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Topology and its Applications 146147 (2005) 105121 www.elsevier.com/locate/topol

Summary: Topology and its Applications 146­147 (2005) 105­121
Obtaining graph knots by twisting unknots
Mohamed Aït Nouh a
, Daniel Matignon b
, Kimihiko Motegi c,,1
a Department of Mathematics, University of California at Santa Barbara, CA 93106, USA
b CMI, Université de Provence, 39, rue Joliot Curie, F-13453 Marseille cedex 13, France
c Department of Mathematics, Nihon University, Tokyo 156-8550, Japan
Received 24 October 2002; received in revised form 27 November 2002; accepted 5 February 2003
Let K be a knot in the 3-sphere S3 and D a disk in S3 meeting K transversely more than once in
the interior. For nontriviality we assume that |D K| 2 over all isotopies of K in S3 - D. Let
KD,n ( S3) be a knot obtained from K by n twisting along the disk D. We prove that if K is a
trivial knot and KD,n is a graph knot, then |n| 1 or K and D form a special pair which we call an
"exceptional pair". As a corollary, if (K,D) is not an exceptional pair, then by twisting unknot K
more than once (in the positive or the negative direction) along the disk D, we always obtain a knot
with positive Gromov volume. We will also show that there are infinitely many graph knots each of
which is obtained from a trivial knot by twisting, but its companion knot cannot be obtained in such
a manner.


Source: Ait Nouh, Mohamed - Mathematics Department, California State University Channel Islands
Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics