Knots Transverse to a Vector Field

We study knots transverse to a fixed vector field V on a 3-manifold M up to the corresponding isotopy relation. Such knots are equipped with a natural framing.  Motivated by questions in contact topology, it is natural to ask whether two V-transverse knots which are isotopic as framed knots and homotopic through V-transverse immersed curves must be isotopic through V-transverse knots.  When M is R^{^3} and V is the vertical vector field the answer is yes.  However, we construct examples which show the answer to this question can be no in other 3-manifolds, specifically S^{^1}-fibrations over surfaces of genus at least 2.  We also give a general classification of knots transverse to a vector field in an arbitrary closed oriented 3-manifold M.  We show this classification is particularly simple when V is the co-orienting vector field of a tight contact structure, or when M is irreducible and atoroidal. Lastly, we apply our results to study loose Legendrian knots in overtwisted contact manifolds, and generalize results of Dymara and Ding-Geiges.   This work is joint with Vladimir Chernov.