Mathematical Sciences: Knot Theory and Algebraic Geometry inthe Large

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In recent years, interesting classes of knotted and linked curves in spaces of dimension 3 have arisen in the study of large-scale phenomena in algebraic geometry: these include quasipositive links (cut out, in a sphere of arbitrary radius in complex 2-space, by a polynomial in two complex variables) and links-at-infinity (where the sphere is "infinitely large"); these links should be very special topologically (e.g., highly asymmetric), but it is known that such classical invariants as the Alexander polynomial cannot detect this. Also recently, new topological invariants of knots and links have come to light: these include the enhanced Milnor number (defined for "fibered" links, using differential topology) and the generalized Jones polynomial (defined for all links, and apparently much more combinatorial in nature); both of these invariants detect some asymmetries, and the Jones polynomial was used to give the first proof that a particular knot (the figure- 8) is not quasipositive. Now there is evidence that the generalized Jones polynomial and the enhanced Milnor number are related. Rudolph plans to study this relationship more closely--among other reasons, in search of the geometry underlying the generalized Jones polynomial. Quasipositive links and links-at-infinity will serve as test cases.