Mathematisches Forschungsinstitut Oberwolfach Geometric Knot Theory
Mathematisches Forschungsinstitut Oberwolfach Report No. 22/2013 DOI: 10.4171/OWR/2013/22 Geometric Knot Theory Organised by Dorothy Buck, London Jason Cantarella, Athens John M. Sullivan, Berlin Heiko von der Mosel, Aachen 28 April – 4 May 2013 Abstract. Geometric knot theory studies relations between geometric prop- erties of a space curve and the knot type it represents. As examples, knotted curves have quadrisecant lines, and have more distortion and more total cur- vature than (some) unknotted curves. Geometric energies for space curves – like the M¨obius energy, ropelength and various regularizations – can be minimized within a given knot type to give an optimal shape for the knot. Increasing interest in this area over the past decade is partly due to various applications, for instance to random knots and polymers, to topological fluid dynamics and to the molecular biology of DNA. This workshop focused on the mathematics behind these applications, drawing on techniques from algebraic topology, differential geometry, integral geometry, geometric measure theory, calculus of variations, nonlinear optimization and harmonic analysis. Mathematics Subject Classification (2010): 57M, 53A, 49Q. Introduction by the Organisers The workshop Geometric Knot Theory had 23 participants from seven countries; half of the participants were from North America. Besides the fifteen main lectures, the program included an evening session on open problems. One morning session was held jointly with the parallel workshop “Progress in Surface Theory” and included talks by Joel Hass and Andre Neves. Classically, knot theory uses topological methods to classify knot and link types, for instance by considering their complements in the three-sphere.
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