Séminaire de Géométrie Tropicale
1er Février 2012, 14h00 salle 1525-502
Sergei Duzhin
(Steklov Mathematical Institute)
Solution to Przytycki's conjecture on paired diagrams
Résumé :
A conjecture stated in 1987
asserts that there exists a knot whose diagram cannot
be presented in the "matched form", that is,
in such a way that its crossings are divided into
pairs of the simplest possible form (as a twice twisted braid).
It is said that John Conway told Prsytycki that he had an example
of such a knot in 1991,
but he did not provide any example.
In June 2011, my student
M.Shkolnikov and myself proved that, e.g., the pretzel knot P(3,3,-3)
does not have such a diagram. The proof is based on the construction of a
special Seifert surface of the knot and on the properties of Alexander
ideals.