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Topology and its Applications 18 (1984) 235-258 North-Hollend
 

Summary: Topology and its Applications 18 (1984) 235-258
North-Hollend
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 one-cells 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 n-bridge 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

  

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

 

Collections: Mathematics