Wednesday, September 14, 2011

Tangled circuit diagrams I

I haven't blogged for some time, but I want to start a series of posts about a paper we (Rosebrugh, Sabadini and I) have written on tangled circuit diagrams (we are just cleaning up final details now). I have given hints in earlier posts but now I would like to give an exposition in blog-post form.

The aim is to describe an algebra with the property that expressions in the algebra may be pictured as circuit diagrams, but circuits in which the tangling of the wires is taken into account.

For example, the following two circuits should be expressions in the algebra, but distinct:
  Another example of two distinct circuits:

Both a trefoil and an unknot should be examples of circuits (with wires only, and no components) and they should be distinct.

But it should be possible to do Dirac's belt trick and prove in the algebra that the following two circuits are equal:

The point of the paper is to define such an algebra, and to prove statements such as those above, using representations of the algebra in more concrete algebras of the same type. A special case will be the use of knot colouring, and knot groups in the study of knots. Some of the more concrete algebras have been used by us previously in modelling various kinds of circuits with state, and systems of parallel processes etc.

Let me begin in this post by at least making the definition of the algebra of tangled cicuits. I will simplify slightly by considering circuits with only one type of wire.
In fact it is not just one algebra; first one must choose a "multigraph" M of components. The edges (components) in a multigraph have not just a source and a target but a number of sources (the input wires) and a number of targets (the output wires).

Definition Given a multigraph M the algebra of tangled circuit diagrams formed from M, denoted TCircD, is the free braided strict monoidal category on M such that the single generating object has a commutative Frobenius algebra structure.

I will stop there for today, but will explain this definition in the next post. I should recount the history of the ideas, and the other research on which they depend but will leave that to a later post - that's one of the things we are cleaning up in the paper now.

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