Unknotting number explained

In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself (crossing switch) to untie it. If a knot has unknotting number

n , then there exists a diagram of the knot which can be changed to unknot by switching n crossings.^[1] The unknotting number of a knot is always less than half of its crossing number.^[2] This invariant was first defined by Hilmar Wendt in 1936.^[3]

Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. The following table show the unknotting numbers for the first few knots:In general, it is relatively difficult to determine the unknotting number of a given knot. Known cases include:

• The unknotting number of a nontrivial twist knot is always equal to one.
• The unknotting number of a

(p,q) -torus knot is equal to (p -1 )(q -1 )/ 2 .^[4]

• The unknotting numbers of prime knots with nine or fewer crossings have all been determined. (The unknotting number of the 10₁₁ prime knot is unknown.)

Other numerical knot invariants

• Crossing number
• Bridge number
• Linking number
• Stick number

See also

• Unknotting problem

Notes and References

1. Book: Adams, Colin Conrad . The knot book: an elementary introduction to the mathematical theory of knots . American Mathematical Society . Providence, Rhode Island . 2004 . 56 . 0-8218-3678-1.
2. .
3. Wendt . Hilmar . Die gordische Auflösung von Knoten . Mathematische Zeitschrift . December 1937 . 42 . 1 . 680–696 . 10.1007/BF01160103.
4. "Torus Knot", Mathworld.Wolfram.com. "(1)/(2) (p -1 )(q -1 ) ".