Polynomial Invariants in the Theory of Knots

9802859Kauffman本课题研究结论中多项式不变量的相关问题，特别是Jones多项式及其在统计力学、量子群和量子场论中的推广方法。关于琼斯多项式的一个中心问题是，它的琐碎性是否意味着它所应用的结的琐碎性。对这个问题的肯定回答将构成我们对三维空间中打结性质的理解的一个突破。由物理思想驱动的数学思想已经成功地产生了结、链和三流形的不变量的混合体。研究者正在将这些想法扩展到四维空间的打结表面和四维流形。在这里，新的不变量的发现本身就构成了四维研究的一个突破。这样的突破将对量子引力物理学产生影响。结的数学研究不仅对其本身很重要，而且对其在分子生物学中的应用以及它与物理学和数学基础的关系也很重要。本项目中指出的研究与这些应用相关联，并使用计算机图形学和说明性写作，以扩大跨学科和教育背景下对这些思想的讨论。通过使用绳结的计算机模型，人们可以研究这样的问题：如果绳结涂上电荷（自我排斥绳结），绳结将如何运动；如果绳结系在给定直径的最小绳上，绳结将如何运动；在绳结中飞行的观察者眼中，绳结互补体的几何形状是如何运动的；DNA分子在细胞环境中运动时是如何运动的。所有这些直接的物理问题，在我们尚未完全理解的层面上，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.***