Mathematical Sciences: Knot Theory and Algebraic Geometry inthe Large

近年来，在研究代数几何中的大规模现象时，出现了3维空间中有趣的纽结和连接曲线类：其中包括拟正环(在复2空间中的任意半径的球面上被两个复变量的多项式割去)和无穷远处的环(其中球面“无限大”)；这些链接在拓扑上应该是非常特殊的(例如，高度不对称)，但众所周知，像Alexander多项式这样的经典不变量无法检测到这一点。最近，纽结和环的新的拓扑不变量也被发现：其中包括增强的米尔诺数(使用微分拓扑学，为“纤维”环定义)和广义琼斯多项式(为所有环定义，显然在本质上更具组合性)；这两个不变量都检测到一些不对称性，琼斯多项式被用来第一次证明一个特定的结(图8)不是准正的。现在有证据表明，广义Jones多项式和增强型Milnor数是相关的。鲁道夫计划更密切地研究这种关系--其中一个原因是寻找广义琼斯多项式背后的几何学。拟正链接和无穷大链接将作为测试用例。

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.