Unknotting number

{{Short description|Minimum number of times a specific knot must be passed through itself to become untied}}

Image:Unknotting trefoil.svg

File:Unknotting Whitehead link.svg 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 $n$, then there exists a diagram of the knot which can be changed to unknot by switching $n$ crossings.{{cite book |author=Adams, Colin Conrad |title=The knot book: an elementary introduction to the mathematical theory of knots |publisher=American Mathematical Society |location=Providence, Rhode Island |year=2004 |pages=56 |isbn=0-8218-3678-1}} The unknotting number of a knot is always less than half of its crossing number.{{citation

| last = Taniyama | first = Kouki

| doi = 10.1142/S0218216509007361

| issue = 8

| journal = Journal of Knot Theory and its Ramifications

| mr = 2554334

| pages = 1049–1063

| title = Unknotting numbers of diagrams of a given nontrivial knot are unbounded

| volume = 18

| year = 2009| arxiv = 0805.3174

}}. This invariant was first defined by Hilmar Wendt in 1936.{{cite journal

|last1=Wendt

|first1=Hilmar

|title=Die gordische Auflösung von Knoten

|journal=Mathematische Zeitschrift

|date=December 1937

|volume=42

|issue=1

|pages=680–696

|doi=10.1007/BF01160103}}

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:

Image:Blue Trefoil Knot.png|Trefoil knot
unknotting number 1

Image:Blue Figure-Eight Knot.png|Figure-eight knot
unknotting number 1

Image:Blue Cinquefoil Knot.png|Cinquefoil knot
unknotting number 2

Image:Blue Three-Twist Knot.png|Three-twist knot
unknotting number 1

Image:Blue Stevedore Knot.png|Stevedore knot
unknotting number 1

Image:Blue 6_2 Knot.png|6₂ knot
unknotting number 1

Image:Blue 6_3 Knot.png|6₃ knot
unknotting number 1

Image:Blue 7_1 Knot.png|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 $(p-1)(q-1)/2$."[http://mathworld.wolfram.com/TorusKnot.html Torus Knot]", Mathworld.Wolfram.com. "$\backslash frac\{1\}\{2\}(p-1)(q-1)$".
• The unknotting numbers of prime knots with nine or fewer crossings have all been determined.{{MathWorld|title=Unknotting Number|urlname=UnknottingNumber}} (The unknotting number of the 10₁₁ prime knot is unknown.)