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Hinckaert knot
The Hinckaert badge.
Information
Family	Hinckaert family
Region	Netherlands
Note(s): Named for Philip Hinckaert, maître d’hôtel to Philip the Handsome in the late 15th century.

The Hinckaert knot, a type of decorative unknot, is a heraldic knot used primarily in Dutch heraldry. It is most notable for its appearance on the Hinckaert family heraldic badge, where a semi-angular form is used as canting arms, a common practice with heraldic badges.

The name “Hinckaert” is delineated as a derivation of hincken, “to limp”, in the badge. Hence the center crutch, and the buckle on the knot, implying that it is a strap used to attach the crutch to the leg.^[1] The dexter “P” and sinister “G” are traditionally translated as standing for Philip (Hinckaert), with whom the knot originated, and his wife, née Gasparine.^[1]

A heraldic knot (referred to in heraldry as simply a knot) is a knot or design incorporating a knot used in European heraldry.^[1] While a given knot can be used on more than one family’s achievement of arms, the family on whose coat the knot originated usually gives its name to the said knot (the exception being the Tristram knot). These knots can be used to charge shields and crests, but can also be used in badges or as standalone symbols of the families for whom they are named (like Scottish plaids). The simplest of these patterns, the Bowen knot, is often referred to as the heraldic knot in symbolism and art outside of heraldry.

Heraklas (Ancient Greek: Ἡρακλᾶς) was a Greek physician of the 1st century AD whose descriptions of surgeons’ knots and slings are preserved in book 48 of Oribasius’ Medical Collections (Ἰατρικαὶ Συναγωγαί, Iatrikai Synagogai) under the title From Heraklas.^[1]

Describing them in detail, Heraklas discussed 16 different knots and slings,^[1] including the earliest known written account of a string figure.^[2] Accompanying illustrations of the knots were added later by Renaissance copyists, but modern analysis of the writings by knot experts has shown many of these early drawings to contain significant errors or misinterpretations.^[3]

The current understanding of Heraklas’ knots results primarily from analysis and identification by Hjalmar Öhrvall, Lawrence G. Miller, and Cyrus L. Day, although slightly differing interpretations and refinements continue to be made.^[1] The table below shows the knots believed to have been described by Heraklas.

• Medicine in ancient Greece

Zeppelin loop
Names	Zeppelin loop, Rosendahl loop
Category	Loop
Related	Zeppelin bend

A zeppelin eye knot, is a secure, jam resistant fixed size loop knot based on the zeppelin bend. It is one of the few eye knots suitable for bungee. It is also special in its ease of untying.

Virtually all bends (i.e. end-to-end joining knots) have a corresponding ‘eye knot’. For example, the Sheet bend (ABoK #1431) has a corresponding eye knot which is none other than the common (#1010) Bowline. The Zeppelin bend is formed from 2 superposed loops of opposite chirality.

There are two versions of the zeppelin eye knot depending on how it is tied.

• The first version where the end seems to be vertical to the main part, and one of the loop sides seems to be the continuation of the main part, while the other loop side seems to continue as the working end out of the loop knot. This version is tied using the clover method, starting with an overhand knot, then letting the working end pass in the following order through
  1. first the eye of the clover (overhand knot) on the main part side along with the main part (thus forming the loop)
  2. then the loop itself opposite the direction of the main part
  3. and last the other eye of the clover knot in the opposite direction of the exiting working end (loop side)
• The other version where the end seems to be the continuation of the main part, and both ends of the loop seems to be vertical to the main part. This version is tied using the clover method, starting with an underhand knot, then letting the working end pass in the following order through
  1. – the same eye to form a simple noose thus forming the loop
  2. – around the edge of the underhand knot, and through the loop of the noose
  3. – around and through the knot along with the main part ( thus passing simultaneously through all 3 of the original underhand knot, the noose and the last round of itself).

• 
  Tying a jamming false Zeppelin loop using the noose method: One starts with an underhand knot
• 
  The working end is brought back in through the same eye to form a simple noose thus forming the loop
• 
  The working end is led around the edge of the underhand knot, and through the loop of the noose
• 
  The working end is led around the knot to reenter and cross it alongside the main part ( thus passing through all 3 loops of underhand knot, noose and its own).

