2 00 6 New lower bound for the distortion of a knotted curve

We prove that distortion of a knotted curve in R is great than 4.76. This improves a result obtained by John M. Sullivan and Elizabeth Denne in [DS]. 1 The definition and results M. Gromov introduced the notion of distortion for submanifolds of R in [G], p.113-114. In the case of a simple closed curve K (closed 1-dimensional submanifold of R) distortion is U(K) = max P,Q∈K |PQ|K |PQ| , where |PQ| is length of the secant PQ, and |PQ|K is length of the shortest arc in K joining P and Q . M. Gromov showed that any closed curve has distortion at least π/2. John M. Sullivan and Elizabeth Denne showed in [DS] that each knotted curve (=a curve represented a nontrivial knot) has distortion > 3.99. (The reader can use the paper [DS] as an introduction to the topic.) To obtain this bound they use a special set of “essential secants” (see Definition 2.2). Using similar arguments we obtain the following Theorem 4.5 Each knotted curve has distortion > 4.76. 2 Background Let K ⊂ R be an oriented closed 1-dimensional submanifold. By γPQ we denote the arc in K joining points P,Q ∈ K following the orientation of K. ∗The second author was partially supported by RFBR grant 05-01-00939a.