 
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 GodaTeragaito 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 EudaveMunoz and by MorimotoSakuma, as well as
twobridge 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 ScharlemannThompson, 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
