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RESEARCH BLOG 6/8/04 TEXAS TOPOLOGY
 

Summary: RESEARCH BLOG 6/8/04
TEXAS TOPOLOGY
Last week, I visited UT Austin. Ken Baker, a student of John Luecke,
had his thesis defence on Wednesday. He has a nice result about Berge
knots, and his thesis has an impressive number of pictures. These
are a special class of knots in S3
which admit a cyclic surgery. If
one considers a genus 2 Heegaard surface S, and a knot k S, then
one obtains an induced integral framing on k from S. This means
that a vector field in S normal to k gives a framing which may wind
a number of times around the meridian, but only once around the
longitude. If one performs Dehn surgery on k with framing induced
by S, then the resulting manifold is obtained by adding a 2-handle to
each genus 2 handlebody, and gluing the resulting manifolds together
along the new toroidal boundary. Berge realized that if the handle
is added along a curve which is not disk-busting in each handlebody
(that is, there is an essential disk whose boundary misses k in each
handlebody, in which case k is called primitive in the handlebody),
then the resulting manifold is a lens space (since handle addition creates
two solid tori which are glued together). In this case, k is called doubly

  

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

 

Collections: Mathematics