Polynomial Invariants in the Theory of Knots

这个项目研究了纽结理论中与多项式不变量有关的问题，特别是琼斯多项式及其推广在统计力学、量子群和量子场论中的方法。关于琼斯多项式的一个中心问题是，它的平凡是否意味着它所应用的纽结的平凡。对这个问题的肯定回答将是我们对三维空间中结的本质的理解的突破。受物理思想启发的数学思想已经成功地产生了结、环和三维流形的不变量的混合体。研究人员正在将这些想法扩展到四维空间中的节状曲面和四维流形。在这里，新不变量的发现本身就构成了对四维研究的突破。这样的突破将对量子引力的物理学产生影响。纽结的数学研究不仅因为它本身，而且因为它在分子生物学中的应用，以及它与物理和数学基础的关系。本项目中的研究与这些应用相联系，并使用计算机图形学和说明性写作来扩大在跨学科和教育背景下对这些想法的讨论。通过使用结的计算机模型，人们可以研究这样的问题：如果结被电荷覆盖(自排斥结)，它将如何表现；如果它被给定直径的最少数量的绳子捆绑，它将会是什么样子；在它周围飞行的观察者如何看到这个结的互补的几何形状；DNA分子在细胞环境中移动时如何表现。所有这些直接的物理问题，在我们还没有完全理解的水平上，都与多项式不变量的更代数的方法有关。结和自然是密不可分的。

9802859Kauffman This project studies problems related to polynomial invariants in knottheory, particularly for the Jones polynomial and its generalizations viamethods in statistical mechanics, quantum groups and quantum field theory.A central problem about the Jones polynomial is whether its trivialityimplies the triviality of the knot to which it is applied. An affirmativeanswer to this question would constitute a breakthrough in ourunderstanding of the nature of knottedness in three-dimensional space.Mathematical ideas motivated by physical ideas have succeeded in producinga medley of invariants of knots, links, and three-manifolds. Theinvestigator is in the process of extending these ideas to knotted surfacesin four-dimensional space and to four-dimensional manifolds. Here thediscovery of new invariants would itself constitute a breakthrough in thestudy of four dimensions. Such a breakthrough would have implications forthe physics of quantum gravity. The mathematical study of knots is important not only for its ownsake, but also for its applications to molecular biology and itsrelationships with physics and the foundations of mathematics. Theresearch indicated in this project is linked with these applications andwith the use of both computer graphics and expository writing to enlargethe discussion of these ideas in both interdisciplinary and educationalcontexts. By using computer models of knots, one can study questions suchas how the knot will behave if it is coated with electrical charge(self-repelling knots), how it will appear if it is tied with a leastamount of rope for a given diameter, how the geometry of the complementof the knot appears to an observer flying about in it, how DNA moleculesbehave when moving in their cellular environment. All these directlyphysical questions are, at levels that we do not yet fully understand,related to the more algebraic methods of the polynomial invariants.Knots and Nature are inextricably entwined.***