The twisting number of a ribbon knot is bounded below by its doubly slice genus

Abstract

The twisting number of a ribbon knot $K$ is the minimal number of tangle replacements on the symmetry axis of $J\#-J$ for any knot $J$ that is required to produce a symmetric union diagram of $K$. We prove that the twisting number is bounded below by the doubly slice genus and produce examples of ribbon knots with arbitrarily high twisting number, addressing a problem of Tanaka. As an application, we determine hitherto unknown doubly slice genera of some knots with 12 crossings.

v2: Added references, including that to a result by Boden--Elmacioglu--Guha--Karimi--Rushworth--Tang--Wang Peng Jun that implies Proposition 2; 9 pages, 5 figures