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Ashley’s stopper knot
Names	Ashley’s stopper knot, Ashley stopper knot, Oysterman’s stopper
Category	Stopper
Origin	Clifford Ashley, c. 1910
ABoK	#526
A/B notation	820

Ashley’s stopper knot, also known as the oysterman’s stopper, is a knot developed by Clifford W. Ashley around 1910. It makes a well-balanced trefoil-faced stopper at the end of the rope, giving greater resistance to pulling through an opening than other common stoppers. Essentially, the knot is a common overhand noose, but with the end of the rope passing through the noose eye, which closes upon it. It may be multiplied to form a larger knot with more than three bights appearing around the knot. It is the result of implementing a double wall knot in one strand.

Ashley developed this knot in trying to duplicate a knot he saw on a boat in a local oyster fishing fleet. When he had a chance to observe the knot up close at a later time he realized it was just a badly water-swollen figure eight stopper knot.^[1]

The oysterman’s stopper…It is a larger knot than the figure-eight, which has but one part around the stem. The oysterman’s stopper knot has three rim parts, and these are quite symmetrical when viewed from the underside. From this view it closely resembles a three-strand wall knot. The end is nipped by a single top part. It is easy to tie and practical to use when the hole that is to be filled is too large for the figure-eight.

1. Form an overhand noose, or simply tie an overhand knot around the standing part as shown.
2. Tighten the overhand portion of the knot around the standing part. Thread the working part through the loop.
3. First close the noose on the working part by pulling on the standing part, then remove any remaining slack in the knot by pulling on the working part. The knot should have a tidy, triangular shape where the standing part enters the knot. (See image at right.)

• List of knots

• Budworth, Geoffrey (2001). The Ultimate Encyclopedia of Knots & Ropework. Anness Publishing LTD. ISBN 9781859679111.

Ashley’s Bend
Names	Ashley’s Bend, Ashley bend
Category	Bend
Origin	The Ashley Book of Knots
Related	Alpine butterfly bend, Hunter’s bend, Zeppelin bend, Butterfly loop, Trident loop
Releasing	Jamming possible
Typical use	temporary joining of similar-sized cords & ropes
ABoK	#1452, #1408

Ashley’s bend is a knot used to securely join the ends of two ropes together. It is similar to several related bend knots which consist of two interlocking overhand knots, and in particular the alpine butterfly bend.^[1] These related bends differ by the way the two constituent overhand knots are interlocked.

The name “Ashley’s bend” is now associated with the knot described in entry #1452 of The Ashley Book of Knots. Clifford Ashley developed this bend and believed it to be original, along with several similar ones. Rather than giving it a name he simply noted the date when he first tied it: “(2/3/34.)”.^[2] Cyrus L. Day, a contemporary of Ashley’s, called the knot by the name “Ashley’s Bend” in his 1947 book The Art of Knotting & Splicing just a few years after the publication of Ashley’s book.^[3][4] Later authors have continued to use this name.^[5][6]

In the 1930s, Ashley performed security tests on a number of bends for the Collins and Aikman company.^[7] The manufacturer wanted a bend that would not slip when tied in mohair, a stiff slippery material. The jerk testing Ashley performed placed his bend, #1452, equal to the barrel knot in exhibiting no slippage at all. All other bends he tested slipped to some extent, and most failed completely in less than 100 loading cycles.^[8]

Most references fail to distinguish the distinct ways in which the two ends of the knot can be dressed. As the two working ends emerge from the knot, they make a sort of vortex that twists the tails in one direction; the tails can be oriented such that they are twisted ever tighter together, or put on the other side of each other in which case the setting of the knot can lead to a jamming state.

This knot belongs to a group of knots formed by interlocking overhand knots. Its purpose is to securely join two ropes of similar thickness. Ashley’s testing confirmed its reliability with minimal slipping tendencies. One needs to dress the knot correctly for easy loosening when expecting a heavy load.

• List of bend knots
• List of knots
• Knot theory

The theory of open knots attempts to describe entanglements in open curves or filaments in a mathematically consistent way and develop tools and algorithms which can categorize the topology of an open curve. In the mathematical field of knot theory, knots are only considered in closed loops. What is colloquially considered a knot, for example a piece of rope tied into an overhand knot, would not be considered a mathematical knot unless the two ends of the rope were connected. Research into open knot theory is motivated by a desire to understand the formation and properties of knots in proteins and DNA molecules, which often do not form closed loops, and to draw closer connections between knot theory and the properties of physical knots.

