Abstract
A new model in topological quantum computing, named Gravitational Topological Quantum Computing (GTQC), is introduced as an alternative respect to the Anyonic Topological Quantum Computing and DNA Computing. In the new model the quantum computer is the quantum space-time itself and the corresponding quantum algorithms refer to the computation of topological invariants for knots, links and tangles. Some applications of GTQC in quantum complexity theory and computability theory are discussed, particularly it is conjectured that the Khovanov polynomial for knots and links is more hard than #P-hard; and that the homeomorphism problem, which is non-computable, maybe can be computed after all via a hyper-computer based on GTQC.
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