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In the branch of mathematics called knot theory, the volume conjecture is an open problem that relates quantum invariants of knots to the hyperbolic geometry of their complements.

Let O denote the unknot. For any knot ${\displaystyle K}$, let ${\displaystyle \langle K\rangle _{N}}$ be the Kashaev invariant of ${\displaystyle K}$, which may be defined as

${\displaystyle \langle K\rangle _{N}=\lim _{q\to e^{2\pi i/N}}{\frac {J_{K,N}(q)}{J_{O,N}(q)}}}$,

where ${\displaystyle J_{K,N}(q)}$ is the ${\displaystyle N}$–Colored Jones polynomial of ${\displaystyle K}$. The volume conjecture states that^[1]

${\displaystyle \lim _{N\to \infty }{\frac {2\pi \log |\langle K\rangle _{N}|}{N}}=\operatorname {vol} (S^{3}\backslash K)}$,

where ${\displaystyle \operatorname {vol} (S^{3}\backslash K)}$ is the simplicial volume of the complement of ${\displaystyle K}$ in the 3-sphere, defined as follows. By the JSJ decomposition, the complement ${\displaystyle S^{3}\backslash K}$ may be uniquely decomposed into a system of tori

${\displaystyle S^{3}\backslash K=\left(\bigsqcup _{i}H_{i}\right)\sqcup \left(\bigsqcup _{j}E_{j}\right)}$

with ${\displaystyle H_{i}}$ hyperbolic and ${\displaystyle E_{j}}$ Seifert-fibered. The simplicial volume ${\displaystyle \operatorname {vol} (S^{3}\backslash K)}$ is then defined as the sum

${\displaystyle \operatorname {vol} (S^{3}\backslash K)=\sum _{i}\operatorname {vol} (H_{i})}$,

where ${\displaystyle \operatorname {vol} (H_{i})}$ is the hyperbolic volume of the hyperbolic manifold ${\displaystyle H_{i}}$.^[1]

As a special case, if ${\displaystyle K}$ is a hyperbolic knot, then the JSJ decomposition simply reads ${\displaystyle S^{3}\backslash K=H_{1}}$, and by definition the simplicial volume ${\displaystyle \operatorname {vol} (S^{3}\backslash K)}$ agrees with the hyperbolic volume ${\displaystyle \operatorname {vol} (H_{1})}$.

The Kashaev invariant was first introduced by Rinat M. Kashaev in 1994 and 1995 for hyperbolic links as a state sum using the theory of quantum dilogarithms.^[2][3] Kashaev stated the formula of the volume conjecture in the case of hyperbolic knots in 1997.^[4]

Murakami & Murakami (2001) pointed out that the Kashaev invariant is related to the colored Jones polynomial by replacing the variable ${\displaystyle q}$ with the root of unity ${\displaystyle e^{i\pi /N}}$. They used an R-matrix as the discrete Fourier transform for the equivalence of these two descriptions. This paper was the first to state the volume conjecture in its modern form using the simplicial volume. They also prove that the volume conjecture implies the following conjecture of Victor Vasiliev:

If all Vassiliev invariants of a knot agree with those of the unknot, then the knot is the unknot.

The key observation in their proof is that if every Vassiliev invariant of a knot ${\displaystyle K}$ is trivial, then ${\displaystyle J_{K,N}(q)=1}$ for any ${\displaystyle N}$.

The volume conjecture is open for general knots, and it is known to be false for arbitrary links. The volume conjecture has been verified in many special cases, including:

• The figure-eight knot (Tobias Ekholm),^[5]
• The three-twist knot (Rinat Kashaev and Yoshiyuki Yokota),^[5]
• The Borromean rings (Stavros Garoufalidis and Thang Le),^[5]
• Torus knots (Rinat Kashaev and Olav Tirkkonen),^[5]
• All knots and links with volume zero (Roland van der Veen),^[5]
• Twisted Whitehead links (Hao Zheng),^[6]
• Whitehead doubles of nontrivial torus knots ${\displaystyle T(p,q)}$ with ${\displaystyle q=2}$ (Hao Zheng).^[6]

Using complexification, Murakami et al. (2002) conjectured that for a hyperbolic knot ${\displaystyle K}$,

${\displaystyle \lim _{N\to \infty }{\frac {2\pi \log \langle K\rangle _{N}}{N}}=\operatorname {vol} (S^{3}\backslash K)+CS(S^{3}\backslash K)}$,

where ${\displaystyle CS}$ is the Chern–Simons invariant of the frame field of the hyperbolic structure of ${\displaystyle K}$. This suggests a relationship between the colored Jones polynomial and complexified Chern–Simons theory.