• List of knots

• Description of the Zeppelin Loop on the Notable Knot Index

Yosemite bowline
Names	Yosemite bowline, Bowline with a Yosemite finish
Category	Loop
Related	bowline
Releasing	Non-jamming
ABoK	#1015, #1436

A Yosemite bowline is a loop knot often perceived as having better security^[1] than a bowline. If the knot is not dressed correctly, it can potentially collapse into a noose,^[1][2][3][4] however testing reveals this alternative configuration to be strong and safe as a climbing tie-in.^[5]

A Yosemite bowline is made from a bowline with the free end wrapped around one leg of the loop and tucked back through the knot, a final round turn and reeve commonly known as a “Yosemite finish.” The knot’s security is enhanced by preventing the bowline capsizing to form a highly dangerous slip knot. Additional safety is achieved by tying with a tail (see below). When finished, the working end forms a figure eight.

Because of the danger of incorrectly dressing the Yosemite bowline and capsizing it even before it is set, it may be safer and less error-prone to use a standard or double bowline with a backup stopper knot added to the tail, such as a double overhand knot tied around the loop.^[3][4]

The Yosemite finish can be applied to other bowline variants, such as the double bowline.

While the knot’s versatility suggests it as a convenient tie-in for attaching a climbing rope to a climber’s harness, the figure-of-eight follow through is the most common choice because it is more widely known and more easily checked.^[6] The Mountaineering Handbook is one of the few texts that suggest that the Yosemite bowline is better for this purpose. Suggested benefits of the bowline include being easier to untie after loading or when wet and frozen, and being possible to tie-in with only one hand.^[7] Testing found it a strong knot for the purpose.^[8]

It is recommended that any knot which is used to attach a rope to a safety harness is always finished with a stopper knot. A stopper knot, while serving to keep the loose end tidy, will only help to prevent failure of the primary knot, and does not act as a secondary safety knot by itself. It is sometimes said that if enough of a tail is left to tie a stopper knot, the stopper becomes unnecessary. The tail should be a minimum of 10cm but depends on the thickness of the rope.^[9]

• List of knots

• Little Campfires (2020-08-01). “How to Tie a Bowline Knot (Plus Variations): Yosemite Bowline”. YouTube.
• TheTautLine (2023-09-11). “Complete Guide to the Bowline Knot and its Most Important Variants: Yosemite Bowline”. YouTube.

In knot theory, there are several competing notions of the quantity writhe, or ${\displaystyle \operatorname {Wr} }$. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amount of “coiling” of a mathematical knot (or any closed simple curve) in three-dimensional space and assumes real numbers as values. In both cases, writhe is a geometric quantity, meaning that while deforming a curve (or diagram) in such a way that does not change its topology, one may still change its writhe.^[1]

In knot theory, the writhe is a property of 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 the link at a point in each component and this direction is followed all the way around each component. For each crossing one comes across while traveling in this direction, if the strand underneath goes from right to left, the crossing is positive; if the lower strand goes from left to right, the crossing is negative. One way of remembering this is to use a variation of the right-hand rule.

Positive crossing		Negative crossing

For a knot diagram, using the right-hand rule with either orientation gives the same result, so the writhe is well-defined on unoriented knot diagrams.

The writhe of a knot is unaffected by two of the three Reidemeister moves: moves of Type II and Type III do not affect the writhe. Reidemeister move Type I, however, increases or decreases the writhe by 1. This implies that the writhe of a knot is not an isotopy invariant of the knot itself — only the diagram. By a series of Type I moves one can set the writhe of a diagram for a given knot to be any integer at all.

Writhe is also a property of a knot represented as a curve in three-dimensional space. Strictly speaking, a knot is such a curve, defined mathematically as an embedding of a circle in three-dimensional Euclidean space, ${\displaystyle \mathbb {R} ^{3}}$. By viewing the curve from different vantage points, one can obtain different projections and draw the corresponding knot diagrams. Its writhe ${\displaystyle \operatorname {Wr} }$ (in the space curve sense) is equal to the average of the integral writhe values obtained from the projections from all vantage points.^[2] Hence, writhe in this situation can take on any real number as a possible value.^[1]

In a paper from 1961,^[3] Gheorghe Călugăreanu proved the following theorem: take a ribbon in ${\displaystyle \mathbb {R} ^{3}}$, let ${\displaystyle \operatorname {Lk} }$ be the linking number of its border components, and let ${\displaystyle \operatorname {Tw} }$ be its total twist. Then the difference ${\displaystyle \operatorname {Lk} -\operatorname {Tw} }$ depends only on the core curve of the ribbon,^[2] and

${\displaystyle \operatorname {Wr} =\operatorname {Lk} -\operatorname {Tw} }$.

