Unknotting number

Trefoil knot without 3-fold symmetry being unknotted by one crossing switch.

Whitehead link being unknotted by undoing one crossing

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 ${\displaystyle n}$, then there exists a diagram of the knot which can be changed to unknot by switching ${\displaystyle n}$ 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 unknotting number is not additive under connected sum, ^[4] although that possibility, implicit in [Wendt,1937 ^[3] ] and explicitly asked by Gordon in 1977 ^[5] and many others, was not resolved until 2025.

The following table show the unknotting numbers for the first few knots:

• 
  Trefoil knot
  unknotting number 1
• 
  Figure-eight knot
  unknotting number 1
• 
  Cinquefoil knot
  unknotting number 2
• 
  Three-twist knot
  unknotting number 1
• 
  Stevedore knot
  unknotting number 1
• 
  6₂ knot
  unknotting number 1
• 
  6₃ knot
  unknotting number 1
• 
  7₁ knot
  unknotting number 3

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 ${\displaystyle (p,q)}$-torus knot is equal to ${\displaystyle (p-1)(q-1)/2}$. ^[6]
• The unknotting numbers of prime knots with nine or fewer crossings have all been determined. ^[7] (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

References

1. Adams, Colin Conrad (2004). The knot book: an elementary introduction to the mathematical theory of knots. Providence, Rhode Island: American Mathematical Society. p. 56. ISBN   0-8218-3678-1.
2. Taniyama, Kouki (2009), "Unknotting numbers of diagrams of a given nontrivial knot are unbounded", Journal of Knot Theory and Its Ramifications, 18 (8): 1049–1063, arXiv: 0805.3174 , doi:10.1142/S0218216509007361, MR   2554334 .
3. Wendt, Hilmar (December 1937). "Die gordische Auflösung von Knoten". Mathematische Zeitschrift. 42 (1): 680–696. doi:10.1007/BF01160103.
4. Brittenham, Mark; Hermiller, Susan (2025). "Unknotting number is not additive under connected sum". arXiv: 2506.24088 [math.GT].
5. Gordon, C. M. (1978). "Some aspects of classical knot theory". In Hausmann, Jean-Claude (ed.). Knot Theory. Vol. 685. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 1–60. doi:10.1007/bfb0062968. ISBN   978-3-540-08952-0. MR   0521730 . Retrieved 2025-09-14.This volume is dedicated to the memory of Christos Demetriou Papakyriakopoulos, 1914–1976.
6. Weisstein, Eric W. "Torus Knot". MathWorld .${\displaystyle {\frac {1}{2}}(p-1)(q-1)}$".
7. Weisstein, Eric W. "Unknotting Number". MathWorld .

External links

• " Three_Dimensional_Invariants#Unknotting_Number ", The Knot Atlas .