Planar Contact Structures with Calabi-Yau Fillings and Topological Quantum Computation
We study planar open books obtained by lifting braids through branched covers of the disk D^2, together with the quantum operations in the Ising representation. We note a criterion for the associated Stein fillings to be Calabi-Yau (CY). Among positive factorizations of a fixed monodromy, every CY factorization has minimal length, and its filling minimizes χand b_2. An application to Baykur's recent examples gives one planar contact 3-manifold with infinitely many non-homeomorphic CY fillings. The cover there is of degree \ge 6. At degree 4 a single CY filling forces every filling to be CY, and the filling is unique in a certain case; at degree \le 3 the filling is unique and is CY under a mild condition. Admissible cuts of the covering disk decompose the openbook into subopenbooks. Every positive factorization then localizes to the pieces, and the CY condition holds exactly when it holds locally. This makes the state space a direct sum of tensor products indexed by the compatible parity choices with at most two qubits in each factor when the pieces have degree \le 4, which is also the range in which the CY condition depends only on the monodromy. For a particular degree 4 cover, the points, lines and flags of the two-qubit doily are realized by the system of its subopenbooks. Fifteen liftable braids there share the same quantum operation and the same Stein filling, and are separated only by which subopenbook systems they admit. A choice of tensor-product structure is thus carried by the lift and not by the braid group representation, which suggests a link between contact topology and quantum entanglement.