Invariants of Knots and 3-Manifolds

小行星9800703 研究者最近和目前的研究是关于链接和3-流形的有限型不变量的概念。 该项目的动机来自几个来源：（i）微扰的陈-西蒙斯理论，（ii）推广3-球面中节点的瓦西里耶夫不变量的概念，（iii）将节点和3-流形的量子群不变量分离成基本部分并重新组装它们。 纽结和三维流形的有限型不变量理论的目标之一是了解这些不变量在纽结和三维流形分类中的作用和强度。 13年前，沃恩·琼斯发现了一个简单但非常有用的多项式，它与三维空间中的每一个结都有关联。 不久之后，爱德华维滕研究了一个三维规范场理论，将琼斯多项式推广到任意三维空间中的节点。 3维和4维的规范理论与我们物理世界的高能描述密切相关（在我们目前感知的能量和距离中）。 此外，结自然地表现为DNA的超螺旋形式（生命的重要成分），以及一些高能基本粒子的轨迹。 因此，目前的研究项目在生物学和高能物理学中具有潜在的应用，但尚未完全理解。 ***

9800703 Garoufalidis The investigator's recent and current research is on the notion of finite type invariants of links and 3-manifolds. The motivation for the project comes from several sources: (i) perturbative Chern-Simons theory, (ii) generalizing the notion of Vassiliev invariants of knots in the 3-sphere, (iii) separating the quantum group invariants of knots and 3-manifolds into elementary pieces and reassembling them. One of the goals of the theory of finite type invariants of knots and 3-manifolds involves understanding the role and the strength of these invariants in the classification of knots and 3-manifolds. Thirteen years ago, Vaughan Jones discovered a simple yet remarkably useful polynomial associated to every knot in 3-dimensional space. Shortly afterwards, Edward Witten studied a 3-dimensional gauge field theory that generalized the Jones polynomial to knots in arbitrary 3-dimensional space. Gauge theories in 3 and 4 dimensions are intimately related to the high energy description of our physical world (in the energies and the distances that we currently perceive). In addition, knots appear naturally as supercoiled forms of DNA (an important ingredient of life), as well as being trajectories of some elementary particles at high energies. Thus, the current research project has potential applications, yet to be fully understood, in biology and in high energy physics. ***