For a fixed surface S, one can ask if there are conditions on open books with pages S that imply maximality of the Euler characteristic of S among all pages of open books encoding the same 3-manifold (or at least imply maximality among those open books that encode the same 3-manifold and support the same contact structure). In this short note we propose an explicit variant of this question with a condition that involves the amount of twisting of the monodromy and the topological type of S, and we construct examples of open books for 3-manifolds that support our choice of condition. In particular, our examples show that the condition on twisting necessarily depends on the topological type of S. We find these examples of open books as the double branched covers of families of closed braids studied by Malyutin and Netsvetaev.

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