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KNOTS OBTAINED BY TWISTING UNKNOTS MOHAMED AIT-NOUH, DANIEL MATIGNON AND KIMIHIKO MOTEGI
 

Summary: KNOTS OBTAINED BY TWISTING UNKNOTS
MOHAMED AĻIT-NOUH, DANIEL MATIGNON AND KIMIHIKO MOTEGI
Abstract. Let K be the unknot in the 3-sphere S3
, and D a disk in S3
meeting K transversely in the interior, at least twice (after all isotopies).
We denote by KD,n a knot obtained from K by n twistings along the
disk D. We describe for which pairs (K, D) and integers n, KD,n is a
torus knot, a satellite knot or a hyperbolic knot.
1. Twisted knots
1.1. Definitions. Let k a knot in S3, and D a disk such that k intersects D
transversely in its interior at least twice, after all isotopies of k in S3 - D.
Let kD,n be the new knot obtained from k by performing n Dehn twists
along D; n = 1 on the figure below.
D
k
1-twist
along D
kD,1
By Ohyama [16], each knot can be obtained from a trivial knot by twisting
along at most two properly chosen disks.

  

Source: Ait Nouh, Mohamed - Mathematics Department, California State University Channel Islands

 

Collections: Mathematics