Skein Theory, Khovanov Homology, and Quantum Doubles of Subfactors

琼斯多项式起源于子因式的研究。琼斯多项式研究中的一个关键因素是用斜率关系和状态和来描述它。其他子因素导致了结和链的其他量子不变量。Pi和U·Haagerup发现了与任何其他已知物体(如量子群)无关的子因子。她将继续研究与这些子因子以及{cal D}和{cal E}子因子有关的量子不变量(通过量子双重结构)，揭示它们背后的斜切关系和态和。另一方面，Khovanov同调理论是S^3中的结环的同调理论，它的欧拉特性产生了与量子群U_q(Sl_N)有关的量子不变量。PI建议继续她的研究，将它们的定义扩展到来自子因子的量子不变量，以及除了S^3之外的流形上的纽结和链环。PI还继续她对子因子的量子偶数的工作和对量子倍数的推广。子因式理论是算符代数理论中最有影响力的学科之一，因为它与拓扑学、表示论和量子物理有关系和应用。然而，它的优点还没有被理解，特别是在拓扑学上。这项研究将使拓扑学家能够接触到子因子论的结果，从而在子因子论和拓扑学之间架起一座新的桥梁，并加强它们之间的合作。到目前为止，子因素理论对于本科生来说还不是很容易理解，甚至对大多数研究生来说也不是很容易理解，因为分析的前提条件很多。建议的研究将揭示子因素的组合结构，这些结构可能在没有强大的分析背景的情况下很容易处理。它将为本科生的研究提供合适的主题，从而促进不同水平和教职员工之间的互动，包括女性和少数族裔。

AbstractAsaedaThe Jones polynomial arose from the study of subfactors. A key element in the study of the Jones polynomial is its description in terms of skein relations and state sums. Other subfactors give rise to other quantum invariants of knots and links. The PI and U. Haagerup discovered subfactors that are not associated to any other known objects like quantum groups. She will continue her study of quantum invariants associated to those subfactors as well as the type {cal D} and {cal E} subfactors (via the quantum double construction) by exposing the skein relations and state sums underlying them. In another direction, Khovanov homology theories are homology theories of knots and links in S^3 whose Euler characteristic yields quantum invariants associated to the quantum groups U_q(sl_n). The PI proposes to continue her study of these by extending their definition to quantum invariants coming from subfactors, and to knots and links lying in manifolds besides S^3. The PI also continue her work on quantum double ofsubfactors and a generalization to quantum multiples. Subfactor theory is among operator algebra theory one of the most influential subjects in the sense that it has relations and applications to topology, representation theory, and quantum physics. However its merit has not been understood, particularly in topology. The proposed research will make the results in subfactor theory accessible to topologists, and thus give a new bridge between subfactor theory and topology, and increase collaboration between them. So far subfactor theory has not been very accessible for undergraduate students, not even for most graduate students due to the amount of prerequisite in analysis. The proposed research will reveal combinatorial structures of subfactors, that may be easily handled without strong background in analysis. It will provide suitable topics for undergraduate research and thus activate interaction among students of various level and faculty, including women and minorities.