Unknotting Number
المؤلف:
Adams, C. C.
المصدر:
The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman
الجزء والصفحة:
...
15-6-2021
5386
Unknotting Number
The smallest number of times 
a knot 
must be passed through itself to untie it. Lower bounds can be computed using relatively straightforward techniques, but it is in general difficult to determine exact values. Many unknotting numbers can be determined from a knot's knot signature. A knot with unknotting number 1 is a prime knot (Scharlemann 1985). It is not always true that the unknotting number is achieved in a projection with the minimal number of crossings.
The following table is from Kirby (1997, pp. 88-89), with the values for 10-139 and 10-152 taken from Kawamura (1998). In the following table, Kirby's (1997, p. 88) value 
has been corrected to reflect the fact that 
is only currently known to be 1 or 2 (Kawauchi 1996, p. 271). The value 
has been computed by Stoimenow (2002). The unknotting numbers for 10-154 and 10-161 can be found using the slice-Bennequin inequality (Stoimenow 1998).
Knots for which the unknotting number is not known are 10-11, 10-47, 10-51, 10-54, 10-61, 10-76, 10-77, 10-79, 10-100 (Cha and Livingston 2008).
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0 |
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2 |
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2 |
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3 |
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2 |
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3 |
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2 |
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2 |
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1 |
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1 |
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1 |
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1 |
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1 |
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2 |
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2 |
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2 |
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2 |
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2 |
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1 |
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2 |
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1 |
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2 |
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2 |
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2 |
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2 |
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3 |
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2 |
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2 |
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3 |
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1 |
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1 |
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2 |
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2 |
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2 |
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1 |
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1 |
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1 |
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1 |
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2 |
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1 |
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2 |
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2 |
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? |
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2 |
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2 |
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1 |
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1 |
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1 |
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? |
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2 |
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1 |
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3 |
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1 |
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3 |
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1 |
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4 |
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2 |
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2 |
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1 |
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2 |
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1 |
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1 |
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2 |
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1 |
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1 |
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2 |
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2 |
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2 |
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1 |
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3 |
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1 |
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2 |
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3 |
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3 |
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1 |
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2 |
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1 |
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2 |
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1 |
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3 |
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1 |
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1 |
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2 |
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1 |
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2 |
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2 |
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2 |
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2 |
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2 |
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2 |
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2 |
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2 |
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3 |
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2 |
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3 |
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? |
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2 |
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1 |
|
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2 |
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3 |
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2 |
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1 |
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? |
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? |
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1 |
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1 |
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2 |
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2 |
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2 |
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1 |
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2 |
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2 |
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2 |
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2 |
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1 |
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2 |
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3 |
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2 |
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3 |
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? |
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2 |
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4 |
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1 |
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3 |
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1 |
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2 |
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2 |
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3 |
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2 |
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2 |
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1 |
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3 |
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2 |
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2 |
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? |
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2 |
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2 |
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1 |
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2 |
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2 |
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2 |
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2 |
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2 |
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1 |
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2 |
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3 |
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2 |
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1 |
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1 |
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1 |
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3 |
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2 |
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1 |
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1 |
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2 |
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3 |
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2 |
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2 |
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? |
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1 |
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1 |
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2 |
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|
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2 |
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1 |
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1 |
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2 |
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2 |
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2 |
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2 |
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2 |
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2 |
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2 |
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1 |
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1 |
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2 |
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2 |
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2 |
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1 |
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1 |
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3 |
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2 |
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1 |
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2 |
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2 |
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2 |
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1 |
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2 |
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2 |
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2 |
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2 |
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2 |
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1 |
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1 |
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2 |
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1 |
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2 |
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2 |
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2 |
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1 |
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2 |
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1 |
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2 |
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2 |
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1 |
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3 |
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1 |
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1 |
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2 |
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3 |
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2 |
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1 |
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2 |
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1 |
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1 |
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? |
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1 |
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2 |
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2 |
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2 |
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1 |
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3 |
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1 |
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2 |
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2 |
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2 |
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4 |
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1 |
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1 |
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2 |
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1 |
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2 |
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2 |
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2 |
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2 |
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1 |
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2 |
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2 |
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2 |
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2 |
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2 |
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4 |
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3 |
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2 |
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1 |
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2 |
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2 |
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2 |
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1 |
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2 |
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2 |
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REFERENCES:
Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 57-64, 1994.
Cha, J. C. and Livingston, C. "Unknown Values in the Table of Knots." 2008 May 16. https://arxiv.org/abs/math.GT/0503125.
Cipra, B. "From Knot to Unknot." What's Happening in the Mathematical Sciences, Vol. 2. Providence, RI: Amer. Math. Soc., pp. 8-13, 1994.
Kawamura, T. "The Unknotting Numbers of 
and 
Are 4." Osaka J. Math. 35, 539-546, 1998.
Kawauchi, A. "Knot Invariants." Appendix F.3 in A Survey of Knot Theory. Boston: Birkhäuser, 1996.
Kirby, R. (Ed.). "Problems in Low-Dimensional Topology." AMS/IP Stud. Adv. Math., 2.2, Geometric Topology (Athens, GA, 1993). Providence, RI: Amer. Math. Soc., pp. 35-473, 1997.
Scharlemann, M. "Unknotting Number One Knots Are Prime." Invent. Math. 82, 37-55, 1985.
Stoimenow, A. "Polynomial Values, the Linking Form, and Unknotting Numbers." https://www.math.toronto.edu/stoimeno/goer.ps.gz. Feb. 10, 2002.
Stoimenow, A. "Positive Knots, Closed Braids and the Jones Polynomial." https://www.math.toronto.edu/stoimeno/pos.ps.gz. Mar. 2, 2002.
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