Abstract:

As an area of study, knot theory has been propelled over the last twenty years by two main viewpoints; Floer theories and Khovanov homology. The aim of the talk will be to introduce Khovanov homology, a field of study which combines algebraic topology with the combinatorics of knot diagrams and has ties to other perspectives on knot theory. Assuming very little knot theory, the talk will have three parts; we will first cover how to build a chain complex from a link diagram in such a way that taking its homology yields a powerful, bigraded invariant that extends a simple, known knot invariant. Next, we will look at how Khovanov homology can be upgraded to a functor from a certain category of links, and ways in which this gives us information about smooth manifolds. Time permitting, we will also explore some surprising applications of this invariant and relations to other knot invariants.

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