Turaev developed the notion of a quantum group by considering the category of tangles (thought of with objects as collections of 2$n$ points and with morphisms being braids between them with cups and caps) and considering the algebraic version of this.
I see to recall that someone has developed a notion of a quantum 2-group, or something like that, by considering the 2-category defined by 2-link movies. However, I have no recollection of where I saw this (I think it was a dissertation or something).
It also seems that the 2-knot category, considered as a standard category, should define some family of algebraic objects - has this been studied, and if so where should I look for information on this?
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7$\begingroup$ You seem to have somewhat inaccurate view of the development of quantum groups. Like any success, they have many fathers, but they seem to have first appeared in something resembling their modern form in the work of the Leningrad school in the early 80's, and their formalization in modern form is usually ascribed to Drinfeld and Jimbo. They appeared from considering various physical models and the interpretation in terms of knots appeared later. $\endgroup$– Ben Webster ♦Commented Nov 19, 2011 at 17:04
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$\begingroup$ I agree with Ben, but point out that Turaev described the category of tangles at the Isle of Thorns in 1986. A week later across the channel either Joyal or Street described the same category as a free braided monoidal category. $\endgroup$– Scott CarterCommented Nov 20, 2011 at 0:38
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$\begingroup$ Yes, I shouldn't have said 'developed.' More like his description of the category of tangles made explicity the relationship to quantum groups. Thanks for pointing out the proper history. $\endgroup$– BlakeCommented Nov 20, 2011 at 5:21
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2 Answers
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Just as the category of tangles may be thought of as a free braided, monoidal category with duals, the 2-category of 2-tangles may be thought of as a free braided monoidal 2-category with duals, which was proved by Baez and Langford. Part of the program of "categorification" is to work out how to find invariants of 2-knots and 4-manifolds. The first step may be Khovanov homology. This should hopefully relate to Lauda's categorification of quantum sl2.
answered Nov 19, 2011 at 13:30
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There are many more details that are needed in Eitan's answer. Since weak 2-groups are determined by group 3-cocycles, one would think that group cocycle conditions are related to 2-knots. They are, but the relation is only in relation to quandle cocycles, see for example, Kabaya and Inoue . Khovanov homology gives very weak invariants of closed surfaces in 4-space. However, Kevin Walker's answer to my question indicates that Khovanov homology does give a braided monoidal 2-category with duals. It also seems likely that some of Jacob Lurie's work might give other neat examples from which invariants of 2-tangles can be constructed.
In work not yet complete, Alissa Crans, Mohamed Elhamdadi and I are relating strict $2$-groups to strict $2$-quandles. We are hoping that weak $2$-quandles would be parametrized by their $3$-cocycles. It seems reasonable that quandle cocycle invariants of classical knots come from a braided monoidal category, and that the associated invariants of knotted surfaces come from braided monoidal $2$-categories with duals.
My own opinion about skein-type invariants of knotted surfaces is that, if they exist and are not trivial, then they might be coming from the Khovonov-Rozansky categorifications of MOY $sl_3$ invariants, theories of Soergel bi-modules, or work of Ben Webster. I also think that there is a nice intermediate category and $2$-category such as this. The category such be described as a Frobenius object in a braided monoidal category. I don't know how to describe the associated $2$-category, but it will represent embedded foams in $4$-space. Embedded foams are analogous to knotted trivalent graphs. For both of these there are invariants that come from cocycles defined for group families of quandles. We are writing this stuff down now.
More generally, it is a good idea to have explicit descriptions of the chain homotopies in a knot homology theory that are induced by the Reidemeister moves.
answered Nov 19, 2011 at 16:54