles of conformations to be used as reconnection input. Figure generated by Reuben Brasher using KnotPlot software.
Applications reemerged as the tools to study knots and their related spaces became more powerful, especially in regards to polynomial invariants by Vaughan F.R. Jones, Louis Kauffman, W.B. Raymond Lickorish, Kenneth Millett, Józef Przytycki, John Horton Conway, and others. Flexible circular chains often appear in nature, from microscopic DNA plasmids to macroscopic loops in solar corona. Such chains entrap rich geometrical and topological complexity that can offer insight into the processes underlying their formation or modification.
In the 1970s, molecular biologists encountered circular DNA molecules that adopted interesting geometrical and topological forms. Budding collaborations among mathematicians, physicists, chemists, and molecular biologists shaped DNA topology as an interdisciplinary field, with tools from mathematical and computational knot theory at its center.
Mathematical representation of a DNA link using the tangle method of De Witt Sumners and Claus Ernst. We characterize the shortest unlinking pathways by local reconnection. Figure generated by Rob Scharein using KnotPlot software.
Although DNA knots and links are considered undesirable in the cellular environment, nontrivial topologies occur frequently. Enzymes use local cleavage and strand passage or local reconnection to simplify DNA topology. Examples include the action of type II topoisomerases and DNA recombinases. Physicists have observed local reconnection events of knotted vortices in fluid flow; their study reveals similar patterns of topology simplification as those observed after DNA recombination.
At the 2018 SIAM Annual Meeting, I will show how mathematicians use techniques from knot theory and low-dimensional topology—aided by computational tools—to identify minimal pathways of unlinking newly-replicated DNA molecules. Our results and numerical methods are not restricted to the biological example, and are applicable to any local reconnection process.
||Mariel Vazquez is a professor in the Department of Mathematics and the Department of Microbiology and Molecular Genetics at the University of California, Davis.