Skein Theory, Khovanov Homology, and Quantum Doubles of Subfactors

琼斯多项式起源于对子因子的研究。琼斯多项式的研究中的一个关键因素是它的描述在绞关系和状态和。其他的子因子产生了其他的量子不变量的纽结和链接。PI和U。Haagerup发现了与任何其他已知对象（如量子群）无关的子因子。她将继续她的量子不变量相关的研究，这些子因子以及类型{cal D}和{cal E}子因子（通过量子双结构）通过暴露的绞关系和状态总和的基础上。 在另一个方向上，霍瓦诺夫同调理论是S^3中纽结和链环的同调理论，其欧拉特征产生与量子群U_q（sl_n）相关联的量子不变量。PI建议继续她对这些的研究，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之外的流形中的纽结和链环。PI还继续她的工作量子倍的子因子和推广到量子倍。 子因子理论是算子代数理论中最有影响力的学科之一，因为它与拓扑学，表示论和量子物理学有联系和应用。然而，它的优点还没有被理解，特别是在拓扑学。本文的研究将使子因子理论的研究成果更容易为拓扑学家所接受，从而在子因子理论和拓扑学之间架起一座新的桥梁，促进二者之间的合作。 到目前为止，由于分析中的先决条件太多，子因素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不是。拟议的研究将揭示子因素的组合结构，这可能是很容易处理，没有强大的背景分析。它将为本科生的研究提供合适的课题，从而激活包括妇女和少数民族在内的各级学生和教师之间的互动。

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.