Preface The original motivation for understanding and classifying knots was due to Lord Kelvin who theorized in the 1880s that atoms were knotted or linked vortex rings in the “ether”, and that different elements were determined by the knot or link type of the vortex ring. By the early 1900s, Kelvin’s theory had been proven wrong. However topologists continued to study the knot theory as an area of pure mathematics. Over the past 20-30 years, knot theory has rekindled its historic ties with biology, chemistry, and physics as a means of creating more sophisticated de- scriptions of the entanglements and properties of natural phenomena—from strings to organic compounds to DNA. For example, DNA knots and links have been implicated in a number of cellular processes since their discovery in the late 1960s. In particular, they have been found during replication and recombination, and as the products of protein actions, notably with topoisomerases, recombinases, and transposases. The variety of DNA knots and links observed makes biologically separating and distinguishing these molecules a critical issue. While DNA knots and links can be visualized via electron microscopy, this process can be both difficult and time-consuming. So topological methods of characterizing and predicting their behavior can be helpful. Chemists have been interested in molecular chirality since Pasteur first de- scribed it in 1848. For example, since the two mirror forms of the same molecule can interact with a host’s metabolism very differently, predicting whether or not a molecule will be chiral is important to pharmaceutical companies as they develop new medications. While the geometry of a rigid molecule determines whether or not it is chiral, for flexible or even partially flexible molecules, knot theory can play a role in determining chirality. In addition to the examples described above, there are many other deep inter- actions between knot theory and various areas of scientific investigation. The 2008 AMS Short Course Applications of Knot Theory, on which this volume is based, was intended to introduce the area of applied knot theory to a broad mathemat- ical audience. The aim of the Short Course and this volume, while not covering all aspects of applied knot theory, is to provide the reader with a mathematical appetizer, in order to stimulate the mathematical appetite for further study of this exciting field. No prior knowledge of topology, biology, chemistry, or physics is assumed. In particular, the first three chapters of this volume introduce the reader to knot theory (by Colin Adams), topological chirality and molecular symmetry (by Erica Flapan), and DNA topology (by Dorothy Buck). The second half of this volume is focused on three particular applications of knot theory. Lou Kauffman presents a chapter on applications of knot theory to physics, Ned Seeman presents a chapter vii
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