<p>Avishy Carmi and Daniel Moskovich have been developing <em>tangle machines</em> very recently, which is a topological model to describe information. There are <a href="http://arxiv.org/abs/1404.2862">two</a> <a href="http://arxiv.org/abs/1404.2863">papers</a> on the arXiv, as well as three introductory posts on the blog "Low Dimensional Topology" :</p>

<p><a href="http://ldtopology.wordpress.com/2014/05/04/low-dimensional-topology-of-information/">http://ldtopology.wordpress.com/2014/05/04/low-dimensional-topology-of-information/</a></p>

CC BY-SA 3.0
2014-05-20T08:42:55.947Z
2014-05-20T08:42:55.947Z
Arnaud
<p>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 &quot;additively&quot; 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 <a href="http://www.math.columbia.edu/~volconf09/notes/kuperbergconf.pdf" rel="nofollow noreferrer">these slides by Kuperberg</a>.</p>
<p>There is also <a href="https://doi.org/10.1142/S0218216507005282" rel="nofollow noreferrer">work of Kauffman and Lomonaco</a> and various papers of Kauffman (e.g. <a href="https://arxiv.org/abs/1001.0354" rel="nofollow noreferrer">this</a>) 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.</p>
<p>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. <a href="https://arxiv.org/pdf/1207.4563v1.pdf" rel="nofollow noreferrer">HERE</a>. 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.</p>

CC BY-SA 4.0
2014-05-26T17:24:04.077Z
2022-06-22T08:37:31.587Z
2022-06-22T08:37:31.587Z
Daniel Moskovich
<p>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.</p>

CC BY-SA 3.0
2014-05-19T07:17:43.923Z
What's a "homomorphism on a manifold"?
Andrej Bauer
CC BY-SA 3.0
2014-05-19T07:25:39.453Z
Also, what's a "braiding computer"?
Huck Bennett
CC BY-SA 3.0
2014-05-19T11:05:15.360Z
You might be interested in tangle machines : see http://ldtopology.wordpress.com/2014/05/04/low-dimensional-topology-of-information/ and the links therein
Arnaud
CC BY-SA 3.0
2014-05-20T00:12:00.020Z
@Amaud Can you post that as an answer that answers my question actually?
Joshua Herman
CC BY-SA 3.0
2014-05-20T09:59:15.143Z
@HuckBennett  After thinking about my question I realized those concepts are irrelevant to the question and reading up on tangle machines.
Joshua Herman
CC BY-SA 3.0
2014-05-20T15:08:13.110Z
the jones knot polynomial & related problems are known to be a neat/deep bridge between TCS/topology. eg see [graph isomorphism/knot problem](http://cstheory.stackexchange.com/questions/12786/approaches-to-gi-inspired-by-knot-problem)
vzn
<cc.complexity-theory><reference-request><ds.data-structures><topology><computational-geometry>
Would a purely topological computational model be useful in decision problems in topology?
Joshua Herman
# Would a purely topological computational model be useful in decision problems in topology?

(Asked by: username_0 on 2014-05-18)

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.

**Tags:** cc.complexity-theory, reference-request, ds.data-structures, topology, computational-geometry
> **Comments:**
> **username_1**: What's a "homomorphism on a manifold"?  
> **username_2**: Also, what's a "braiding computer"?  
> **username_3**: You might be interested in tangle machines : see http://ldtopology.wordpress.com/2014/05/04/low-dimensional-topology-of-information/ and the links therein  
> **username_0**: @Amaud Can you post that as an answer that answers my question actually?  
> **username_0**: @HuckBennett  After thinking about my question I realized those concepts are irrelevant to the question and reading up on tangle machines.  
> **username_4**: the jones knot polynomial & related problems are known to be a neat/deep bridge between TCS/topology. eg see [graph isomorphism/knot problem](http://cstheory.stackexchange.com/questions/12786/approaches-to-gi-inspired-by-knot-problem)  

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## 2 answers

### Answer by username_5 on 2014-05-26. Score: 8 | Accepted answer

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](http://www.math.columbia.edu/~volconf09/notes/kuperbergconf.pdf).

There is also [work of Kauffman and Lomonaco](https://doi.org/10.1142/S0218216507005282) and various papers of Kauffman (e.g. [this](https://arxiv.org/abs/1001.0354)) 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](https://arxiv.org/pdf/1207.4563v1.pdf). 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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### Answer by username_3 on 2014-05-20. Score: 9

Avishy Carmi and Daniel Moskovich have been developing *tangle machines* very recently, which is a topological model to describe information. There are [two](http://arxiv.org/abs/1404.2862) [papers](http://arxiv.org/abs/1404.2863) 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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