Category: Knots

Half-Windsor knot

The half-Windsor knot, also known as the single Windsor knot,^[1] is a way of tying a necktie which produces a neat, triangular knot. It is larger than the four-in-hand knot and Pratt knot, but smaller than the Windsor knot. The half-Windsor is derived from the Windsor in that it is only brought up around the loop on one side rather than both. It works well with light- and medium-weight fabrics.

According to The 85 Ways to Tie a Tie, the knot is tied thus:

• Li Ro Ci Lo Ri Co T (knot 7)

with a common self-releasing variation being

• Li Ro Ci Ro Li Co T (knot 8)

• Windsor knot – a bulkier knot
• Four-in-hand knot
• List of knots

In the mathematical theory of knots, a knot is tame if it can be “thickened”, that is, if there exists an extension to an embedding of the solid torus ${\displaystyle S^{1}\times D^{2}}$ into the 3-sphere. A knot is tame if and only if it can be represented as a finite closed polygonal chain. In knot theory and 3-manifold theory, often the adjective “tame” is omitted. Smooth knots, for example, are always tame.

Knots that are not tame are called wild and can have pathological behavior.
Every closed curve containing a wild arc is a wild knot.^[1]
It has been conjectured that every wild knot has infinitely many quadrisecants.^[2]

As well as their mathematical study, wild knots have also been studied for their potential for decorative purposes in Celtic-style ornamental knotwork.^[3]

• Wild arc
• Alexander horned sphere
• Eilenberg–Mazur swindle, a technique for analyzing connected sums using infinite sums of knots

In the mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables m and l.

A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One tool used to answer such questions is a knot polynomial, which is computed from a diagram of the knot and can be shown to be an invariant of the knot, i.e. diagrams representing the same knot have the same polynomial. The converse may not be true. The HOMFLY polynomial is one such invariant and it generalizes two polynomials previously discovered, the Alexander polynomial and the Jones polynomial, both of which can be obtained by appropriate substitutions from HOMFLY. The HOMFLY polynomial is also a quantum invariant.

The name HOMFLY combines the initials of its co-discoverers: Jim Hoste, Adrian Ocneanu, Kenneth Millett, Peter J. Freyd, W. B. R. Lickorish, and David N. Yetter.^[1] The addition of PT recognizes independent work carried out by Józef H. Przytycki and Paweł Traczyk.^[2]

The polynomial is defined using skein relations:

${\displaystyle P(\mathrm {unknot} )=1,\,}$

${\displaystyle \ell P(L_{+})+\ell ^{-1}P(L_{-})+mP(L_{0})=0,\,}$

where ${\displaystyle L_{+},L_{-},L_{0}}$ are links formed by crossing and smoothing changes on a local region of a link diagram, as indicated in the figure.

The HOMFLY polynomial of a link L that is a split union of two links ${\displaystyle L_{1}}$ and ${\displaystyle L_{2}}$ is given by

${\displaystyle P(L)={\frac {-(\ell +\ell ^{-1})}{m}}P(L_{1})P(L_{2}).}$

See the page on skein relation for an example of a computation using such relations.

This polynomial can be obtained also using other skein relations:

${\displaystyle \alpha P(L_{+})-\alpha ^{-1}P(L_{-})=zP(L_{0}),\,}$

${\displaystyle xP(L_{+})+yP(L_{-})+zP(L_{0})=0,\,}$

${\displaystyle P(L_{1}\#L_{2})=P(L_{1})P(L_{2}),\,}$, where # denotes the knot sum. In other words, the HOMFLY polynomial of a composite knot is the product of the HOMFLY polynomials of its components.

${\displaystyle P_{K}(\ell ,m)=P_{{\text{Mirror Image}}(K)}(\ell ^{-1},m)}$, so the HOMFLY polynomial can often be used to distinguish between two knots of different chirality. However there exist chiral pairs of knots that have the same HOMFLY polynomial, e.g. knots 9₄₂ and 10₇₁ together with their respective mirror images.^[3]

The Jones polynomial, V(t), and the Alexander polynomial, ${\displaystyle \Delta (t)\,}$ can be computed in terms of the HOMFLY polynomial (the version in ${\displaystyle \alpha }$ and ${\displaystyle z}$ variables) as follows:

${\displaystyle V(t)=P(\alpha =t^{-1},z=t^{1/2}-t^{-1/2}),\,}$

${\displaystyle \Delta (t)=P(\alpha =1,z=t^{1/2}-t^{-1/2}),\,}$

• Kauffman, L.H., “Formal knot theory”, Princeton University Press, 1983.
• Lickorish, W.B.R. “An Introduction to Knot Theory”. Springer. ISBN 0-387-98254-X.

