Friday, February 11, 2011

Knots, groups, cospans and spans

In the last few days I have clarified the ideas described in the last post (partially responding to objections by Aurelio Carboni), introducing further structure which John Armstrong does not seem to have noticed.

The clarification which I will describe makes evident that the two invariants described by John Armstrong (i) the fundamental group of (the complement of) a knot, and (ii) the number of colourings of a knot, are both examples of a single phenomenon, and both can be extended beyond tangles.

The abstract context is this: consider a group object G in a category C with finite limits. In case (i) the group is the free group on one generator which has a group structure in the dual of the category of (finitely presented) groups. In case (ii) the group object is any group in Sets.
Then consider the full subcategory Sp(C,G)  of Span(C) whose objects are the powers of G.

This category Sp(C,G)  has the obvious monoidal structure arising from products, but a subcategory has a braiding different from the usual symmetry arising from conjugation in the group G. The braiding is GxG <- GxG->GxG, where the first arrow is the identity and the second (x,y) |-> (y, inverse(y)xy). (9/3/2011: Notice that in the original post I claimed that this was a braiding for the whole of Sp(C,G) but the braiding is natural only for a subcategory: the error was pointed out by Aurelio Carboni.)

In addition the object G has a Frobenius algebra structure  which is braided commutative (both multiplication and comultiplication).

The mutiplication of G in Sp(C,G) is GxG <- GxG -> G, the first arrow being the identity, and the second is multiplication of G in C. The identity of G in Sp(C,G) is  I <- I -> G the second arrow being the identity of G in C. The comultiplication and coidentity in Sp(C,G) are just the transposes of the multiplication and identity in Sp(C,G).
The equations for a Frobenius algebra, first introduced by Carboni and Walters in 1987, are easily checked - note they require that G be a group. The operations are braided-symmetric. (Notice that Span(C) has also quite different Frobenius algebra structures on the objects  arising from the diagonal map and projection, which was the structure studied 1987 on relations.)

As a consequence the subcategory of Sp(C,G) is self-dual compact closed and braided, and the braiding goes well with the compact closed structure, so tangles may be represented there.

The calculation of the fundamental group of the (complement of the) trefoil correspond to the case in which G is the free group on one generator in the dual of groups. If instead the letters in the diagrams had been elements of a group in Sets the limit of the diagram (the subset of a power of the group satisfying the equations) is the number of colourings of the trefoil. The limit may also be calculated by the expression in Span.