In a paper from 1959,^[4] Călugăreanu also showed how to calculate the writhe Wr with an integral. Let ${\displaystyle C}$ be a smooth, simple, closed curve and let ${\displaystyle \mathbf {r} _{1}}$ and ${\displaystyle \mathbf {r} _{2}}$ be points on ${\displaystyle C}$. Then the writhe is equal to the Gauss integral

${\displaystyle \operatorname {Wr} ={\frac {1}{4\pi }}\int _{C}\int _{C}d\mathbf {r} _{1}\times d\mathbf {r} _{2}\cdot {\frac {\mathbf {r} _{1}-\mathbf {r} _{2}}{\left|\mathbf {r} _{1}-\mathbf {r} _{2}\right|^{3}}}}$.

Since writhe for a curve in space is defined as a double integral, we can approximate its value numerically by first representing our curve as a finite chain of ${\displaystyle N}$ line segments. A procedure that was first derived by Michael Levitt^[5] for the description of protein folding and later used for supercoiled DNA by Konstantin Klenin and Jörg Langowski^[6] is to compute

${\displaystyle \operatorname {Wr} =\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {\Omega _{ij}}{4\pi }}=2\sum _{i=2}^{N}\sum _{j<i}{\frac {\Omega _{ij}}{4\pi }}}$,

where ${\displaystyle \Omega _{ij}/{4\pi }}$ is the exact evaluation of the double integral over line segments ${\displaystyle i}$ and ${\displaystyle j}$; note that ${\displaystyle \Omega _{ij}=\Omega _{ji}}$ and ${\displaystyle \Omega _{i,i+1}=\Omega _{ii}=0}$.^[6]

To evaluate ${\displaystyle \Omega _{ij}/{4\pi }}$ for given segments numbered ${\displaystyle i}$ and ${\displaystyle j}$, number the endpoints of the two segments 1, 2, 3, and 4. Let ${\displaystyle r_{pq}}$ be the vector that begins at endpoint ${\displaystyle p}$ and ends at endpoint ${\displaystyle q}$. Define the following quantities:^[6]

${\displaystyle n_{1}={\frac {r_{13}\times r_{14}}{\left|r_{13}\times r_{14}\right|}},\;n_{2}={\frac {r_{14}\times r_{24}}{\left|r_{14}\times r_{24}\right|}},\;n_{3}={\frac {r_{24}\times r_{23}}{\left|r_{24}\times r_{23}\right|}},\;n_{4}={\frac {r_{23}\times r_{13}}{\left|r_{23}\times r_{13}\right|}}}$

Then we calculate^[6]

${\displaystyle \Omega ^{*}=\arcsin \left(n_{1}\cdot n_{2}\right)+\arcsin \left(n_{2}\cdot n_{3}\right)+\arcsin \left(n_{3}\cdot n_{4}\right)+\arcsin \left(n_{4}\cdot n_{1}\right).}$

Finally, we compensate for the possible sign difference and divide by ${\displaystyle 4\pi }$ to obtain^[6]

${\displaystyle {\frac {\Omega }{4\pi }}={\frac {\Omega ^{*}}{4\pi }}{\text{sign}}\left(\left(r_{34}\times r_{12}\right)\cdot r_{13}\right).}$

In addition, other methods to calculate writhe can be fully described mathematically and algorithmically, some of them outperform method above (which has quadratic computational complexity, by having linear complexity).^[6]

DNA will coil when twisted, just like a rubber hose or a rope will, and that is why biomathematicians use the quantity of writhe to describe the amount a piece of DNA is deformed as a result of this torsional stress. In general, this phenomenon of forming coils due to writhe is referred to as DNA supercoiling and is quite commonplace, and in fact in most organisms DNA is negatively supercoiled.^[1]

Any elastic rod, not just DNA, relieves torsional stress by coiling, an action which simultaneously untwists and bends the rod. F. Brock Fuller shows mathematically^[7] how the “elastic energy due to local twisting of the rod may be reduced if the central curve of the rod forms coils that increase its writhing number”.

