Mathematical Sciences: Polynomial Invariants in the Theory of Knots

这个项目关注纽结理论中的多项式不变量， 特别是琼斯多项式及其推广。 沿着 运用纽结理论，将项目与相关的 组合学问题，广义自旋网络，量子 群、统计力学和三维流形的不变量。 在 特别是，它解决了缠结理论的再耦合理论， Temperley-Lieb代数及其与 图嵌入、SL（2）量子群和Turaev-Viro 3-流形的不变量 这个再耦合理论给出了一个 量子6 j系数的完全缠结理论版本. 团结的根源，它允许建设图拉耶夫-维罗 在纯纽结理论的基础上不变。 这些主题将是 在Witten-Reshetikhin-Turaev不变量的背景下探索 3-流形 关于链接多项式的其他问题将是 考虑，如统计力学模型的属性 Alexander-Conway多项式和与四个- 维拓扑学 纽结是相当基本的几何物体， 有趣的性质是拓扑性质。 我们的意思是， 几何结并没有以有趣的方式不同，如果其中一个 可以在不切割的情况下变形成另一个样子， 解开它，只是通过推动它的字符串重新排列， 交叉路口 拓扑学家说它们是两种不同的几何 相同拓扑结的实现。 然而， 当一个复杂的几何结 在拓扑上与另一个不同，而不仅仅是一个 不同的几何实现。 这个问题可以通过以下方式解决： 计算某些数或多项式，这些数或多项式被称为 “拓扑不变量”，这意味着它们总是具有相同的 相同拓扑的不同几何实现的值 结 这个问题将被简化为纯代数，如果 一个不变量，它也总是有不同的几何值， 实现不同的拓扑结，但生活并非如此 简单-没有一个单一的不变量达到这个理想，甚至所有的 已知的不变量放在一起。 因此， 研究新的不变量，其中一些最有用的是那些 近年来受到量子物理学的启发。在 特别是，纽结理论在生物学中的应用， DNA链已经利用了这些新的不变量。

This project concerns polynomial invariants in knot theory, particularly the Jones polynomial and its generalizations. Along with knot theory, the project is integrally involved with related problems in combinatorics, generalized spin networks, quantum groups, statistical mechanics, and invariants of 3-manifolds. In particular, it addresses a tangle-theoretic recoupling theory for the Temperley-Lieb algebra and its relationship with invariants of graph embeddings, the SL(2) quantum group, and the Turaev-Viro invariant of 3-manifolds. This recoupling theory gives a completely tangle-theoretic version of quantum 6j coefficients at roots of unity, and it permits the construction of the Turaev-Viro invariant on a purely knot-theoretic basis. These topics will be explored in the context of the Witten-Reshetikhin-Turaev invariants of 3-manifolds. Other problems about link polynomials will be considered, such as the properties of statistical mechanical models for the Alexander-Conway polynomial and relations with four- dimensional topology. Knots are rather elementary geometric objects whose really interesting properties are topological. By this we mean that two geometric knots do not differ in an interesting way if one of them can be transformed to look just like the other without cutting or untying it, just by pushing its string about to rearrange the crossings. Topologists say that they are two different geometric realizations of the same topological knot. Nevertheless, it is not a trivial matter to recognize when one complicated geometric knot is topologically different from another, rather than just a different geometric realization. This problem can be addressed by computing certain numbers or polynomials which are called "topological invariants," meaning that they always have the same value for different geometric realizations of the same topological knot. The problem would be reduced to pure algebra if there were one invariant which also always had different values for geometric realizations of different topological knots, but life is not so simple -- no single invariant achieves this ideal, nor even all the known invariants taken together. It is therefore valuable to investigate new invariants, some of the most useful being those inspired in recent years by ideas from quantum physics. In particular, applications of knot theory to the biology of long strands of DNA have drawn upon knowledge of these newer invariants.