Minimum number of times a specific knot must be passed through itself to become untied
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 , then there exists a diagram of the knot which can be changed to unknot by switching 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:
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 -torus knot is equal to .[4]
The unknotting numbers of prime knots with nine or fewer crossings have all been determined.[5] (The unknotting number of the 1011 prime knot is unknown.)
^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.
^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.
^Wendt, Hilmar (December 1937). "Die gordische Auflösung von Knoten". Mathematische Zeitschrift. 42 (1): 680–696. doi:10.1007/BF01160103.
^"Torus Knot", Mathworld.Wolfram.com. "".
^Weisstein, Eric W. "Unknotting Number". MathWorld.
In the mathematical area of knot theory, the unknottingnumber of a knot is the minimum number of times the knot must be passed through itself (crossing...
Unknotting may refer to: Unknottingnumber, the minimum number of times the knot must be passed through itself to untie it Unknotting problem, a mathematical...
showed that the unknotting problem is in the complexity class NP. Hara, Tani & Yamamoto (2005) claimed the weaker result that unknotting is in AM ∩ co-AM;...
150. Kawauchi credits this result to Kondo, H. (1979), "Knots of unknottingnumber 1 and their Alexander polynomials", Osaka J. Math. 16: 551-559, and...
bridge number of K is one less than the sum of the bridge numbers of K1 and K2. Crossing number Linking number Stick numberUnknottingnumber Adams, Colin...
single curve is regular homotopic to a standard circle (any knot can be unknotted if the curve is allowed to pass through itself). The fact that it is homotopic...
half-twists (72 knot) Six half-twists (81 knot) All twist knots have unknottingnumber one, since the knot can be untied by unlinking the two ends. Every...
He gave the first proof of the classical theorem that knots with unknottingnumber one are prime. He used hard combinatorial arguments for this. Simpler...
The stick number is the minimal number of segments needed to represent a knot as a linkage, and a stuck unknot is a particular unknotted linkage that...
MR 2170571 Hass, Joel; Lagarias, Jeffrey C. (2001), "The number of Reidemeister moves needed for unknotting", Journal of the American Mathematical Society, 14...
crossing number of a knot, the minimum number of crossings in its diagrams. Chapter four discusses another invariant, the unknottingnumber, the minimum...
and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed. Most commonly...
last of Ravenel's conjectures in stable homotopy theory to be resolved. Unknotting problem: can unknots be recognized in polynomial time? Volume conjecture...
an oriented link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings. A direction is assigned to...
(Fig. 3). The range of reactions includes: DNA supercoil relaxation, unknotting of single-stranded circles, and decatenation, provided at least one partner...
mathematical theory of knots, the stick number is a knot invariant that intuitively gives the smallest number of straight "sticks" stuck end to end needed...
a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface...
bubble conjecture, for proving that the unknotting problem is in NP, and for giving an exponential bound on the number of Reidemeister moves needed to reduce...
is (Hass 1998). The special case of recognizing the unknot, called the unknotting problem, is of particular interest (Hoste 2005). In February 2021 Marc...
alternating if it has an alternating diagram. Many of the knots with crossing number less than 10 are alternating. This fact and useful properties of alternating...
been removed. Tait conjectured that in certain circumstances, crossing number was a knot invariant, specifically: Any reduced diagram of an alternating...
trefoil is the first nontrivial knot, and is the only knot with crossing number three. It is a prime knot, and is listed as 31 in the Alexander-Briggs notation...
a matter of days. Knot theory Knot (mathematics) List of prime knots Unknotting problem Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeff (1998), "The first...
Global Information
Trefoil knot
Figure-eight knot
Cinquefoil knot
Three-twist knot
Stevedore knot
6₂ knot
6₃ knot
7₁ knot