Drawing a straight line from one end of a curve to another effectively closes it into a loop. After this virtual closure, the knot can be classified by computing an invariant such as the Alexander polynomial. If the two ends of a knotted curve filament are close to each other or well separated from the highly knotted portion, this direct virtual closure will not introduce new crossings into a diagram. However, there are configurations where direct closure can virtually turn a knot into a slipknot and effectively erase the knot, or introduce additional complexity that is not present in the initial curve. In such cases, it may be beneficial to connect the ends of the curve by virtual lines to an external surface enclosing the curve, such as a sphere with a large radius, and then connecting the ends of two lines along the surface so that they do not interfere with the curve itself. An algorithm known as minimally interfering closure will determine whether the two ends of an open polygonal knot are closer to each other or to the convex hull of the knot, and connect them by whichever path is shorter.^[1]

The choice of virtual closure scheme will influence which type of knot a curve is determined to be consistent with. A more general method known as stochastic closure chooses many uniformly distributed points on the surface of a large sphere enclosing the curve and connects the ends to each of those points and computes the knot type at each closure. This yields a distribution of different knot types at different regions around the sphere which can be visualized as a map.^[2] This is primarily used for analyzing knotted proteins.

Rather than attempting to map an open curve onto a specific closed knot, concepts have been developed to classify open entanglements. One such concept is the knotoid which is a generalization of a knot diagram which includes the two ends of the curve, first described by Turaev in 2010.^[3] When a Reidemeister move is applied to a diagram of a knot, the knot topology cannot change. If a Reidemeister move on a knotoid diagram moves part of the curve over one of the ends, it will change the type of knotoid and is considered “forbidden.” In this sense, a knotoid can be envisioned as a knotted piece of string on ground with its ends attached to two vertical pegs; the knotoid type will not change unless the string is lifted and unwrapped around the pegs. Like knots, knotoids can be classified based on their crossing number, and invariants such as polynomials have been derived to distinguish them. An open curve in three dimensions will be consistent with different knotoids depending on the surface that it is projected onto. Similar to stochastic closure, the full picture of a curve’s topological complexity must be determined by sampling the knotoids of many projections. It is possible to compute the minimum number of “forbidden” knotoid moves (passing a curve in the diagram over one of the ends) to reach a trivial crossing-free knotoid,^[4] which provides a measure of complexity of an open curve similar to the unknotting number.

Virtual knots are another generalization of knot diagrams. Whereas knotoids deal with ambiguous closure, virtual knots deal with ambiguous crossings. Where one part of a knot diagram passes over or under another, two parts of a virtual knot diagram may meet at a point, called a virtual crossing. A diagram of an open knot may be treated as a virtual knot by connecting its to ends with a line and creating a virtual crossing at each point the end intersects the diagram.

Definitions of knot invariants that categorize the topology of closed curves can be generalized to describe open curves. An example is the space writhe which is an extension of the Gauss linking number, and describes how many times a curve will cross over itself when viewed from different directions.^[5] More entangled and twisted curves will typically have a higher space writhe, however an unentangled curve such as a helix will also have a high space writhe. Similarly, the Gauss linking integral can be computed from two open curves, such as two strands in a hair braid, to determine how many times one curve winds around another. Other knot invariants have also been extended to open curves, including the second Vassiliev invariant^[6] and the Jones polynomial.^[7]

Many of the techniques used to categorize open entangled curves have been applied to the study of knotted proteins. This includes a categorization of knotted protein structures based on stochastic closure,^[2] using knotoids,^[4] virtual knots,^[8] the space writhe,^[9] and open versions of the second Vassiliev invariant ^[10] and Jones polynomial.^[11] Beyond simply categorization, a goal of this research is to understand the formation and stability of these knotted proteins. Similar analysis has also been applied to DNA, which does not have a stable native state like proteins do. The most common tool used to determine the topology of simulated DNA molecules is the Alexander molecule combined with chain closure, which has been used to detect knots in simulated DNA in virus capsids,^[12] human chromosomes,^[13] as well as simpler models of polymers,^[14] of which DNA is an example. Beyond the study of biomolecules, tools from open knot theory have been applied to physical ropes, for example in determining the most effective way to tie two pieces of rope together by comparing the writhe within each type of knot to the force required to pull two tied ropes apart.^[15]

In the mathematical area of group theory, Artin groups, also known as Artin–Tits groups or generalized braid groups, are a family of infinite discrete groups defined by simple presentations. They are closely related with Coxeter groups. Examples are free groups, free abelian groups, braid groups, and right-angled Artin–Tits groups, among others.

