Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 98,Number 1, September 1986
A REMARK ON COMPANIONSHIP AND PROPERTY P

MARTIN SCHARLEMANN

ABSTRACT. If some nontrivial Dehn surgery on 7 yields a homotopy bsphere
and n is a companion of 7, then some nontrivial Dehn surgery on n yields a
homotopy bsphere.
DEFINITION. A knot n is a companion of a knot 7 in S3 if lies in a tubular
neighborhood %(n)of n, but not in any 3-cell contained in %(n). A knot is
simple if its only companions are itself and the unknot.
In [Go] it is shown that if some nontrivial Dehn surgery on a nontrivial knot
yields a homotopy 3-sphere, then some nontrivial Dehn surgery on a nontrivial
simple knot yields a homotopy 3-sphere. A more natural stronger statement is the
following:
THEOREM If some nontrivial Dehn surgery on 7 yields a homotopy 3-sphere1.
and n is a companion of 7, then some nontrivial Dehn surgery on n yields a homo-
topy 3-sphere.

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

Collections: Mathematics