Big Dreams for Small Creatures: Ilana and Eugene Rosenberg’s path to the Hologenome Theory

A biographical sketch of the Hologenome Theory

Symmetric union presentations for 2-bridge ribbon knots

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

The search for non-symmetric ribbon knots

We present the results of Axel Seeliger's tabulation of symmetric union presentations for ribbon knots with crossing numbers 11 and 12 and exhibit possible examples for ribbon knots which are not representable as symmetric unions. In addition, we give a complete atlas of band diagrams for prime ribbon knots with 11 and 12 crossings.Comment: Version 2, to appear in Experimental Mathematics. Changes with respect to Version 1: fix of three typos in the Appendix (a minus sign was missing for 11a316 and the parameters for 12a473 and 12a887 needed to be interchanged), includes Piccirillo's result, enhancement of Paragraph 2.2, some further improvements (style and notation