The groups are named after Emil Artin, due to his early work on braid groups in the 1920s to 1940s,^[1] and Jacques Tits who developed the theory of a more general class of groups in the 1960s.^[2]

An Artin–Tits presentation is a group presentation ${\displaystyle \langle S\mid R\rangle }$ where ${\displaystyle S}$ is a (usually finite) set of generators and ${\displaystyle R}$ is a set of Artin–Tits relations, namely relations of the form ${\displaystyle stst\ldots =tsts\ldots }$ for distinct ${\displaystyle s,t}$ in ${\displaystyle S}$, where both sides have equal lengths, and there exists at most one relation for each pair of distinct generators ${\displaystyle s,t}$. An Artin–Tits group is a group that admits an Artin–Tits presentation. Likewise, an Artin–Tits monoid is a monoid that, as a monoid, admits an Artin–Tits presentation.

Alternatively, an Artin–Tits group can be specified by the set of generators ${\displaystyle S}$ and, for every ${\displaystyle s,t}$ in ${\displaystyle S}$, the natural number ${\displaystyle m_{s,t}\geqslant 2}$ that is the length of the words ${\displaystyle stst\ldots }$ and ${\displaystyle tsts\ldots }$ such that ${\displaystyle stst\ldots =tsts\ldots }$ is the relation connecting ${\displaystyle s}$ and ${\displaystyle t}$, if any. By convention, one puts ${\displaystyle m_{s,t}=\infty }$ when there is no relation ${\displaystyle stst\ldots =tsts\ldots }$ . Formally, if we define ${\displaystyle \langle s,t\rangle ^{m}}$ to denote an alternating product of ${\displaystyle s}$ and ${\displaystyle t}$ of length ${\displaystyle m}$, beginning with ${\displaystyle s}$ — so that ${\displaystyle \langle s,t\rangle ^{2}=st}$, ${\displaystyle \langle s,t\rangle ^{3}=sts}$, etc. — the Artin–Tits relations take the form

${\displaystyle \langle s,t\rangle ^{m_{s,t}}=\langle t,s\rangle ^{m_{t,s}},{\text{ where }}m_{s,t}=m_{t,s}\in \{2,3,\ldots ,\infty \}.}$

The integers ${\displaystyle m_{s,t}}$ can be organized into a symmetric matrix, known as the Coxeter matrix of the group.

If ${\displaystyle \langle S\mid R\rangle }$ is an Artin–Tits presentation of an Artin–Tits group ${\displaystyle A}$, the quotient of ${\displaystyle A}$ obtained by adding the relation ${\displaystyle s^{2}=1}$ for each ${\displaystyle s}$ of ${\displaystyle R}$ is a Coxeter group. Conversely, if ${\displaystyle W}$ is a Coxeter group presented by reflections and the relations ${\displaystyle s^{2}=1}$ are removed, the extension thus obtained is an Artin–Tits group. For instance, the Coxeter group associated with the ${\displaystyle n}$-strand braid group is the symmetric group of all permutations of ${\displaystyle \{1,\ldots ,n\}}$.

• ${\displaystyle G=\langle S\mid \emptyset \rangle }$ is the free group based on ${\displaystyle S}$; here ${\displaystyle m_{s,t}=\infty }$ for all ${\displaystyle s,t}$.
• ${\displaystyle G=\langle S\mid \{st=ts\mid s,t\in S\}\rangle }$ is the free abelian group based on ${\displaystyle S}$; here ${\displaystyle m_{s,t}=2}$ for all ${\displaystyle s,t}$.
• ${\displaystyle G=\langle \sigma _{1},\ldots ,\sigma _{n-1}\mid \sigma _{i}\sigma _{j}\sigma _{i}=\sigma _{j}\sigma _{i}\sigma _{j}{\text{ for }}\vert i-j\vert =1,\sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}{\text{ for }}\vert i-j\vert \geqslant 2\rangle }$ is the braid group on ${\displaystyle n}$ strands; here ${\displaystyle m_{\sigma _{i},\sigma _{j}}=3}$ for ${\displaystyle \vert i-j\vert =1}$, and ${\displaystyle m_{\sigma _{i},\sigma _{j}}=2}$ for ${\displaystyle \vert i-j\vert >1}$.

