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Pseudo-Anosov homeomorphisms not arising from branched covers

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 Publication date 2017
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In this paper we provide a negative answer to a question of Farb about the relation between the algebraic degree of the stretch factor of a pseudo-Anosov homeomorphism and the genus of the surface on which it is defined.



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By using double branched covers, we prove that there is a 1-1 correspondence between the set of knotoids in the 2-sphere, up to orientation reversion and rotation, and knots with a strong inversion, up to conjugacy. This correspondence allows us to study knotoids through tools and invariants coming from knot theory. In particular, concepts from geometrisation generalise to knotoids, allowing us to characterise invertibility and other properties in the hyperbolic case. Moreover, with our construction we are able to detect both the trivial knotoid in the 2-sphere and the trivial planar knotoid.
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