Mathematical Sciences: Problems in Knot Theory and Low- Dimensional Topology: Applications of Quasipositive Knots and Surfaces

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9504832 Rudolph Recent research on manifolds of dimension 3 and 4 both draws upon, and has striking consequences for, the theory of complex curves in complex surfaces. Topologically, complex curves in the unit ball in complex 2-space are simply represented as quasipositive surfaces, which are combinatorially defined and readily manipulable. Earlier, Dr. Rudolph combined quasipositivity with results from gauge theory to obtain estimates showing that many knots are much more complicated (in certain specific senses) than classical knot invariants can detect; he is now working to extend such estimates (using results from monopole theory). Other work in progress includes development of skein theory for Legendrian knots, an attack on a question of Harer about fibered links (are they all stably Hopf-plumbed?), and (with Michel Boileau) a study of Stein-fillable 3-manifolds. Knots and links are superficially simple objects which turn up in contexts as diverse as physics, theoretical meteorology, and biology. Mathematically, a rich source of knots is the theory of equations in a small number of variables (the simplest mathematical link, which looks like two rings in a piece of chain, comes from the simple equation xy = 0). Knot theory--the study of aspects of the geometry of knots and links which are insensitive to continuous deformations--has always had a lot to give to, and take from, the qualitative side of the theory of equations. Quasipositive knot theory takes advantage of extra structure in certain equations to obtain stronger results about their associated knots; potential applications range from quantum gravity to unraveling the molecular structure of DNA. ***