The Unapologetic Mathematician

Mathematics for the interested outsider

Some of my own stuff

I’m talking tomorrow in the Geometry, Symmetry, and Physics seminar here at Yale about the work that spun out of my realization of March 16. This ties into knot colorings, but goes far beyond that starting point. I won’t be able to say this with just the tools I’ve developed in the main line of my writings, so like the Atlas stuff it may not be comprehensible (yet!) to anyone beyond professionals.

So here’s the most general statement. Let \mathcal{C} be an algebraic category and X a co-\mathcal{C} object in the category of pointed topological pairs up to homotopy. Write P_n for the plane with n marked points, and C for the cube. Then every tangle T:m\rightarrow n gives rise to a cospan in \mathcal{C}:

\hom(X,P_m)\rightarrow\hom(X,(C,T))\leftarrow\hom(X,P_n)

where the \hom-objects are taken in the category of pointed pairs up to homotopy. This then gives rise to an anafunctor from the comma category (\hom(X,P_m),\mathcal{C}) to the comma category (\hom(X,P_n),\mathcal{C}). This assignment is a monoidal functor from the category of tangles to the category of (categories, anafunctors). When \mathcal{C} is the category of quandles, this functor categorifies the extension to tangles of the coloring number invariant of knots and links.

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April 26, 2007 - Posted by | Knot theory

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