Abstract
We consider a class of suspensions of diffeomorphisms of the annulus as flows in the orientable 3-manifoldT2x I. Using results of Birman & Williams (Topology22, 47‒82 (1983);Contemp. Math. 20, 1‒60 (1983)), we construct a knotholder or template that carries the set of periodic orbits of the flow. We define rotation numbers and show that any orbit of periodqand rotation numberp/qcan be arranged as a positive braid onpstrands. This yields existence and uniqueness results for families of resonant torus knots (p-braids that are (p,q)-torus knots of periodq>pwhich correspond to order-preserving (Birkhoff-) periodic orbits of the diffeomorphism. We show that all otherq-periodicp-braids have higher genus, and we establish bounds on the genera of such knots. We obtain existence and uniqueness results for a number of other, non-resonant, torus knots, including non-order-preserving (q+s,q)-torus knots of rotation number 1.
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