Planar Contact Structures with Calabi-Yau Fillings and Topological Quantum Computation

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