In knot theory, a virtual knot is a generalization of knots in 3-dimensional Euclidean space, R³, to knots in thickened surfaces ${\displaystyle \Sigma \times [0,1]}$ modulo an equivalence relation called stabilization/destabilization. Here ${\displaystyle \Sigma }$ is required to be closed and oriented. Virtual knots were first introduced by Kauffman (1999).

In the theory of classical knots, knots can be considered equivalence classes of knot diagrams under the Reidemeister moves. Likewise a virtual knot can be considered an equivalence of virtual knot diagrams that are equivalent under generalized Reidemeister moves. Virtual knots allow for the existence of, for example, knots whose Gauss codes which could not exist in 3-dimensional Euclidean space. A virtual knot diagram is a 4-valent planar graph, but each vertex is now allowed to be a classical crossing or a new type called virtual. The generalized moves show how to manipulate such diagrams to obtain an equivalent diagram; one move called the semi-virtual move involves both classical and virtual crossings, but all the other moves involve only one variety of crossing.

A classical knot can also be considered an equivalence class of Gauss diagrams under certain moves coming from the Reidemeister moves. Not all Gauss diagrams are realizable as knot diagrams, but by considering all equivalence classes of Gauss diagrams we obtain virtual knots.

A classical knot can be considered an ambient isotopy class of embeddings of the circle into a thickened 2-sphere. This can be generalized by considering such classes of embeddings into thickened higher-genus surfaces. This is not quite what we want since adding a handle to a (thick) surface will create a higher-genus embedding of the original knot. The adding of a handle is called stabilization and the reverse process destabilization. Thus a virtual knot can be considered an ambient isotopy class of embeddings of the circle into thickened surfaces with the equivalence given by (de)stabilization.

Some basic theorems relating classical and virtual knots:

• If two classical knots are equivalent as virtual knots, they are equivalent as classical knots.
• There is an algorithm to determine if a virtual knot is classical.
• There is an algorithm to determine if two virtual knots are equivalent.

There is a relation among the following.

• Virtual equivalence of virtual 1-knot diagrams, which is a set of virtual 1-knots.
• Welded equivalence of virtual 1-knot diagrams
• Rotational welded equivalence of virtual 1-knot diagrams
• Fiberwise equivalence of virtual 1-knot diagrams

• Boden, Hans; Nagel, Matthias (2017). “Concordance group of virtual knots”. Proceedings of the American Mathematical Society. 145 (12): 5451–5461. arXiv:1606.06404. doi:10.1090/proc/13667. S2CID 119139769.
• Carter, J. Scott; Kamada, Seiichi; Saito, Masahico (2002). “Stable equivalence of knots on surfaces and virtual knot cobordisms. Knots 2000 Korea, Vol. 1 (Yongpyong)”. J. Knot Theory Ramifications. 11 (3): 311–322.
• Carter, J. Scott; Silver, Daniel; Williams, Susan (2014). “Invariants of links in thickened surfaces”. Algebraic & Geometric Topology. 14 (3): 1377–1394. arXiv:1304.4655. doi:10.2140/agt.2014.14.1377. S2CID 53137201.
• Dye, Heather A (2016). An Invitation to Knot Theory : Virtual and Classical (First ed.). Chapman and Hall/CRC. ISBN 9781315370750.
• Goussarov, Mikhail; Polyak, Michael; Viro, Oleg (2000). “Finite-type invariants of classical and virtual knots”. Topology. 39 (5): 1045–1068. arXiv:math/9810073. doi:10.1016/S0040-9383(99)00054-3. S2CID 8871411.
• Kamada, Naoko; Kamda, Seiichi (2000). “Abstract link diagrams and virtual knots”. Journal of Knot Theory and Its Ramifications. 9 (1): 93–106. doi:10.1142/S0218216500000049.
• Kauffman, Louis H. (1999). “Virtual knot theory” (PDF). European Journal of Combinatorics. 20 (7): 663–690. doi:10.1006/eujc.1999.0314. ISSN 0195-6698. MR 1721925. S2CID 5993431.
• Kauffman, Louis H.; Manturov, Vassily Olegovich (2005). “Virtual Knots and Links”. arXiv:math.GT/0502014.
• Kuperberg, Greg (2003). “What is a virtual link?”. Algebraic & Geometric Topology. 3: 587–591. arXiv:math/0208039. doi:10.2140/agt.2003.3.587. S2CID 16803280.
• Manturov, Vassily (2004). Knot Theory. CRC Press. ISBN 978-0-415-31001-7.
• Manturov, Vassily Olegovich (2004). “Virtual knots and infinite dimensional Lie algebras”. Acta Applicandae Mathematicae. 83 (3): 221–233. doi:10.1023/B:ACAP.0000038944.29820.5e. S2CID 124019548.
• Turaev, Vladimir (2008). “Cobordism of knots on surfaces”. Journal of Topology. 1 (2): 285–305. arXiv:math/0703055. doi:10.1112/jtopol/jtn002. S2CID 17888102.

