• Source: Unknotting number

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


{\displaystyle n}

, then there exists a diagram of the knot which can be changed to unknot by switching



n


{\displaystyle n}

crossings. The unknotting number of a knot is always less than half of its crossing number. This invariant was first defined by Hilmar Wendt in 1936.
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
)


{\displaystyle (p,q)}

-torus knot is equal to



(
p
−
1
)
(
q
−
1
)

/

2


{\displaystyle (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




References






External links


"Three_Dimensional_Invariants#Unknotting_Number", The Knot Atlas.

Kata Kunci Pencarian:

• Daftar masalah matematika yang belum terpecahkan
• Unknotting number
• Unknotting
• Unknot
• Unknotting problem
• Alexander polynomial
• Crossing number (knot theory)
• List of knot theory topics
• Linking number
• Bridge number
• Martin Scharlemann