流体纽结的多项式拓扑不变量的研究

本项目着重于流体力学的拓扑方面，目标是获得比目前国际通用的缠绕数方法更强的流体纽结拓扑辨识工具，如多项式及其他拓扑不变量。这一研究很重要，原因在于纽结的拓扑与能量密切相关。缠绕数方法是1969-1992年由剑桥大学Moffatt和Ricca（本项目主要参与人）发展起来的，在处理拓扑守恒系统时表现较好；但对拓扑非守恒现象（如磁重联和纽结碎裂）则暴露出严重不足，尤其是在拓扑辨识能力和能量计算一致性方面。为解决这个问题，最近本项目申请人与Ricca合作发展出一套纽结多项式方法。该方法引起理论界广泛关注，目前已获多项荣誉，包括多个国际高等级学术会议全会报告（Keynote talk）等。本项目是这一工作的继续和向更高层次的发展。预期结果包括：基于流体螺旋度，构造HOMFLYPT纽结多项式、Kontsevich积分和Vassiliev拓扑不变量；对已建立的流体纽结多项式理论进行进一步实践检验。

This Research Project lies in the area of topological fluid mechanics, focusing on topological characterizing tools for fluid knots. We aim to achieve polynomials and other knot topological invariants that are more powerful than linking numbers, the commonly-used tool of today. The method of linking numbers was developed in 1969-1992 by Moffatt from Cambridge and his student Ricca (major Participant of this Project). It works well for topologically conserved systems, but fails in treating non-conserved phenomena such as magnetic reconnections and breaking processes of complex knots. It is weak in topological characterization and in energy and other dynamical computations. In order to solve this problem, very recently Ricca and I (Chief Investigator of this Project, Liu) developed a method of polynomial topological invariants for fluid knots, which has proved to be powerful and received much interest from the international society of fluid dynamics. The academic honors awarded to this method include a cover-page publication of Journal of Physics A and inclusion in the 2012 Highlights (annual collection) of the journal, as well as several keynote talks in highly-ranking international conferences held by Cambridge U, Moscow U, UC Santa Barbara, International Union of Theoretical & Applied Mechanics (IUTAM), the European Science Foundation (ESF), etc....The present project is a continuation and development of our preceding work. Expected outcome includes constructing the HOMFLYPT knot polynomial, Kontsevich integral and Vassiliev invariants based on fluid helicity, as well as designing experiments to further test our established theory of polynomial topological invariants for fluid knots.