• A Table of Virtual Knots
• Elementary explanation with diagrams

The cutting of the Gordian Knot is an Ancient Greek legend associated with Alexander the Great in Gordium in Phrygia, regarding a complex knot that tied an oxcart. Reputedly, whoever could untie it would be destined to rule all of Asia. In 333 BC, Alexander was challenged to untie the knot. Instead of untangling it laboriously as everyone expected, he dramatically cut through it with his sword. This is used as a metaphor for inventing an unexpected method to solve a seemingly intractable problem.

The Phrygians had no king, but an oracle at Telmissus (the ancient capital of Lycia) decreed that the next man to enter the city driving an ox-cart should become king. A peasant farmer named Gordias drove into town on an ox-cart and was immediately declared king.^[a] Out of gratitude, his son Midas dedicated the ox-cart^[1] to the Phrygian god Sabazios (whom the Greeks identified with Zeus) and tied it to a post with an intricate knot of cornel bark (Cornus mas). The knot was later described by Roman historian Quintus Curtius Rufus as comprising “several knots all so tightly entangled that it was impossible to see how they were fastened”.^[2]

The ox-cart still stood in the palace of the former kings of Phrygia at Gordium in the fourth century BC when Alexander the Great arrived, at which point Phrygia had been reduced to a satrapy, or province, of the Persian Empire. An oracle had declared that any man who could unravel its elaborate knots was destined to rule over all of Asia.^[2] Alexander the Great wanted to untie the knot but struggled to do so before reasoning that it would make no difference how the knot was loosed. Sources from antiquity disagree on his solution. In one version of the story, he drew his sword and sliced it in half with a single stroke.^[2] However, Plutarch and Arrian relate that, according to Aristobulus,^[b] Alexander pulled the linchpin from the pole to which the yoke was fastened, exposing the two ends of the cord and allowing him to untie the knot without having to cut through it.^[3][4] Some classical scholars regard this as more plausible than the popular account.^[5] Literary sources of the story include Arrian (Anabasis Alexandri 2.3), Quintus Curtius (3.1.14), Justin’s epitome of Pompeius Trogus (11.7.3), and Aelian’s De Natura Animalium 13.1.^[6]

Alexander the Great later went on to conquer Asia as far as the Indus and the Oxus, thus partially fulfilling the prophecy.

The knot may have been a religious knot-cipher guarded by priests and priestesses. Robert Graves suggested that it may have symbolised the ineffable name of Dionysus that, knotted like a cipher, would have been passed on through generations of priests and revealed only to the kings of Phrygia.^[7]

Unlike popular fable, genuine mythology has few completely arbitrary elements. This myth taken as a whole seems designed to confer legitimacy to dynastic change in this central Anatolian kingdom: thus Alexander’s “brutal cutting of the knot … ended an ancient dispensation.”^[7]

The ox-cart suggests a longer voyage, rather than a local journey, perhaps linking Alexander the Great with an attested origin-myth in Macedon, of which Alexander is most likely to have been aware.^[8] Based on this origin myth, the new dynasty was not immemorially ancient, but had widely remembered origins in a local, but non-priestly “outsider” class, represented by Greek reports equally as an eponymous peasant^[9] or the locally attested, authentically Phrygian^[10] in his ox-cart. Roller (1984) separates out authentic Phrygian elements in the Greek reports and finds a folk-tale element and a religious one, linking the dynastic founder (with the cults of “Zeus” and Cybele).^[11]

Other Greek myths legitimize dynasties by right of conquest (compare Cadmus), but in this myth the stressed legitimising oracle suggests that the previous dynasty was a race of priest-kings allied to the unidentified oracular deity.

