Skein Theory, Khovanov Homology, and Quantum Doubles of Subfactors

【摘要】琼斯多项式是从子因子的研究中产生的。研究琼斯多项式的一个关键因素是用绞丝关系和状态和来描述它。其他子因子产生结和链的其他量子不变量。PI和U.哈格鲁普发现了与量子群等其他已知物体无关的子因子。她将继续研究与这些子因子相关的量子不变量以及类型{cal D}和{cal E}子因子（通过量子双重构造），通过揭示串结关系和它们背后的状态和。在另一个方向上，Khovanov同调理论是S^3中结点和链接的同调理论，其欧拉特性产生与量子群U_q（sl_n）相关的量子不变量。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S^3以外流形中的结和链接。PI还继续她在子因子的量子倍和量子倍的推广方面的工作。子因子理论是算子代数理论中最具影响力的学科之一，因为它与拓扑学、表示理论和量子物理都有联系和应用。然而，它的优点尚未被理解，特别是在拓扑学方面。本研究将使子因子理论的研究成果能够为拓扑学家所接受，从而在子因子理论和拓扑学之间架起一座新的桥梁，增加它们之间的协作。到目前为止，子因子理论对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是如此，因为分析的先决条件太多了。该研究将揭示子因子的组合结构，这可能很容易处理，没有强大的分析背景。它将为本科研究提供合适的主题，从而激活不同层次的学生和教师之间的互动，包括女性和少数民族。

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.