Artin–Tits monoids are eligible for Garside methods based on the investigation of their divisibility relations, and are well understood:

• Artin–Tits monoids are cancellative, and they admit greatest common divisors and conditional least common multiples (a least common multiple exists whenever a common multiple does).
• If ${\displaystyle A^{+}}$ is an Artin–Tits monoid, and if ${\displaystyle W}$ is the associated Coxeter group, there is a (set-theoretic) section ${\displaystyle \sigma }$ of ${\displaystyle W}$ into ${\displaystyle A^{+}}$, and every element of ${\displaystyle A^{+}}$ admits a distinguished decomposition as a sequence of elements in the image of ${\displaystyle \sigma }$ (“greedy normal form”).

Very few results are known for general Artin–Tits groups. In particular, the following basic questions remain open in the general case:

– solving the word and conjugacy problems — which are conjectured to be decidable,

– determining torsion — which is conjectured to be trivial,

– determining the center — which is conjectured to be trivial or monogenic in the case when the group is not a direct product (“irreducible case”),

– determining the cohomology — in particular solving the ${\displaystyle K(\pi ,1)}$ conjecture, i.e., finding an acyclic complex whose fundamental group is the considered group.

Partial results involving particular subfamilies are gathered below. Among the few known general results, one can mention:

• Artin–Tits groups are infinite countable.
• In an Artin–Tits group ${\displaystyle \langle S\mid R\rangle }$, the only relation connecting the squares of the elements ${\displaystyle s,t}$ of ${\displaystyle S}$ is ${\displaystyle s^{2}t^{2}=t^{2}s^{2}}$ if ${\displaystyle st=ts}$ is in ${\displaystyle R}$ (John Crisp and Luis Paris ^[3]).
• For every Artin–Tits presentation ${\displaystyle \langle S\mid R\rangle }$, the Artin–Tits monoid presented by ${\displaystyle \langle S\mid R\rangle }$ embeds in the Artin–Tits group presented by ${\displaystyle \langle S\mid R\rangle }$ (Paris^[4]).
• Every (finitely generated) Artin–Tits monoid admits a finite Garside family (Matthew Dyer and Christophe Hohlweg^[5]). As a consequence, the existence of common right-multiples in Artin–Tits monoids is decidable, and reduction of multifractions is effective.

Several important classes of Artin groups can be defined in terms of the properties of the Coxeter matrix.

• Free partially commutative monoid
• Artinian group (an unrelated notion)
• Non-commutative cryptography
• Elementary abelian group

• Charney, Ruth (2007), “An introduction to right-angled Artin groups”, Geometriae Dedicata, 125 (1): 141–158, arXiv:math/0610668, doi:10.1007/s10711-007-9148-6, MR 2322545
• Godelle, Eddy; Paris, Luis (2012), Basic questions on Artin–Tits groups, CRM Series, vol. 14, Ed. Norm., Pisa, pp. 299–311, arXiv:1105.1048, doi:10.1007/978-88-7642-431-1_13, ISBN 978-88-7642-430-4, MR 3203644
• McCammond, Jon (2017), “The mysterious geometry of Artin groups”, Winter Braids Lecture Notes, 4 (Winter Braids VII (Caen, 2017)): 1–30, doi:10.5802/wbln.17, MR 3922033
• Flores, Ramon; Kahrobaei, Delaram; Koberda, Thomas (2019). “Algorithmic problems in right-angled Artin groups: complexity and applications”. Journal of Algebra. 519: 111–129. arXiv:1802.04870. doi:10.1016/j.jalgebra.2018.10.023. MR 3874519.

