Core curves of triangulated solid tori

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We show that in any triangulation of a solid torus, there is a pre-core curve that lies in the 2-skeleton and that intersects the interior of each face in at most 10 straight arcs. By definition, a pre-core curve is a simple closed curve that becomes a core curve when a collar is attached to the boundary of the solid torus. This theorem imposes restrictions on the possible Riemannian metrics on a solid torus. It also has applications in knot theory.