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RESEARCH BLOG 5/6/03 I haven't heard much about Perelman's lectures at Stoneybrook, only
 

Summary: RESEARCH BLOG 5/6/03
I haven't heard much about Perelman's lectures at Stoneybrook, only
that his work he's written so far seems to be holding up well, and he is
not discussing the collapsing part of his argument which is unwritten,
but is the type of thing on which he is an expert.
At the U. of Arkansas Spring Lecture Series, Marty Scharlemann
spoke about his solution to the Goda-Teragaito conjecture, which gives
a complete classification of tunnel number one, genus one knots in S3
.
That is, knots which have a genus 2 Heegaard decomposition, such that
the knot bounds a punctured torus. There are satellite examples, which
were classified by Eudave-Munoz and by Morimoto-Sakuma, as well as
two-bridge knots which are plumbings of unknotted annuli. His argu-
ment makes much use of thin position for graphs in S3
and other tech-
niques developed by Scharlemann-Thompson, and thus seems rather
special to knots. But it would be interesting to obtain a better under-
standing of tunnel number one knots and links in general. Marc Lack-
enby has classified tunnel number one alternating knots, using almost
normal surfaces. I reckon his arguments could be generalize to classify

  

Source: Agol, Ian - Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

 

Collections: Mathematics