# Connections and Reconnections: A Link Between Mathematics, Physics, and DNA

*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.*

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.

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.

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. |