• “Jones-Conway polynomial”, Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. “HOMFLY Polynomial”. MathWorld.
• “The HOMFLY-PT Polynomial“, The Knot Atlas.

Whitehead link
Braid length	5
Braid no.	3
Crossing no.	5
Hyperbolic volume	3.663862377
Linking no.	0
Unknotting no.	1
Conway notation	[212]
A–B notation	52 1
Thistlethwaite	L5a1
Last / Next	L4a1 / L6a1
Other
alternating

In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings, from the overlay of a circle and a figure-eight shaped loop.

Alternating link diagram

Alternative diagram, symmetric by 3d rotation around a vertical line in the plane of the drawing^[1]

A common way of describing this knot is formed by overlaying a figure-eight shaped loop with another circular loop surrounding the crossing of the figure-eight. The above-below relation between these two unknots is then set as an alternating link, with the consecutive crossings on each loop alternating between under and over. This drawing has five crossings, one of which is the self-crossing of the figure-eight curve, which does not count towards the linking number. Because the remaining crossings have equal numbers of under and over crossings on each loop, its linking number is 0. It is not isotopic to the unlink, but it is link homotopic to the unlink.

Although this construction of the knot treats its two loops differently from each other, the two loops are topologically symmetric: it is possible to deform the same link into a drawing of the same type in which the loop that was drawn as a figure eight is circular and vice versa.^[2] Alternatively, there exist realizations of this knot in three dimensions in which the two loops can be taken to each other by a geometric symmetry of the realization.^[1]

In braid theory notation, the link is written

${\displaystyle \sigma _{1}^{2}\sigma _{2}^{2}\sigma _{1}^{-1}\sigma _{2}^{-2}.\,}$

Its Alexander polynomial is

${\displaystyle \Delta (t)=t^{3/2}-3t^{1/2}+3t^{-1/2}-t^{-3/2},}$

since ${\displaystyle {\begin{pmatrix}1&0&0\\-1&1&0\\0&1&-1\end{pmatrix}}}$ is a possible Seifert matrix, or because of its Conway polynomial, which is

${\displaystyle \nabla (z)=z^{3}.}$

Its Jones polynomial is

${\displaystyle V(t)=t^{-{3 \over 2}}\left(-1+t-2t^{2}+t^{3}-2t^{4}+t^{5}\right).}$

This polynomial and ${\displaystyle V(1/t)}$ are the two factors of the Jones polynomial of the L10a140 link. Notably, ${\displaystyle V(1/t)}$ is the Jones polynomial for the mirror image of a link having Jones polynomial ${\displaystyle V(t)}$.

The hyperbolic volume of the complement of the Whitehead link is 4 times Catalan’s constant, approximately 3.66. The Whitehead link complement is one of two two-cusped hyperbolic manifolds with the minimum possible volume, the other being the complement of the pretzel link with parameters (−2, 3, 8).^[3]

Dehn filling on one component of the Whitehead link can produce the sibling manifold of the complement of the figure-eight knot, and Dehn filling on both components can produce the Weeks manifold, respectively one of the minimum-volume hyperbolic manifolds with one cusp and the minimum-volume hyperbolic manifold with no cusps.

The Whitehead link is named for J. H. C. Whitehead, who spent much of the 1930s looking for a proof of the Poincaré conjecture. In 1934, he used the link as part of his construction of the now-named Whitehead manifold, which refuted his previous purported proof of the conjecture.^[4]

• Solomon’s knot
• Weeks manifold
• Whitehead double

• “L5a1 knot-theoretic link“, The Knot Atlas.
• Weisstein, Eric W., “Whitehead link”, MathWorld

Group-based cryptography is a use of groups to construct cryptographic primitives. A group is a very general algebraic object and most cryptographic schemes use groups in some way. In particular Diffie–Hellman key exchange uses finite cyclic groups. So the term group-based cryptography refers mostly to cryptographic protocols that use infinite non-abelian groups such as a braid group.

