Mathematics Colloquia and Seminars

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A somewhat long time ago, in a land not too far away, the Jones polynomial was invented by sentences in passive voice. It was a good knot invariant, but it was soon found to be derivable by methods understood by middle-school students. The extent of this problem went unnoticed until the world was overrun by swarms of mutants. Thus begins the story of the Khovanov Homology, a single prophet-hero that gave birth to a slew of modern homological knot invariants, each one less intelligible than the last.

I will start with basic knot theory to define the Jones polynomial as a state sum, then define and explore Khovanov homology, with a bit of both specific and general computations. I would like to present a proposition about the homology of the knot mirror and to explore the homology of some simple classes of knots. If all goes as planned, most of it will be understandable, and all of it should be at least somewhat interesting, to anyone knowing elementary abstract algebra, though familiarity with basic homology will be (very) useful. In addition to the technical details, I will try to give some perspective as to where this stands in the web of mathematics.