 
Summary: RESEARCH BLOG 9/27/04
GEODESIC SURFACES
Chris Leininger posted a paper to the ArXiv which shows that there
is a sequence of hyperbolic knots in S3
which have closed embedded
surfaces, such that the surface may be taken to be arbitrarily close to
being geodesic as the sequence approaches . Menasco and Reid have
conjectured that there is no knot in S3
which has a totally geodesic
embedded closed surface in its complement. In fact, Leininger finds
a hyperbolic link in S3
which contains a geodesic surface in its com
plement, such that one may perform infinitely many Dehn fillings on
the link to obtain a sequence of knots, which immediately implies his
theorem. Such a link has a sublink (with one fewer components) which
is the boundary of a surface in S3
whose components are disks and an
nuli. A theorem of Gordon implies that this is the only way one could
possibly obtain an infinite collection of hyperbolic knots by surgery on
a fixed link. Recently, Adams and Schoenfeld have shown that their
