The 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 of subfactors 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.
