Symmetric union presentations for 2-bridge ribbon knots

Abstract

Symmetric unions have been defined as generalizations of Kinoshita-Terasaka's construction in 1957. They are given by diagrams which look like the connected sum of a knot and its mirror image with additional twist tangles inserted near the symmetry axis. Because all symmetric unions are ribbon knots, we can ask how big a subfamily of ribbon knots they form. It is known that all 21 ribbon knots with crossing number less or equal 10 are symmetric unions. In this talk we extend our knowledge about symmetric unions: we prove that the family of symmetric unions contains all known 2-bridge ribbon knots. The question, however, whether the three families of 2-bridge ribbon knots, found by Casson and Gordon in 1974, are a complete list of all 2-bridge ribbon knots, is still open.Comment: 13 pages (notes for a talk at the Joint Meeting of AMS and DMV at Mainz, 2005-06-18