9,258 research outputs found
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
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