Lorenz knots, T–links and twisted torus links

A Lorenz knot is the isotopy class of any periodic orbit in the flow on R 3 given by the Lorenz differential equations. By studying them on the Lorenz template, Lorenz knots were shown to include certain known classes, but a complete description remained elusive. Twisted torus links are obtained by twisting a subset of strands of a torus link. We generalize these to a class of positive repeated twisted torus links, which we call T–links. We prove that T–links are exactly the links on the Lorenz template, so Lorenz knots and T–knots coincide. In addition, T-links are in one-one correspondence with certain minimal braid-index representatives. There are applications to both Lorenz knots and to twisted torus links: On one hand, we prove that any Lorenz knot can be embedded on the Lorenz template in at most finitely many ways, up to trivial stabilizations, and gain some insight into the number of such embeddings. On the other hand, we establish previously unnoticed symmetries of twisted torus links, identify precisely all the positive twisted torus links that are torus links, and prove certain useful braid index formulas.