Offset overhand bend
Names	Offset overhand bend, European Death knot (EDK), offset water knot, flat overhand bend, thumb knot, thumb bend, Creeler’s knot, openhand knot
Category	Bend
Origin	Ancient
Related	Overhand knot, water knot
Releasing	Jamming
Typical use	sewing, weaving, baling, climbing, rappelling
ABoK	246, 359, 1236, 1410, 1557, 1558, 3789

The offset overhand bend (OOB, ABoK No. 1410) is a knot used to join two ropes together end-to-end. It is formed by holding two rope ends next to each other and tying an overhand knot in them as if they were a single line. Due to its common use in several fields, this bend has become known by many names, such as thumb knot, openhand knot,^[1] one-sided overhand knot or flat overhand bend (FOB), though the terms “one-sided” and “flat” are considered incorrect.^[2]

The term ‘offset’ refers to the knot core being displaced from the axis of tension. This geometry allows the knot to more easily translate around an edge – particularly a 90 degree edge.

Long used by weavers to join the ends of yarn, the offset water knot is very old. It was one of the knots likely identified among the possessions of Ötzi the Iceman, who dates from 3300 BC.^[3]

The knot is also tied in a slipped form by mechanical balers to bind straw and hay, but this bend is not practical to use as a binding knot when tied by hand.^[1]

In climbing and mountaineering

For mountaineers/climbers, there tends to be a strong preference for using knots that are perceived to be relatively easy to tie – even when fatigued or in a less than optimal frame of mind – and so #1410 (Offset overhand bend) is favored. Climbers/canyoners need to retrieve their ropes after an abseil/rappel descent. The ability to retrieve ropes after an abseil descent is crucial – and in many cases, two ropes need to be joined to increase the distance that can be descended in one ‘pitch’. The resulting knot that unites the two ropes needs be secure and stable, have a small footprint, and be resistant to jamming.

There is controversy over its safety, as it can fail by capsizing under high loads,^[4][5][6][7] and some American climbers refer to it as the European death knot, abbreviated to EDK, with some sources recommending against its use.^[8][9] Failure of this knot has been implicated in some accidents and near-misses – although post accident retrieval of ropes for examination are usually inconclusive because the ropes have separated (and hence there is no remnant knot to examine).^[10][11][12]

Many sources argue that the name ‘EDK’ is a misnomer, and the knot is safe for abseiling / rappelling, since this does not generate as high forces as a fall. The nominal load during abseiling/rappelling is one person – generally around 1.0kN (approximately 100kg). If the system is configured so the ropes are doubled through the anchor, the joining knot will only be subjected to 50% of the load (ie approximately 0.5kN) – which is well below the instability threshold. With due diligence given to dressing and setting the knot, the risk of capsizing is highly unlikely.^[13]

Several sources recommend adding a second overhand as close as possible to the first (a stacked overhand or double overhand) for most situations, which maintains most of the benefits, while preventing it from capsizing.^{[9][13][12][11][14][15][16][17]} This doubles the overall footprint of the knot, which might increase its likelihood of getting stuck in cracks, but does not harm its ability to pass over edges. There are several different choices of offset knots – all offering varying levels of advantages/disadvantages. Another option is wrapping the strands a second time before passing the tail through (a two-rope version of ABoK #516, also called a double overhand^[8] or flat doubled overhand bend^[9]) but again, it increases the overall footprint.

Easily formed in most lines, the offset overhand bend is jam resistant at nominal loads of one person (approximately 100kg). In EN892 climbing ropes, the jamming threshold is thought to be in the vicinity of 3.0kN (300kg). The instability threshold is thought to be above 4.0kN (400kg) – that is, a capsizing event becomes increasingly probable as loads exceed 400kg. It is critically important to pay close attention to dressing and cinching of the knot before attempting to abseil. That is, climbers must exercise due diligence when tying this knot – by pulling firmly on each of the four rope segments – which is necessary to achieve a properly compacted and cinched dressing state.^[18]

Despite questions about this knot’s security, it does present some advantages for use in rappels. Because the knot is offset from the axis of tension, it can translate more easily over uneven surfaces and 90 degree edges than other knots; and it is quickly tied and readily untied. Since a stuck rope on a multi-pitch descent can be catastrophic for climbers, these advantages, along with ease of tying, have led to its popularity. As with all knots used in life critical applications, the tails must be of sufficient minimum length (never less than 200mm in offset knots), and be diligently dressed and fully tightened by pulling individually on all four rope segments.^[19]

An interesting yet overlooked fact is that #1410 (offset overhand bend), can be rotated to induce a choking effect to trap and crush the tails. Virtually all testers appear to only examine this knot in its mid-rotation state. It is theorized that this mid-rotation state is in fact the orientation where the structure is most vulnerable to capsizing. In addition, when tying the offset overhand bend using different rope diameters, the thinner diameter rope must be positioned underneath the larger diameter rope. This tactic further inhibits any likelihood of capsizing.

