Connections between Khovanov - and Heegaard Floer - type Homology Theories

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The principal investigator will probe the connection between Heegaard Floer homology and Khovanov homology, two theories, inspired by ideas in physics, that have transformed the landscape of low-dimensional topology during the past decade. The project will focus on one partially-understood connection--namely, the relationship between Khovanov theories of tangles and Heegaard Floer theories of their double-branched covers, first discovered by Peter Ozsvath and Zoltan Szabo and later reinterpreted, using Andras Juhasz's relative version of Heegaard Floer homology for sutured manifolds, by the principal investigator and Stephan Wehrli. The naturality of the connection under various TQFT-type operations suggests a path for developing Khovanov-type invariants for a wider class of objects in low-dimensional topology which should, in turn, yield new applications.The broad aim of the present project is to improve our understanding of the topology of 3- and 4-dimensional spaces, i.e., the properties of these spaces that remain unchanged under stretching and contracting (but not under tearing and gluing). Topological ideas underpin the development of efficient computer chips and information networks. The shapes of molecules and proteins determine their electrical properties and biological functions. Basing quantum computing algorithms on large-scale features of a quantum system minimizes their susceptibility to random error. Moreover, knot theory, the study of loops imbedded in 3-dimensional space, has become increasingly important in our understanding of how DNA behaves in cells.