Would a purely topological computational model be useful in decision problems in topology? 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.
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/