Nagle Lecture Series
2006 Nagle Lecture
Louis H. Kauffman
Unknots, Collapsing Tangles, and DNA Recombination
February 16, 2006
Description of the Talk
Magicians often present their audience with a knotted rope that miraculously unties itself. The secret to this trick is not always in a sleight-of-hand, but rather in topology! One can make 鈥渒nots鈥� that look knotted but are really not knotted. How can we recognize if a knot is really knotted? This is the fundamental question in knot theory. In this talk we begin with a discussion of the basics of knot theory and some very intriguing questions about the complexity of diagrams for unknots. We follow this path and find ourselves in the subject of rational tangles (certain weaving patterns that correspond to rational numbers) and some elementary number theory. Returning, we find that we have constructed infinitely many unknot diagrams that are hard to untie in the sense that they have to be made more complicated before they simplify. We find the smallest such unknot and we apply these ideas to DNA. The DNA molecule can start in an unknotted state and get knotted by the repeated application of recombination enzymes. The theory of tangles and knots applies to unlocking the mechanisms of DNA recombination. This work is done in collaboration with Sofia Lambropoulou of NTUA, Athens, Greece.
Description of the Speaker
Louis H. Kauffman is a professor of Mathematics at University of Illinois at Chicago. He received his Ph.D. from Princeton, and has worked at many places as a visiting professor and researcher, including the University of Zaragoza in Spain, the University of Iowa in Iowa City, the Institute Hautes Etudes Scientifiques in Bures Sur Yevette, France, the Institute Henri Poincar茅 in Paris, France, the Universidad de Pernambuco in Recife, Brazil, and the Newton Institute in Cambridge England. He is the founding editor and one of the managing editors of the Journal of Knot Theory and its Ramifications, and editor of the World Scientific Book Series On Knots and Everything. He is the author of the books 鈥淔ormal Knot Theory鈥�, 鈥淥n Knots鈥�, 鈥淭emperley Lieb Recoupling Theory鈥�, and 鈥淚nvariants of 3-Manifolds鈥� (Princeton University Press), and 鈥淜nots and Physics鈥� (World Scientific Pub. Co.). He has been a prominent leader in Knot Theory, one of the most active research areas in mathematics today. His discoveries include a state sum model for the Alexander-Conway Polynomial, the bracket state sum model for the Jones polynomial, the Kauffman polynomial, and Virtual Knot Theory. Many important concepts in the field bear his name. His publication list numbers over 170 and continues to grow, and with intriguing new results and concepts. He continues to inspire young mathematicians in the field.