Topological quantum information, virtual Jones polynomials and Khovanov homology

Details Topological quantum information, virtual Jones polynomials and Khovanov homology Topological quantum information, virtual Jones polynomials and Khovanov homology

Dec 2011

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2011njph...13l5007k&link_type=abstract

New Journal of Physics, Volume 13, Issue 12, pp. 125007 (2011).

Physics

Scientific paper

In this paper, we give a quantum statistical interpretation of the bracket polynomial state sum , the Jones polynomial V K(t) and virtual knot theory versions of the Jones polynomial, including the arrow polynomial. We use these quantum mechanical interpretations to give new quantum algorithms for these Jones polynomials. In those cases where the Khovanov homology is defined, the Hilbert space C(K) of our model is isomorphic with the chain complex for Khovanov homology with coefficients in the complex numbers. There is a natural unitary transformation U:C(K) → C(K) such that = Trace(U), where denotes the evaluation of the state sum model for the corresponding polynomial. We show that for the Khovanov boundary operator ∂:C(K) → C(K), we have the relationship ∂U + U∂ = 0. Consequently, the operator U acts on the Khovanov homology, and we obtain a direct relationship between the Khovanov homology and this quantum algorithm for the Jones polynomial.

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