Informational notes

Citations

• Media related to Gordian Knot at Wikimedia Commons
• The dictionary definition of Gordian knot at Wiktionary

Vibration-proof hitch
Category	Hitch
Related	Clove hitch, Ground-line hitch, Snuggle hitch

The vibration-proof hitch is a knot used for fastening a line or rope to a solid object. This particular hitch is designed to tighten when subjected to vibration and functions best when the object is fairly large compared to the diameter of the rope. Knot expert Geoffrey Budworth credits the knot to Amory Bloch Lovins.^[1]

• List of knots

Good luck knot
Names	Good luck knot, Shamrock knot[1]
Category	Decorative
ABoK	2436

The Good luck knot^[a],^[2][3][4] also known as the Chrysanthemum Knot^[b][5] and One Mind Knot^[c],^[6] can be seen in images carved on a statue of the East Asian Goddess of Mercy, Guanyin, which was created between AD 557 and 588, and later found in a cave in northwest China.^[3]

• List of knots
• Chinese knotting

• Tying video

Versatackle knot

A versatackle^[1] is a self-locking tensioning structure implemented in cordage. It consists of two loops with the rope passed back and forth between them. It is functionally similar to the trucker’s hitch; however, unlike the trucker’s hitch, the versatackle is self-locking under tension.

The pressure, friction, and heat that may be generated by the running end moving through the loops can accelerate wear at the loops.

• 
  Make a loop knot on one end of the rope and another loop knot in the middle, just shorter than the area to be bound. (An overhand loop knot can be used here, but a Butterfly Knot works better because it doesn’t jam when strained and it’s easy to untie.)
• 
  Pass the second working end through the loop knot in the first working end.
• 
  Bring the second working end up through the loop knot tied in the middle of the rope.
• 
  Repeat until there are two or three complete passes (two or three ropes in each loop).
• 
  Pull on the second working end, and work out the slack to tighten.

• List of knots

• Versatackle knot

German Armed Forces Badge of Marksmanship
Schützenschnur
Type	Military Lanyard
Awarded for	Marksmanship
Description	Three classes: gold, silver and bronze
Presented by	the Federal Republic of Germany
Eligibility	Soldiers of the German armed forces and Allied nations
Status	Currently awarded
Established	27 January 1894 (historic) 16 July 1954 (current)

The German Armed Forces Badge for Weapons Proficiency (German: Schützenschnur) is a decoration of the Bundeswehr, the armed forces of the Federal Republic of Germany.

The decoration is awarded to German military personnel of all grades but is only allowed to be worn by enlisted members. The German armed forces regulations point out that “the Schützenschnur is a decoration for weapons proficiency for enlisted soldiers.” Officers can receive the award, although it is not currently authorized to be worn on their uniforms. Foreign military members may also be awarded the badge. The German military regulation on officers still applies, permitting only enlisted members to wear the badge.

The history of the Schützenschnur dates back to the Eighty Years’ War where Spanish troops were ordered to hang any Dutch person who carried a musket. Therefore Spanish musketeers began to carry ropes which were often carried over one shoulder.
Awarding a cord as a decoration began in the early 18th century in Prussia under Frederick William I of Prussia.

With the reorganization of the Prussian Army under Gerhard von Scharnhorst the Schützenschnur became an official military award.

The Reichswehr and later the Wehrmacht adapted the Schützenschnur as an award for proficiency in marksmanship. The award existed in 12 different levels with different versions for infantry and armored troops.^[1]

In 1957 the Bundesgrenzschutz introduced the Schützenschnur.

A similar decoration existed within the East German National People’s Army and the Border Troops of the German Democratic Republic.