• Shpilrain–Zapata public-key protocols
• Magyarik–Wagner public key protocol
• Anshel–Anshel–Goldfeld key exchange
• Ko–Lee et al. key exchange protocol

• Non-commutative cryptography

• Paul, Kamakhya; Goswami, Pinkimani; Singh, Madan Mohan. (2022). “ALGEBRAIC BRAID GROUP PUBLIC KEY CRYPTOGRAPHY”, Jnanabha, Vol. 52(2) (2022), 218-223. ISSN 0304-9892 (Print) ISSN 2455-7463 (Online)

• Cryptography and Braid Groups page (archived version 7/17/2017)

Ground-line hitch
Names	Ground-line hitch, picket-line hitch, miller’s knot, sack knot, bag knot
Category	Hitch
Category 2	Binding
Related	Miller’s knot, clove hitch, snuggle hitch, ossel hitch, vibration-proof hitch, Turk’s head
Releasing	Non-jamming
ABoK	#154, #277, #278, #390, #1243, #1676, #1680

The ground-line hitch is a type of knot used to attach a rope to an object. Worked-up and dressed properly, it is more secure than the simpler clove hitch and has less tendency to jam, but does not respond well to swinging. It can also be used as a simple binding knot and is classed among several knots known as the miller’s knot.^[1] The Ground-line hitch is also the start of a three-lead four-bight Turk’s head.^[2]

The knot is named for its use to attach a net to the groundline, a weighted or lead cored rope on the bottom of the net (especially a gillnet).

• List of binding knots
• List of knots

• An Introduction to the Theory of Hitches and Knots

West Country whipping
Category	Whipping
Related	Sailmaker’s whipping
Typical use	Whipping
ABoK	#3458

The West Country whipping is a quick practical whipping knot, a method of using twine to secure the end of a rope to prevent it fraying. It has several advantages: it can be tied without a needle; it is simple to understand and remember; if the whipping fails, the loose ends can usually be re-tied to temporarily prevent the rope’s end from fraying.

West Country whipping was the name given by Biddlecombe in 1848 to this particular practice, but most subsequent seamanship books, including the British Admiralty Manual of Seamanship, have modified the name to West County whipping…I have not seen this whipping used but it has this advantage: if any part breaks it will be a very long while before the whole whipping lets go. The break will be evident and the whipping can be replaced in time.

Half knots are tied alternately behind and in front of the rope until the width of the band of twine approaches the diameter of the rope. A reef (square) knot, or better a series of reef (square) knots, completes the whipping. If a needle is available this string of reef (square) knots can be pulled through the rope to bury the ends. Alternatively, a short bight of another rope can be laid first and used to pull the rope ends through. If the rope is a stranded rope, the ends can usually be pulled through without a needle.

The sailmaker’s whipping is the yardstick for comparison, for its durability. There are two approaches to forming the frapping turns, the source of the durability, both of which are harder to understand and remember compared to the West Country whipping.

• List of knots

• Grog. “West Country Whipping”. Animated Knots. Retrieved May 5, 2013.

The traditional wedding cord, also known as the “wedding lasso”, is a piece of paraphernalia used in some Catholic wedding ceremonies. It is actually a representation of a loop of rosary beads made out of white satin or silk. During the wedding proper, this is traditionally formed into a figure-of-eight shape, and then placed around the neck areas of the bride and the groom after they have made their wedding vows, and are already kneeling on pillows for the pronouncement of a wedding prayer. This cord symbolizes lifetime unity or the everlasting union of the bride and groom when they officially become husband and wife, as well as a symbol of marital protection; while the loops formed signifies their love for one another. After the wedding, this marital twine is typically kept by the bride as a wedding souvenir. Use of the traditional wedding cord for weddings is common in Hispanic countries such as Mexico, the Philippines, and Spain.

After shrouding the bride and groom with the wedding veils, a pair of wedding participants is assigned in placing the wedding cord around the couple, with the groom being the first to be “lassoed” or “looped” by it at the shoulder area.^[1] The cord is held in place by means of pins. In other wedding ceremonies, the wedding cord is tied around the couple’s wrists. The wedding cord stays on and around the couple until the wedding mass or religious service is finished. Then, it is removed by the same pair of wedding participants who were assigned to place the loop around the couple.^[2] The significance of the “lassoing” is to symbolize the unification of the couple in matrimony for their entire lives.^[3]

