Tuesday, June 14, 2011

Tangled Circuits

The work Nicoletta and I did with Bob Rosebrugh during his visit concerned tangled circuits; that is, adding to our model of circuits the possibility that the tangling of the wires might be recorded. This is related to the previous post here.
In that post I described, or better hinted at, a new category in which tangles might be represented, which I will now call TRel. Let me describe that category here.

Take a group G. The objects of the category are (formal) powers of G. An arrow from G^m to G^n is a relation R from G^m to G^n satisfying two properties:
1) (x_1, ..., x_m)R(y_1, ... ,y_n) implies that for all g in G
(g^(-1)x_1g, .... , g^(-1)x_mg)R(  g^(-1)y_1g, .... ,   g^(-1)y_mg)
2) (x_1  , .... , x_m)R(  y_1  , .... , y_m ) implies x_1...x_m(y1...yn)^(-1) is in Z(G) (the center of G).

I will give some more details later, but this category is a braided monoidal category, and the object G has the structure of a commutative Frobenius algebra. This means that G has a fortiori the structure that the generator of the category of tangles has. Hence there is a monoidal functor Tangles -> TRel. There is a similar category TSpan based on spans instead of relations, and the functor Tangles -> TSpan yields the colouring of knots in the group G.
Bob Rosebrugh has just been speaking about this at FMCS. in Calgary.

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