To earn the award one must successfully shoot weapons from all three classes:

1. Pistol (current service pistol is the P8)
2. Rifle (there are several rifles in service; the standard rifle is the G36)
3. Heavy weapon (for example machine gun (MG3) or antitank launcher (Panzerfaust 3)

The awarded grade is determined by the lowest weapon qualification (e.g. if you qualify all gold and one bronze, you are awarded the bronze.).

Class 1 (Bronze)

Class 2 (Silver)

Class 3 (Gold)

• German Armed Forces Badge for Weapons Proficiency in Bronze (Schützenschnur in Bronze) is awarded for shooting with the rifle and the pistol and the machine pistol with at least two scores in at least bronze (medical service at least one score in at least bronze). The admissible categories of weapon depends on the branch of the service member. The category of heavy weapons (most commonly the machine gun) is not mandatory to earn the bronze badge. The rifle, pistol and machine pistol are the only weapons that require minimally score of bronze.
• German Armed Forces Badge for Weapons Proficiency in Silver (Schützenschnur in Silber) is awarded for shooting by a service member with his designated “light” weapon (pistol, rifle or machine pistol) and one of the “heavy” weapons (machine gun or Panzerfaust) with all scores at least in silver.
• German Armed Forces Badge for Weapons Proficiency in Gold (Schützenschnur in Gold) is awarded for shooting by a service member with his designated “light” weapon (pistol, rifle or machine pistol) and one of the “heavy” weapons (machine gun or Panzerfaust) with all scores at least in gold.

The number of exercises depends on the chosen (or ordered) weapon and the grade of the badge. A member of the medical branch, for example, can reach the bronze badge by two exercises with the pistol. A paratrooper needs for the gold badge one exercise with the G36 rifle (or three with the G3 rifle) AND two with the MG3 machine gun (or two with the Panzerfaust).

The German Armed Forces Badge for Weapons Proficiency in Gold is awarded with the number 5, 10, 15 etc. for annually retaking.

• Awards and decorations of the German Armed Forces
• German Armed Forces Badge for Military Proficiency
• Authorized foreign decorations of the United States military
• The Military Marching Badge (Norwegian Foot March)

Unlink
2-component unlink
Common name	Circle
Crossing no.	0
Linking no.	0
Stick no.	6
Unknotting no.	0
Conway notation	–
A–B notation	02 1
Dowker notation	–
Next	L2a1
Other
, tricolorable (if n>1)

In the mathematical field of knot theory, an unlink is a link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane.^[1]

The two-component unlink, consisting of two non-interlinked unknots, is the simplest possible unlink.

• An n-component link L ⊂ S³ is an unlink if and only if there exists n disjointly embedded discs D_i ⊂ S³ such that L = ∪_i∂D_i.
• A link with one component is an unlink if and only if it is the unknot.
• The link group of an n-component unlink is the free group on n generators, and is used in classifying Brunnian links.

• The Hopf link is a simple example of a link with two components that is not an unlink.
• The Borromean rings form a link with three components that is not an unlink; however, any two of the rings considered on their own do form a two-component unlink.
• Taizo Kanenobu has shown that for all n > 1 there exists a hyperbolic link of n components such that any proper sublink is an unlink (a Brunnian link). The Whitehead link and Borromean rings are such examples for n = 2, 3.^[1]

• Linking number

• Kawauchi, A. A Survey of Knot Theory. Birkhauser.

Garda hitch
Category	Hitch
Related	Italian/Munter hitch
Typical use	Rock climbing, Mountaineering

The Garda Hitch, also known as the Alpine Clutch, is a type of climbing knot that can only be moved in one direction. It is often used in climbing and mountaineering, such as in pulley systems to haul loads up a cliff. However, the Garda Hitch has some drawbacks, including being difficult to release under load, difficult to inspect, and adding significant friction to a pulley system. It can be challenging to determine which direction the rope will move freely and which direction it will lock just by looking at it. To tie a Garda Hitch, you need two similar carabiners, and it works best with two identical oval carabiners. While “D” carabiners can also be used, there is a risk of them unclipping.^[1]

Attach the two oval carabiners to a sling or cord with both of their gates facing up and out in the same direction.^[2] To use the Garda Hitch with “D” carabiners, attach them to a sling or cord so that they are side by side with the gates facing downward and outward in the same direction. Then, clip the rope through both gates and loop the right side of the rope through the left carabiner. This will create a simple twist around the two carabiners. By pulling on the rope ends, you can see that the hitch locks in one direction but moves freely in the other. It takes some practice to determine which end of the rope will move freely when pulled; this is the end that comes out between the carabiners, “spreading” them apart. The other end will pull the carabiners together, squeezing the rope between them and locking it in place.

