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A free-group valued invariant of free knots
Preview AbstractThe aim of the present paper is to construct series of invariants of free knots (flat virtual knots, virtual knots) valued in free groups (and also free products of cyclic groups).
Homologically visible closed geodesics on complete surfaces
Preview AbstractIn this paper, we give multiple situations when having one or two geometrically distinct closed geodesics on a complete Riemannian cylinder, a complete Möbius band or a complete Riemannian plane leads to having infinitely many geometrically distinct closed geodesics. In particular, we prove that any complete cylinder with isolated closed geodesics has zero, one or infinitely many homologically visible closed geodesics; this answers a question of Alberto Abbondandolo.