• DNA supercoiling
• Linking number
• Ribbon theory
• Twist (mathematics)
• Winding number

• Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1

Windsor knot

The Windsor knot, sometimes referred to as a full Windsor (or misleadingly as a double Windsor) to distinguish it from the half-Windsor, is a knot used to tie a necktie. As with other common necktie knots, the Windsor knot is triangular, and the wide end of the tie drapes in front of the narrow end. The Windsor is a wider knot than most common knots, and while not truly symmetric is more balanced than the common four-in-hand knot. The Windsor’s width makes it especially suited to be used with a spread or cutaway collar.

The knot is named after the Duke of Windsor. He is sometimes credited with its invention^[1] alongside his London shirtmaker.^[2] It is however the case that the Duke achieved the wide knot that was his signature by wearing ties of thicker cloth that produced a wider knot from the conventional four-in-hand, and hence the Windsor knot was likely invented to emulate the Duke’s wide knots using ties of normal thickness.^[3]

It is also the only accepted knot for the SD tie of the RAAF and AAFC in Australia.^[4]

The Windsor is notably favored by many United States politicians, such as Donald Trump.^[5]

In the 1999 book The 85 Ways to Tie a Tie, by Thomas Fink and Yong Mao, the Windsor knot is knot 31 and described in that book’s notation as:

• Li Co Ri Lo Ci Ro Li Co T

This notation encodes the following series of steps:

1. Start with the tie draped over the neck, with the seam inward and the wide end of the tie to the right.
2. Cross the wide end over the narrow end.
3. Bring the wide end inward and up so that it passes under the intersection and out under the neck.
4. Bring the wide end over to the right.
5. Bring the wide end inward and left so that it passes under the intersection and out to the left.
6. Bring the wide end up to the center.
7. Bring the wide end inward and down so that it passes under the intersection and out to the right.
8. Bring the wide end over to the left.
9. Bring the wide end inward and up so that it passes under the intersection and out under the neck.
10. Bring the wide end down and thread it between the front-most horizontal segment and the rest of the knot. Pull both ends gently to tighten.

Common variations on the Windsor include:

• Li Co Li Ro Ci Lo Ri Co T (knot 32) (the “Persian Knot”)
• Li Co Ri Lo Ci Lo Ri Co T (knot 33)
• Li Co Li Ro Ci Ro Li Co T (knot 35)

• List of knots

Halter hitch
Category	Hitch
Related	Falconer’s knot, Slippery hitch, Siberian hitch, Noose
Releasing	Quick-release
Typical use	tethering animals
ABoK	#243, #1715, #1804, #1826

The halter hitch is a type of knot used to connect a rope to an object. As the name implies, an animal’s lead rope, attached to its halter, may be tied to a post or hitching rail with this knot. The benefit of the halter hitch is that it can be easily released by pulling on one end of the rope, even if it is under tension. Some sources show the knot being finished with the free end running through the slipped loop to prevent it from working loose or being untied by a clever animal, still allowing easy but not instant untying.^[1][2]

• 
  Halter hitch 1 : Place rope behind, through or around anchor object. Form a loop in the working part of the rope.
• 
  Halter hitch 2 : Pull a bight of the working part behind the standing part and then through the loop formed in first step.

The halter hitch can be derived from the Noose knot by turning the working end into a bight.

The halter hitch is topologically the same knot as the Falconer’s knot, i. e. a slipped overhand knot around the main part.^[3] The falconer has to tie the same knot one handed, throwing the end around the anchor object (the perch), gripping it with a scissoring fingers act, pulling the bight from opposite side of the main part using the back of the thumb.