• List of knots
• List of hitch knots

• http://www.chockstone.org/TechTips/GardaHitch.htm

In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algorithms. A major unresolved challenge is to determine if the problem admits a polynomial time algorithm; that is, whether the problem lies in the complexity class P.

First steps toward determining the computational complexity were undertaken in proving that the problem is
in larger complexity classes, which contain the class P. By using normal surfaces to describe the Seifert surfaces of a given knot, Hass, Lagarias & Pippenger (1999) 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; however, later they retracted this claim.^[1] In 2011, Greg Kuperberg proved that (assuming the generalized Riemann hypothesis) the unknotting problem is in co-NP,^[2] and in 2016, Marc Lackenby provided an unconditional proof of co-NP membership.^[3]

In 2021, Lackenby announced an unknot recognition algorithm which he claimed ran in quasi-polynomial time.^[4] As of October 2025, the result has not been published in the peer-reviewed literature.

The unknotting problem has the same computational complexity as testing whether an embedding of an undirected graph in Euclidean space is linkless.^[5]

Several algorithms solving the unknotting problem are based on Haken’s theory of normal surfaces:

• Haken’s algorithm uses the theory of normal surfaces to find a disk whose boundary is the knot. Haken originally used this algorithm to show that unknotting is decidable, but did not analyze its complexity in more detail.
• Hass, Lagarias, and Pippenger showed that the set of all normal surfaces may be represented by the integer points in a polyhedral cone and that a surface witnessing the unknottedness of a curve (if it exists) can always be found on one of the extreme rays of this cone. Therefore, vertex enumeration methods can be used to list all of the extreme rays and test whether any of them corresponds to a bounding disk of the knot. Hass, Lagarias, and Pippenger used this method to show that the unknottedness is in NP; later researchers such as Burton (2011a) refined their analysis, showing that this algorithm can be useful (though not polynomial time), with its complexity being a low-order singly-exponential function of the number of crossings.
• The algorithm of Birman & Hirsch (1998) uses braid foliations, a somewhat different type of structure than a normal surface. However to analyze its behavior they return to normal surface theory.

Other approaches include:

• The number of Reidemeister moves needed to change an unknot diagram to the standard unknot diagram is at most polynomial in the number of crossings.^[6] Therefore, a brute force search for all sequences of Reidemeister moves can detect unknottedness in exponential time.
• Similarly, any two triangulations of the same knot complement may be connected by a sequence of Pachner moves of length at most doubly exponential in the number of crossings.^[7] Therefore, it is possible to determine whether a knot is the unknot by testing all sequences of Pachner moves of this length, starting from the complement of the given knot, and determining whether any of them transforms the complement into a standard triangulation of a solid torus. The time for this method would be triply exponential; however, experimental evidence suggests that this bound is very pessimistic and that many fewer Pachner moves are needed.^[8]
• Any arc-presentation of an unknot can be monotonically simplified to a minimal one using elementary moves.^[9] So a brute force search among all arc-presentations of not greater complexity gives a single-exponential algorithm for the unknotting problem.
• Residual finiteness of the knot group (which follows from geometrization of Haken manifolds) gives an algorithm: check if the group has non-cyclic finite group quotient. This idea is used in Kuperberg’s result that the unknotting problem is in co-NP.
• Knot Floer homology of the knot detects the genus of the knot, which is 0 if and only if the knot is an unknot. A combinatorial version of knot Floer homology allows it to be computed (Manolescu, Ozsváth & Sarkar 2009).
• Khovanov homology detects the unknot according to a result of Kronheimer and Mrowka.^[10] The complexity of Khovanov homology at least as high as the #P-hard problem of computing the Jones polynomial, but it may be calculated in practice using an algorithm and program of Bar-Natan (2007). Bar-Natan provides no rigorous analysis of his algorithm, but heuristically estimates it to be exponential in the pathwidth of a crossing diagram, which in turn is at most proportional to the square root of the number of crossings.

Understanding the complexity of these algorithms is an active field of study.

