论文标题
组合结论和琼斯多项式
Combinatorial Knot Theory and the Jones Polynomial
论文作者
论文摘要
本文是对沃恩·琼斯（Vaughan Jones）的工作和影响的记忆。这是对结理论和低维拓扑的显着突破的阐述，这是他的作品所促进的。本文回忆起琼斯多项式的成立，以及作者发现琼斯多项式的括号多项式模型。然后，我们描述了结理论中的一些发展，这些发展受琼斯多项式的启发，并涉及这种不变的变化和概括。该论文是以个人漫游的形式编写的，目的是显示与琼斯多项式有关的不同数学主题。可以根据组合拓扑，统计力学，谎言代数，Hopf代数，量子场理论，类别理论等来解释这种不变。在每种情况下，琼斯不变式都是这些数学和物理环境的模式和连接的关键示例。值得注意的是，沃恩·琼斯（Vaughan Jones）对他的多项式的发现在多大程度上触及了许多数学，物理学，自然科学以及我们许多数学生活。
This paper is a memory of the work and influence of Vaughan Jones. It is an exposition of the remarkable breakthroughs in knot theory and low dimensional topology that were catalyzed by his work. The paper recalls the inception of the Jones polynomial and the author's discovery of the bracket polynomial model for the Jones polynomial. We then describe some of the developments in knot theory that were inspired by the Jones polynomial and involve variations and generalizations of this invariant. The paper is written in the form of a personal odyssey and with the intent to show different mathematical themes that arise in relation to the Jones polynomial. This invariant can be interpreted in relation to combinatorial topology, statistical mechanics, Lie algebras, Hopf algebras, quantum field theory, category theory and more. In each case the Jones invariant appears as a key example for patterns and connections of these mathematical and physical contexts. It is remarkable to what extent Vaughan Jones' discovery of his polynomial has touched so much mathematics, physics, natural science and so many of our mathematical lives.
