Unknotting number

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

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. The unknotting number of a knot is always less than half of its crossing number.

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:

• 
  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 $(p,q)$-torus knot is equal to $(p-1)(q-1)/2$.
• 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

External links

• "Three_Dimensional_Invariants#Unknotting_Number", The Knot Atlas.