The offset figure-eight bend, a similar knot using the figure-eight knot, has been used in the belief that its greater size and complexity brings more security. But testing and more than one fatal failure indicate the figure-eight variant to be less secure, more prone to capsize at lower loads, and in capsizing uses more of the ends than does a capsizing overhand bend.^[13][19] Moreover, while there is one proper dressing of the overhand bend, there are a couple of dressings for the offset figure eight bend.^[2]

• List of bend knots
• List of knots

Artillery loop
Names	Artillery loop, Artilleryman’s knot, Manharness knot,[1] Manharness loop, Harness loop,[2] Harness hitch,[2] Belayer’s hitch,[3] Sandy Douglass knot[4]
Category	Loop
Related	Farmer’s loop, Alpine butterfly knot, Span loop, Marlinespike hitch
Releasing	Non-jamming
Caveat	Must have load, may slip unexpectedly under tension creating a running knot or noose
ABoK	#153, #428, #532,[5] #1050, #1051

The artillery loop^[1] is a knot with a loop on the bight for non-critical purposes. The artillery loop must have the loop loaded or it will slip and contract easily. It is an inferior knot to the alpine butterfly knot,^[2] possibly dangerously so, in that it can be yanked out of shape and turn into a running knot or noose.^[6]

Budworth states that this knot is often described as being best suited to take a load on only one of the ends, but reliable information on which end is difficult to find.^[7]

• 
  Artillery loop step by step
• 
  Finished Artillery loop

The name harness loop derives from the fact that the knot was used when assisting horses on difficult terrain.^[7] Similarly, the name artillery loop or artilleryman’s hitch derives from the fact that it was used when hauling field artillery into position.^[7]

• Harness bend
• List of knots

• Artillery Loop

Offset figure-eight bend
Names	Offset figure-eight bend, flat figure-eight bend[1], abnormal figure-eight bend[2]

The offset figure-eight bend is a poor knot that has been implicated in the deaths of several rock climbers.^{[1][3][4][5][6]} The knot may capsize (invert) under load, as shown in the figure, and this can happen repeatedly.^[7] Each inversion reduces the lengths of the tails. Once the tails are used up completely, the knot comes undone.

More secure knots for this purpose are the Flemish bend (the “figure eight bend”), (doubled) offset overhand bend, or double fisherman’s knot.

• List of knots

In Ancient Egyptian society, hair was an embodiment of identity.  It could carry religious and erotic significance and portray information about gender, age, and social status.^[1] During the New Kingdom, more elaborate hairstyles for men and women, incorporating curls and plaits, began to be favored over the traditional, simple hairstyles of the Old and Middle Kingdoms.^[2]

Nubian wigs, which Ancient Egyptians grew fond of during the Amarna period, were meant to mimic the short curly hair that Nubian tribespeople wore.^[3][4] Egyptologists believe that the Nubian wig was adopted by Queen Nefertiti after witnessing the hairstyle being worn by Nubians in the Pharaoh’s army.^[5] Though there has been a discussion on what qualifies as a Nubian wig, some arguing layered wigs known as “duplex” styles that include curls and plaits may also be Nubian wigs. Still, many refer to this as a Nubian style and not a Nubian wig.^[4]

In general, wigs in Ancient Egypt were almost entirely confined to the elite due to their price. Even when wigs were made of inexpensive materials like plant fibers, the sophisticated craftsmanship required to make wigs proved costly.^[2][6]

One can see Nubian wigs in the form of reliefs, statues, and paintings. They are a prime factor in determining the gender of figures in Ancient Egyptian art, as royal women exclusively wear them.^[7]  It is characterized by its short bushy appearance with rows of curls that frame the brow and sides of the face.  The feature that requires the most attention is located on the neck.^[7] In Nubian wigs, the hair is cut to expose the nape of the neck, which distinguishes it from a similar headdress where the nape of the neck is not exposed, and the hair falls towards the shoulders.  Royal men exclusively wear this alternative style and can be seen in the left image titled Walk in the Garden.^[7]

