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