On the other hand, the ritual for the cord of three strands is performed by the bride and the groom. The groom holds the end of the cord that has a metal ring, while the bride braids the strands together. The braiding is done while an explanation of the significance of the braiding ritual is being read, or while a wedding music is being played, or while a wedding song is being chanted. The resulting braid is kept in place temporarily by a rubber band, and then permanently by a gold thread. The loop can signify the sacramental union itself or simply the, “yoke of marriage.”^[4]

This Hispanic tradition in Spanish was approved by the U.S. Conference of Catholic Bishops in 2010. In September 2016, an English language version was approved and placed in the English Order of Celebrating Matrimony along with the arras.^[5]

Gripping sailor’s hitch
Category	Hitch
Related	Sailor’s hitch, rolling hitch, Icicle hitch
Releasing	Non-jamming
Typical use	Tie one rope to another rope, boom, spar, shaft, etc., and pull lengthwise.

The gripping sailor’s hitch^[a] is a secure, jam-proof friction hitch used to tie one rope to another, or a rope to a pole, boom, spar, etc., when the pull is lengthwise along the object. It will even grip a tapered object, such as a marlin spike, in the direction of taper, similar to the Icicle hitch, and it is much superior to the rolling hitch for that purpose.^[1][2]

• 
  Make 5 turns around the object at opposite side of to the pull direction of the standing part, then cross the standing part to the pull direction and make one more turn
• 
  Cross back over the standing part in front, as you change the turn direction to opposite the wraps, come through from the back, and pass under the standing part (following the pen in pic).
• 
  Tighten up before loading…
• 
  When pulled to the side opposite the 5 turns, this hitch will hold…

• List of knots
• Sailor’s hitch
• Sailor’s knot
• List of friction hitch knots

• Video: Tying the Gripping Sailor’s Hitch

Water knot
Names	Water knot, Tape knot, Ring bend, Grass knot, Overhand follow through
Category	Bend
Related	Overhand knot, beer knot, overhand bend
Typical use	To join webbing for climbing
Caveat	Ends should be left long, knot should be tightened and inspected before each use. Difficult to untie.
ABoK	#296, #1412, #343
Instructions	animatedknots.com

The water knot (also tape knot, ring bend, grass knot, or overhand follow-through) is a knot frequently used in climbing for joining two ends of webbing together, for instance when making a sling.

It is tied by forming an overhand knot in one end and then following it with the other end, feeding in the opposite direction.

The ends should be left at least 7.5 centimetres (3.0 in) long and the knot should be “set” by tightening it with full body weight. The ends can be knotted, taped or lightly sewn to the standing parts to help prevent them from creeping back into the knot.^[1]

The knot can be used for joining flat materials such as leather or tape.^[3]

Once tied, for additional security each end should be tied in a double overhand stopper knot around the other standing end.

Some testing has shown that the water knot, in certain conditions, can slip very slightly but very consistently, with cyclic loading and unloading at relatively low forces; it is the tail on the exterior that slips (this would be the blue tail in the image presented here). In tests using 9/16 in (14.3 mm) tubular nylon webbing, repeated loading and unloading with 250 lbs (113 kg) caused one of the 3 in (76 mm) tails to work back into the knot in just over 800 loading cycles. Another test showed similar results for Spectra tape (but not for new, 1-inch tubular nylon). And yet the knot can be loaded to rupture without slippage. These results validate the need to leave adequate tails and inspect water knots before each use. With single overhand knot safeties on either end, the combination eventually seized and the slipping stopped.^[4]

Although used extensively in climbing and caving, there is some opinion that the water knot is unsafe. According to Walter Siebert, several deaths have been reported due to failure of this knot (although, as in many failed-knot cases, the actual mechanism of failure is unknown, and only conjecture can be inferred). He demonstrates in a video how easily the knot can pull loose if snagged.^[5] Siebert references an article from Pit Schubert in 1995 that details many deaths investigated where the water knot was used with webbing and failed. Schubert drew the conclusion after reviewing the remaining webbing and the sites where these falls took place that the knot can open if it catches on an edge or any protrusion.

However, these analyses fail to note that this uncommon vulnerability can lead to trouble only if (a) the knot will move much under load, so as to pull out enough tail to fail, and (b) the exterior strand is loaded from the top, resulting in a downwards pull by the interior strand (the red one, as shown here) that pulls it away from the snagged exterior strand.

To remove these failure conditions, orientate the knot in the opposite way –interior strand up, exterior strand down– and place it high so as to minimize sideways movements.^[6]

In Germany, the knot is sometimes called Todesknoten, which means death knot.^[7]

• List of bend knots
• List of knots