Homological Algebra Methods in Topology and Combinatorics

George Washington University, Washington DC

Investigators

Abstract

Over the past two decades, low dimensional topology has seen a great deal of studies in two types of invariants: gauge theory type invariants in dimension four and combinatorial type invariants in dimension three. While both sides have deep connections with physics, they share little common techniques and have rather different flavors. This picture could change though, with recent work due to Khovanov and Ozsvath-Szabo. In 1999, Khovanov introduced a graded homology theory for knots, and proved that its graded Euler characteristic is the Jones polynomial. This has turned out to be a far reaching generalization of the Jones polynomial. Furthermore, there is strong evidence that Khovanov theory, along with the Ozsvath-Szabo theory, could bridge the connection between gauge theory type and combinatorial type invariants. Motivated by Khovanov's work, the PI, with his student Laure Helme-Guizon, has established a graded homology theory for graphs which yields the chromatic polynomial when taking Euler characteristic. The PI intends to further his investigation on these homology theories, both for knots and for graphs. Some of the specific problems are: understanding their geometric meanings, studying their behavior under various cut and paste operations, constructing homology theories for various other polynomials of knots and graphs, and investigating relations with other invariants in low dimensional topology. Low dimensional topology studies the shapes of three and four dimensional spaces. These dimensions are of particular interests to mankind because of the dimensions of our space and our space-time. A specific subfield in low dimensional topology is knot theory, which studies the knottedness in our three dimensional space. Knots are worthwhile to study not only because they are fundamental in 3-dimensional spatial structure, but also because of its connection to areas outside mathematics. For example, biochemists have discovered knotted DNA molecule (1980s) and knotted proteins (2004). It is also intimately related to the study of graph theory, an area interesting to mathematicians, computer scientists, and others. Over the past two decades, there have been a flourish of new invariants in low dimensional topology, boosted by ideas from gauge theory, quantum algebras, and mathematical physics. In particular, a new invariant for knots, developed by Khovanov using ideas in homological algebra, has sparked a great deal of interest recently. An analogous theory for graphs has since been developed by the PI and his student. This project aims to investigate these new invariants, with a particular emphasis on the homological algebra methods for knots and graphs.