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Topology Vol. 27, No. 4, pp. 519-540. 1988 Pnnted in Great Bnram
 

Summary: Topology Vol. 27, No. 4, pp. 519-540. 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 half-twisted 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

  

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

 

Collections: Mathematics