### Tangled circuit diagrams II

Instead of proceeding linearly with the exposition I decided today to point out an open problem.

We are able to distinguish using representations (invariants) the circuits mentioned in Tangled circuit diagrams I, but other very simple examples escape our methods (or perhaps the expressions we are trying to distinguish are actually the same and we can't see the proof).

For example, we can prove that the rotation through 720 degrees in the diagram:

is equal to no rotation (Dirac's belt trick).

However we suspect that rotation through 360 degrees is not equivalent to no rotation, but we cannot decide. Our invariants don't distinguish but we don't see a proof of equality.

Just to make something precise about the definition of tangled circuit diagrams - the commutative property of commutative Frobenius algebras is commutativity with respect to the braid twist.

I will be putting up an arxiv version of the paper in the next week.

We are able to distinguish using representations (invariants) the circuits mentioned in Tangled circuit diagrams I, but other very simple examples escape our methods (or perhaps the expressions we are trying to distinguish are actually the same and we can't see the proof).

For example, we can prove that the rotation through 720 degrees in the diagram:

is equal to no rotation (Dirac's belt trick).

However we suspect that rotation through 360 degrees is not equivalent to no rotation, but we cannot decide. Our invariants don't distinguish but we don't see a proof of equality.

Just to make something precise about the definition of tangled circuit diagrams - the commutative property of commutative Frobenius algebras is commutativity with respect to the braid twist.

I will be putting up an arxiv version of the paper in the next week.

Labels: category theory, computing

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