 
Summary: Topology and its Applications 18 (1984) 235258
NorthHollend
235
TUNNEL NUMBER ONE KNOTS SATISFY THE
POENARU CONJECTURE
Martin SCHARLEMANN*
Department of Mathematics, University of Ca/(fornia, Santa Barbara, CA 93106, USA
Received 15 February 1984
It is shown that tunnel number one knots satisfy the Poenaru conjecture and so have Property
R. As a sidelight they are also shown to be doubly prime.
AMS Subj. Class.: Primary 57M25
tunnel number Poenaru conjecture
knots Property R
doubly prime
Introduction
The tunnel number of a PL knot K is the minimum number of PL onecells which
must be attached in order that the regular neighborhood of the resulting complex
has complement a handlebody [2]. It is easy to see that nbridge knots have tunnel
numbers (n  1) and torus knots have tunnel number one. More difficult is finding
knots which have higher tunnel number. If a knot has tunnel number one, it will