Wigs were composed of various materials such as human hair, wool, plant fibers, and horsehair.^[2]  The most expensive wigs were made of human hair or black sheep wool, or both.^[2] In addition to the hair, false or human, Ancient Egyptians used beeswax and resin to hold the style in place on a mesh cap.^[8]

One wig specifically, titled wig by The British Museum, has been studied extensively. Efforts to study other wigs are scarce due to the fragile nature or incompleteness of the wigs after thousands of years. Though this wig is the “duplex” style that men wore and is referenced above, its condition gives historians insight into how Ancient Egyptians created wigs.^[8]

To begin the process of making a wig, a wigmaker must have first collected hair.  After a wigmaker collected enough hair, the hair was washed and separated into individual locks with about 400 strands per lock.^[6] Construction could then begin on a wooden wig mount very similar to what modern wigmakers would use.  The wig achieved the mesh base by laying hair down vertically and horizontally across each other.  The base layer was kept in place by knotting and folding the hair back over itself and then was further reinforced by a mixture of two-thirds beeswax and one-third conifer resin.^[6] To hook curls to the mesh layer, the curl was looped around the mesh and then fastened by fifteen individual hairs, called a “sub-strand” that was tired around the stem of the curl.  Wigs could take up to 200 hours to complete and could take even longer if plaits were styled on the wooden wig mount, as they probably were in ancient times.^[6]

In mathematics, particularly in topology and knot theory, Arnold invariants are invariants introduced by Vladimir Arnold in 1994^[1] for studying the topology and geometry of plane curves. The three main invariants—${\displaystyle J^{+}}$, ${\displaystyle J^{-}}$, and ${\displaystyle St}$—provide ways to classify and understand how curves can be deformed while preserving certain properties.^[2]

The fundamental context for Arnold invariants comes from the Whitney–Graustein theorem, which states that any two immersed loops (smooth curves in the plane) with the same rotation number can be deformed into each other through a series of continuous transformations.^[3] These transformations can be broken down into three elementary types: direct self-tangency moves (where two portions of the curve become tangent with aligned directions, either creating or eliminating two self-intersection points), inverse self-tangency moves (similar to direct moves, but the tangent directions are opposite), and triple point moves (where three portions of the curve intersect at a single point).^[4]

The ${\displaystyle J^{+}}$ and ${\displaystyle J^{-}}$ invariants keep track of how curves change under these transformations and deformations. The ${\displaystyle J^{+}}$ invariant increases by 2 when a direct self-tangency move creates new self-intersection points (and decreases by 2 when such points are eliminated), while ${\displaystyle J^{-}}$ decreases by 2 when an inverse self-tangency move creates new intersections (and increases by 2 when they are eliminated). Neither invariant changes under triple point moves. A fundamental relationship between these invariants is that their difference equals the total number of self-intersection points in the curve. That is,

${\displaystyle J^{+}(c)-J^{-}(c)={\text{number of self-intersection points of }}c}$.^[5]

Mathematicians Oleg Viro and Eugene Gutkin discovered an explicit formula for calculating ${\displaystyle J^{-}}$:

${\displaystyle J^{-}(c)=1-\sum _{R}{\text{wind}}(c,R)^{2}+\sum _{q}{\text{meanwind}}(c,q)^{2}}$^[5]

where ${\displaystyle R}$ ranges over the regions into which ${\displaystyle c}$ divides the plane, ${\displaystyle {\text{wind}}(c,R)}$ is the winding number around a point in region ${\displaystyle R}$, and ${\displaystyle {\text{meanwind}}(c,q)}$ is the mean winding number at each self-intersection point ${\displaystyle q}$. For example, a curve with ${\displaystyle k}$ curls in standard form has ${\displaystyle J^{+}=-2k}$ and ${\displaystyle J^{-}=-3k}$, while a simple circle has ${\displaystyle J^{+}=J^{-}=0}$.^[4]

In 2002, mathematicians Catarina Mendes de Jesus and Maria Carmen Romero Fuster introduced the concepts of bridges and channels for plane curves to facilitate the calculation of Arnold invariants.^[6] A bridge consists of introducing a rectangle in the complement of the curve in the plane while respecting orientations, decomposing a given curve into two smaller curves with known invariants. The invariant of the original curve can then be obtained as a function of the invariants of these two component curves and the index of the bridge relative to the original curve. This decomposition technique is particularly powerful for analyzing curves with double points.

