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FIBERED KNOTS AND PROPERTY 2R MARTIN SCHARLEMANN AND ABIGAIL THOMPSON
 

Summary: FIBERED KNOTS AND PROPERTY 2R
MARTIN SCHARLEMANN AND ABIGAIL THOMPSON
Abstract. It is shown, using sutured manifold theory, that if
there are any 2-component counterexamples to the Generalized
Property R Conjecture, then any knot of least genus among com-
ponents of such counterexamples is not a fibered knot.
The general question of what fibered knots might appear as a
component of such a counterexample is further considered; much
can be said about the monodromy of the fiber, particularly in the
case in which the fiber is of genus two.
1. Introductory remarks
Recall the famous Property R theorem, proven in a somewhat stronger
form by David Gabai [Ga2]:
thm:PropR Theorem 1.1 (Property R). If 0-framed surgery on a knot K S3
yields S1
S2
then K is the unknot.
There is a natural way of trying to generalize Theorem 1.1 to links
in S3
. In fact, there are several ways in which it can be generalized,

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics