Topological quantum computation of the Dold-Thom functor
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A possible topological quantum computation of the Dold-Thom functor is presented. The method that will be used is the following: a) Certain 1+1-topological quantum field theories valued in symmetric bimonoidal categories are converted into stable homotopical data, using a machinery recently introduced by Elmendorf and Mandell; b) we exploit, in this framework, two recent results (independent of each other) on refinements of Khovanov homology: our refinement into a module over the connective k-theory spectrum and a stronger result by Lipshitz and Sarkar refining Khovanov homology into a stable homotopy type; c) starting from the Khovanov homotopy the Dold-Thom functor is constructed; d) the full construction is formulated as a topological quantum algorithm. It is conjectured that the Jones polynomial can be described as the analytical index of certain Dirac operator defined in the context of the Khovanov homotopy using the Dold-Thom functor. As a line for future research is interesting to study the corresponding supersymmetric model for which the Khovanov-Dirac operator plays the role of a supercharge. © 2014 SPIE.
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