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If one were to develop a purely topological computational model based upon the equivalence of information in knots and the model would perform transformations of that information. This would be the way that model would compute. Would such a model be useful in understanding computational problems in topology or would algebraic models suffice? If one already exists please provide a reference.

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    $\begingroup$ What's a "homomorphism on a manifold"? $\endgroup$ – Andrej Bauer May 19 '14 at 7:17
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    $\begingroup$ Also, what's a "braiding computer"? $\endgroup$ – Huck Bennett May 19 '14 at 7:25
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    $\begingroup$ You might be interested in tangle machines : see ldtopology.wordpress.com/2014/05/04/… and the links therein $\endgroup$ – Arnaud May 19 '14 at 11:05
  • $\begingroup$ @Amaud Can you post that as an answer that answers my question actually? $\endgroup$ – Joshua Herman May 20 '14 at 0:12
  • $\begingroup$ @HuckBennett After thinking about my question I realized those concepts are irrelevant to the question and reading up on tangle machines. $\endgroup$ – Joshua Herman May 20 '14 at 9:59
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I'm not sure whether this qualifies as a purely topological computational model, but there is a topological approach to anyonic quantum computation within the framework of which Aharonov-Jones-Landau and Freedman-Kitaev-Wang proved that a quantum computer can "additively" approximate the Jones polynomial at a root of unity in polynomial time. Furthermore, by Freedman-Larsen-Wang, such an approximation for certain roots of unity is universal for quantum computation. Unfortunately, such an approximation does not appear to be useful in topology, although for certain braids it is useful for anyonic quantum computation. See these slides by Kuperberg.

There is also work of Kauffman and Lomonaco and various papers of Kauffman (e.g. this) on anyonic topological quantum computation, which presents quantum algorithms to compute various topological invariants. Maybe this is the closest thing to what you are looking for.

Regarding tangle machines, the goal of the theory is to provide a diagrammatic formalism for networks of information manipulations (computations) that is more flexible than graphical notations based on labeled directed graphs. Equivalent tangle machines represent computations which are equivalent in the sense that each one can be fully simulated by the other, but which may differ locally in terms of performance. Note also that there is an alternative approach to the same issue by way of higher category theory, e.g. HERE. It isn't in these preprints, but tangle machines can represent any Turing machine computation, and also any recurrent first-order neural network. But I'm not sure how the theory would be useful to understand computational problems specific to topology.

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Avishy Carmi and Daniel Moskovich have been developing tangle machines very recently, which is a topological model to describe information. There are two papers on the arXiv, as well as three introductory posts on the blog "Low Dimensional Topology" :

http://ldtopology.wordpress.com/2014/05/04/low-dimensional-topology-of-information/

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