An important theorem regarding this decomposition states that a curve with ${\displaystyle n}$ double points is a tree-like curve if and only if it admits a decomposition into exactly n curves of types ${\displaystyle K_{0}}$ and ${\displaystyle K_{2}}$ with bridges having no double points, or a decomposition into exactly ${\displaystyle n+1}$ curves of type ${\displaystyle K_{1}}$ (isotopic to the circle) with bridges having double points.^[7] This result proved a conjecture originally proposed by Arnold regarding the formulas for families of tree-like curves. The bridge and channel technique provides a systematic method for computing Arnold invariants for plane curves in terms of simpler curves with at most one double point.

• Plane curve
• Knot invariant
• Whitney–Graustein theorem
• Differential topology

• Santa Rosa, Lílian Neves (2010). Arnold Invariants of Plane Curves. Master’s Thesis, Federal University of Viçosa.
• Mendes de Jesus, C. Topological Invariants of Generic Maps from Oriented Compact Surfaces to the Plane. Doctoral Thesis, PUC-RIO, 2001.
• Moraes, Simone M.; Sánchez, Catarina M. J. “Invariants of Closed Plane Curves”. Proceeding Series of the Brazilian Society of Applied and Computational Mathematics, Vol. 3, N. 1, 2015.
• Lagemann, Anna Marie; von der Mosel, Heiko (Thesis advisor); Hryniewicz, Umberto (Thesis advisor); Reiter, Philipp (Thesis advisor). Variational approach to study Arnold’s invariants of immersed planar curves RWTH Aachen University (2023) PhD Thesis

• Vladimir Arnold: “Plane curves, Their Invariants, Perestroikas and Classifications“. In Singularities and bifurcations, Vol. 21 of Adv. Soviet Math., Amer. Math. Soc., Providence, RI, 1994, pp. 33–91, with an appendix by F. Aicardi.

Noose
A noose knot tied in kernmantle rope
Names	Noose, running knot
Category	Loop
Related	Slip knot, overhand knot, double overhand noose, hangman’s knot, running bowline, arbor knot
Releasing	Non-jamming
Typical use	Animal snares, knitting, hanging device, self tightening end loop
ABoK	#8, #43, #1114,[1] #1789, #1803, #1825
Instructions

A noose is a loop at the end of a rope in which the knot tightens under load and can be loosened without untying the knot. The knot can be used to secure a rope to a post, pole, or animal but only where the end is in a position that the loop can be passed over.

The knot is tied by forming a turn in the end of a rope, and then passing a bight in the standing part through. The noose knot is a slipped version of the overhand knot.

The knot most closely associated with execution is the hangman’s knot, which is also known as the “hangman’s noose”. Tying is similar to the original noose, but many turns are wrapped around the loop. The reason for this was to make the hanging more humane, as it would break the person’s neck, killing the person instantly, rather than strangling them to death. A similar method is also commonly used for suicide.

In the United States, a noose is sometimes left as a message in order to intimidate people, as it was the main object used in segregation era lynchings.^[2][3] In 2022, a bill to make lynching a federal hate crime was passed.^[4] It is illegal to display a noose in a threatening manner in Virginia,^[5] New York and Connecticut.^[6]

Austin Reed Edenfield, a former student of the University of Mississippi, pled guilty in 2016 to a federal civil-rights crime, acknowledging that he and Graeme Phillip Harris had tied a noose and a flag of Georgia around the neck of a statue honoring James Meredith, the university’s first African-American student.^[7] Harris was sentenced to prison and Edenfield to probation and community service.^[8]

In September 2019, Andrew M. Smith, a University of Illinois student, was arrested for placing a noose in a campus elevator. “The incident [came] just months after black employees filed a class-action lawsuit against the campus, alleging they faced racial harassment and were exposed to threats of racial violence, such as nooses, swastikas, KKK garb, racist graffiti, and confederate flags.”^[9] He was sentenced to supervision, public service, and a $75 fine.^[10]

In November 2022, a noose was found on an Obama Presidential Center construction site.^[11]

In July 2025, a noose was found during construction at the New Nissan Stadium.^[12]

• List of knots

• Jack Shuler, The Thirteenth Turn: A History of the Noose, Public Affairs, 2014, ISBN 9781610391368