 
Summary: Topology Vol. 27, No. 4, pp. 519540. 1988
Pnnted in Great Bnram
oo4e9383'88 $3.00+.00
0 1988 Pergamon Press plc
A PROJECTIVE PLANE IN Iw4WITH THREE CRITICAL POINTS
IS STANDARD. STRONGLY INVERTIBLE KNOTS HAVE
PROPERTY P
STEVEN BLEILER and MARTIN ScmRLmfmN t
(Received in revisedform 9 February 1988)
LET P be the projective plane in [w4obtained by capping off the boundary of an unknotted
Miibius band in Iw3x (0) with an unknotted disk in Iw3x [O, co). Here we show that any
smoothly imbedded projective plane in [w4on which some projection Iw4+lRhas three non
degenerate critical points is isotopic to P. The proof is based on a combinatorial solution to
Problem 1.2B of [4]. In particular, if a band is attached to an unknot so that the result is an
unknot, then the band is isotopic to the trivial halftwisted band. One consequence is that
strongly invertible knots have property P (see [I]). Together with [23, this further implies
that pretzel knots (indeed all symmetric knots) have property P.
The solution of 1.2B uses the techniques of [S], [6] and [I], with careful distinction made
between the two sides of the planar surfaces used in those arguments. Here is the philosophy:
It was pointed out in [l] that the techniques of [S] and [6] were inadequate, because a