• 
  Falconer’s knot 1 : Pinching fingers from below, hooking thumb from above
• 
  Falconer’s knot 2 : Hand rotated counterclockwise in a “GO AWAY” sign from below
• 
  Falconer’s knot 3 : End bight scissored between fingers to thumb loop
• 
  Falconer’s knot 4 : End bight slipped through loop around thumb
• 
  Falconer’s knot 5 : Tightened by pulling main part, pushing the knot
• 
  Falconer’s knot 6 : Locked with free end through slip

The halter hitch is similar to other slipped hitches that wrap the main part with small differences:

• The Siberian hitch is one where the bight for the slip is twisted one more time i.e. a slipped Figure-eight knot around the main part. Stronger, traditionally used to tie a horse or reindeer by evenks in Siberia, tied with a method suitable for tying while wearing a glove.
• The Slippery hitch and Buntline hitch are essentially a slipped Clove hitch around the main part. Slippery hitch has the slip placed under the last turn away from the main part, while Buntline hitch has the slip placed under the last turn towards the main part. Easy to tie, secure, easy to untie, Slippery hitch is used several in a row on square-rigged ships for securing the gaskets that bind stowed sails to the yards on top, while Buntline hitch, stronger but more difficult to untie, is used slipped to secure the bottom of open sail.
• The half hitch with slip is one with no extra bight, no extra turn, just a slip inside the half hitch, much weaker than all of the above hitches.

• 
  Siberian hitch
• 
  Slippery hitch
• 
  Untightened slipped buntline hitch
• 
  Half hitch with slip

• List of knots

• Discussion of halter hitch, including tying diagram

Half hitch
A half hitch tied around a pole
Category	Hitch
Origin	Ancient
Related	Two half-hitches, Clove hitch, Munter hitch, single hitch
Releasing	Non-jamming
Typical use	As part of other knots
ABoK	#50, #1662, #1663, #1026, #1717, #1780

The half hitch is a simple hitch knot, where the working end of a line is brought over and under the standing part. Insecure on its own, it is a valuable component of a wide variety of useful and reliable hitches, bends, and knots.

The half hitch is tied with one end of a rope which is passed around an object and secured to its own standing part with a single hitch.

Securing an additional single hitch to the rope’s standing part produces the related knot two half-hitches.^[1]: 283Alternatively, a half hitch may be made secure on its own by placing the final crossing opposite to the turn around the working end. This locks the end in place, and holds fast as long as the hitch is loaded by a steady pull.^[1]: 290 A half hitch in this configuration is sometimes used to tie strings to the bridge of a classical guitar.

Another instance where a half hitch stands on its own without additional embellishment is when added to a timber hitch to help stabilize a load in the direction of pull. A timber hitch is tied on the far end of the load to bind it securely and a half hitch made at the forward end to serve as a guide for the rope. In this instance, the half hitch combined with a timber hitch is known as a killick hitch or kelleg hitch.^[2]

The knot is attractive to the eye and so is used decoratively for French whipping which is also known as half hitch whipping.^[3]

• List of knots
• Buntline hitch

A whipping knot or whipping is a binding of marline twine or whipcord around the end of a rope to prevent its natural tendency to fray.

Some whippings are finished cleanly, as by drawing the bitter end of the cordage beneath the whipping itself. Others are tied off or have the end(s) of the twine sewn through the rope. According to The Ashley Book of Knots, “The purpose of a whipping is to prevent the end of a rope from fraying … A whipping should be, in width, about equal to the diameter of the rope on which it is put … [Two sailmaker’s whippings], a short distance apart, are put in the ends of every reef point, where the constant ‘whipping’ against the sail makes the wear excessive; this is said to be the source of the name whipping.”^[1] The other type of stopping knot is a seizing knot.

Whipping is suitable for synthetic and natural stranded and braided lines, including 3-strand rope, 4-strand cable and 8-strand multiplait, as well as concentric and braided constructions.

Multiple turns of twine (sometimes called small stuff for smaller lines) or heavier whipcord (for large diameter cables and ropes) are tightly wrapped around a rope’s cut end to prevent its fibers from unlaying.

Usually one end of the whipping cord is looped along the rope to be whipped, and the remaining cord wound tightly over the loop. Finally the loose end of the wound whipping is passed through the loop so that both ends may be drawn securely inside the winding.^[2]

Whippings may also be applied by hand or using a palm and needle, and either simply tied off or made neat and permanent by reeving the twine’s cut ends into or behind the whipping, sewing them to adjacent strands, or through the rope itself.

In applications where a lot of flexing is expected, the whipping may be impregnated with dilute spar varnish or superglue.

• Bend knot
• List of knots
• Rope splicing
• Strangle knot

• Grog. “Sailmaker’s Whipping”. Animated Knots.
• Grog. “Sailor’s (Simple) Whipping”. Animated Knots.
• Grog. “West Country Whipping”. Animated Knots.