• Algorithmic topology
• Unknotting number
• Hard unknot

• Bar-Natan, Dror (2007), “Fast Khovanov homology computations”, Journal of Knot Theory and Its Ramifications, 16 (3): 243–255, arXiv:math.GT/0606318, doi:10.1142/S0218216507005294, MR 2320156, S2CID 17036344.
• Birman, Joan S.; Hirsch, Michael (1998), “A new algorithm for recognizing the unknot”, Geometry and Topology, 2: 178–220, arXiv:math/9801126, doi:10.2140/gt.1998.2.175, S2CID 17776505.
• Burton, Benjamin A. (2011a), “Maximal admissible faces and asymptotic bounds for the normal surface solution space” (PDF), Journal of Combinatorial Theory, Series A, 118 (4): 1410–1435, arXiv:1004.2605, doi:10.1016/j.jcta.2010.12.011, MR 2763065, S2CID 11461722.
• Burton, Benjamin (2011b), “The Pachner graph and the simplification of 3-sphere triangulations”, Proc. 27th ACM Symposium on Computational Geometry, pp. 153–162, arXiv:1011.4169, doi:10.1145/1998196.1998220, ISBN 978-1-4503-0682-9, S2CID 382685.
• Dynnikov, Ivan (2006), “Arc-presentations of links: monotonic simplification”, Fundamenta Mathematicae, 190: 29–76, arXiv:math/0208153, doi:10.4064/fm190-0-3, S2CID 14137437.
• Haken, Wolfgang (1961), “Theorie der Normalflächen”, Acta Mathematica, 105: 245–375, doi:10.1007/BF02559591.
• Hara, Masao; Tani, Seiichi; Yamamoto, Makoto (2005), “Unknotting is in AM ∩ co-AM”, Proc. 16th ACM-SIAM Symposium on Discrete algorithms (SODA ’05), Society for Industrial and Applied Mathematics, pp. 359–364, ISBN 978-0-89871-585-9.
• Hass, Joel; Lagarias, Jeffrey C.; Pippenger, Nicholas (1999), “The computational complexity of knot and link problems”, Journal of the ACM, 46 (2): 185–211, arXiv:math/9807016, doi:10.1145/301970.301971, MR 1693203, S2CID 125854.
• Hass, Joel; Lagarias, Jeffrey C. (2001), “The number of Reidemeister moves needed for unknotting”, Journal of the American Mathematical Society, 14 (2): 399–428, arXiv:math/9807012, doi:10.1090/S0894-0347-01-00358-7, MR 1815217, S2CID 15654705.
• Kawarabayashi, Ken-ichi; Kreutzer, Stephan; Mohar, Bojan (2010), “Linkless and flat embeddings in 3-space and the unknot problem” (PDF), Proc. ACM Symposium on Computational Geometry (SoCG ’10), pp. 97–106, doi:10.1145/1810959.1810975, ISBN 978-1-4503-0016-2, S2CID 12290801.
• Kronheimer, Peter; Mrowka, Tomasz (2011), “Khovanov homology is an unknot-detector”, Publications Mathématiques de l’IHÉS, 113 (1): 97–208, arXiv:1005.4346, doi:10.1007/s10240-010-0030-y, S2CID 119586228
• Kuperberg, Greg (2014), “Knottedness is in NP, modulo GRH”, Advances in Mathematics, 256: 493–506, arXiv:1112.0845, doi:10.1016/j.aim.2014.01.007, MR 3177300, S2CID 12634367.
• Lackenby, Marc (2015), “A polynomial upper bound on Reidemeister moves”, Annals of Mathematics, Second Series, 182 (2): 491–564, arXiv:1302.0180, doi:10.4007/annals.2015.182.2.3, MR 3418524, S2CID 119662237.
• Lackenby, Marc (2021), “The efficient certification of Knottedness and Thurston norm”, Advances in Mathematics, 387 107796, arXiv:1604.00290, doi:10.1016/j.aim.2021.107796, S2CID 119307517.
• Manolescu, Ciprian; Ozsváth, Peter S.; Sarkar, Sucharit (2009), “A combinatorial description of knot Floer homology”, Annals of Mathematics, Second Series, 169 (2): 633–660, arXiv:math/0607691, Bibcode:2006math……7691M, doi:10.4007/annals.2009.169.633, MR 2480614, S2CID 15427272.
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