 
Summary: RESEARCH BLOG 8/31/04
John Berge wrote me mentioning that there was a mistake in blog
6/8/04, in the definition of a primitive curve in the boundary of a han
dlebody. A primitive curve is supposed to represent a conjugacy class
of a generator in the free fundamental group of the handlebody. Thus,
the correct definition is that the curve is primitive if there is a merid
ian disk for the handlebody which intersects the curve exactly once. I
mistakenly said that, for a curve on the boundary of a genus 2 han
dlebody, a primitive curve is defined as being disjoint from a meridian
disk (which is only a necessary condition). In fact, if this holds, then
the curve will be a multiple of a generator in the fundamental group
of the handlebody (this is special to the genus 2 case). Maybe in this
case, this condition should be called multiprimitive. Adding a handle
along a multiprimitive curve results in a connect sum of a lens space
and a solid torus (where we allow S3
and S2
× S1
to be lens spaces).
If a knot in a genus 2 Heegaard surface of a manifold is multiprimitive
in each handlebody, then surgery along the framing induced from the
