Knots in Washington XXI: Skein Modules, Khovanov Homology and Hochschild Homology

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AbstractAward: DMS-0555648Principal Investigator: Jozef H. PrzytyckiThis award provides partial support for participants of the 21stconference in the very successful ``Knots in Washington''series. This meeting is devoted to Skein Modules, KhovanovHomology and its relation to Hochschild Homology. ``Khovanovhomology'' is a new development in quantum topology that emergedin the last several years. It is a generalization of the Jonesand Homflypt polynomials of links to homologies of certain chaincomplexes. These homologies turn out to be significantlystronger than the original invariants. Khovanov's ideas were alsoapplied to the polynomial invariants of graphs, as well asKauffman bracket skein module of some 3-manifolds. One of themost recent developments is Przytycki's observation that Khovanovhomology (or its comultiplication-free variant developed byHelme-Guizon and Rong) can be interpreted as Hochschild homologyof underlying algebras. This shows how seemingly distant branchesof mathematics can be put together.Low dimensional topology studies shapes of three and fourdimensional spaces. These dimensions are of particular interestto us because they are the dimensions of our space and ourspace-time. Knot theory is a subfield of low dimensionaltopology, which studies the knottedness in our space. One of themain goals of knot theory is to distinguish knotted objects. Thisis often done by means of the so called "knot invariants,"functions that replace geometric objects, such as knots, withthose that are easier to compare. Over the past two decades,there have been a flourish of new invariants in low dimensionaltopology, boosted by ideas from gauge theory, quantum algebras,and mathematical physics. In particular, a new invariant forknots, developed by Khovanov using ideas from homologicalalgebra, has sparked a great deal of interest recently. Thepurpose of the Knots in Washington Conferences is to bringtogether the researchers of the field, including establishedmathematicians as well as graduate students and recent PhDs, todiscuss the state of the art of the subject.