Category: Knots

Figure-eight knot
Names	Figure-eight knot, figure-of-eight knot, Savoy knot, Flemish knot, double stopper
Category	Stopper
Efficiency	80%
Origin	Ancient
Related	Stevedore knot, figure-eight loop, figure-eight follow through, directional figure eight
Releasing	Jamming
Typical use	General-purpose stopper knot. Replaces the common overhand knot in many uses.
ABoK	#420 #520 #570
Conway Notation	2 2
A/B notation	41
Instructions

The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in sailing, rock climbing and caving as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under strain, often requiring the rope to be cut, the figure-eight will also jam, but is usually more easily undone than the overhand knot.

The figure-eight or figure-of-eight knot is also called (in books) the Flemish knot. The name figure-of-eight knot appears in Lever’s Sheet Anchor; or, a Key to Rigging (London, 1808). The word “of” is nowadays usually omitted. The knot is the sailor’s common single-strand stopper knot and is tied in the ends of tackle falls and running rigging, unless the latter is fitted with monkey’s tails. It is used about ship wherever a temporary stopper knot is required. The figure-eight is much easier to untie than the overhand, it does not have the same tendency to jam and so injure the fiber, and is larger, stronger, and equally secure.

The stevedore knot is an extension of simple figure-eight knot with an additional turn before the end is finally tightened.

• In heraldry, this knot is known as Savoy knot.^[5]
• In the United States Navy, a figure-of-eight badge was formerly worn by enlisted men who had successfully completed the apprentice rating.^[6]
• In The Scout Association in the United Kingdom, awards for gallantry and long service are represented by a cloth figure-of-eight knot emblem in various colours.^[7]

• Figure-eight loop
• Figure-eight knot (mathematics)
• List of knots

• Adams, Colin C. (1994). The knot book: an elementary introduction to the mathematical theory of knots. W. H. Freeman.

In mathematics, a transverse knot is a smooth embedding of a circle into a three-dimensional contact manifold such that the tangent vector at every point of the knot is transverse to the contact plane at that point.

Any Legendrian knot can be C⁰-perturbed in a direction transverse to the contact planes to obtain a transverse knot. This yields a bijection between the set of isomorphism classes of transverse knots and the set of isomorphism classes of Legendrian knots modulo negative Legendrian stabilization.

• Geiges, Hansjörg (2008). An introduction to contact topology; Volume 109 of Cambridge studies in advanced mathematics. Cambridge University Press. p. 94. ISBN 978-0-521-86585-2.

• J. Epstein, D. Fuchs, and M. Meyer, Chekanov–Eliashberg invariants and transverse approximations of Legendrian knots, Pacific J. Math. 201 (2001), no. 1, 89–106.

In knot theory, a branch of mathematics, a knot or link ${\displaystyle K}$
in the 3-dimensional sphere ${\displaystyle S^{3}}$ is called fibered or fibred (sometimes Neuwirth knot in older texts, after Lee Neuwirth) if there is a 1-parameter family ${\displaystyle F_{t}}$ of Seifert surfaces for ${\displaystyle K}$, where the parameter ${\displaystyle t}$ runs through the points of the unit circle ${\displaystyle S^{1}}$, such that if ${\displaystyle s}$ is not equal to ${\displaystyle t}$
then the intersection of ${\displaystyle F_{s}}$ and ${\displaystyle F_{t}}$ is exactly ${\displaystyle K}$.

Fibered knots and links arise naturally, but not exclusively, in complex algebraic geometry. For instance, each singular point of a complex plane curve can be described
topologically as the cone on a fibered knot or link called the link of the singularity. The trefoil knot is the link of the cusp singularity ${\displaystyle z^{2}+w^{3}}$; the Hopf link (oriented correctly) is the link of the node singularity ${\displaystyle z^{2}+w^{2}}$. In these cases, the family of Seifert surfaces is an aspect of the Milnor fibration of the singularity.

A knot is fibered if and only if it is the binding of some open book decomposition of ${\displaystyle S^{3}}$.

• (−2,3,7) pretzel knot

• Harer, John (1982). “How to construct all fibered knots and links”. Topology. 21 (3): 263–280. doi:10.1016/0040-9383(82)90009-X. MR 0649758.
• Gompf, Robert E.; Scharlemann, Martin; Thompson, Abigail (2010). “Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures”. Geometry & Topology. 14 (4): 2305–2347. arXiv:1103.1601. doi:10.2140/gt.2010.14.2305. MR 2740649.

Transom knot
Category	Lashing
Related	Strangle knot, Constrictor knot, Square lashing
Releasing	Jamming
Typical use	Light-duty right-angle lashing
ABoK	#385, #1182, #1255, #3372

The transom knot is a simple lashing knot used to secure two linear objects, such as spars, at right angles to each other.

While often described in relation to the constrictor knot, the underlying structure of the transom knot is the strangle knot.^[1][2] The introduction of a second, perpendicular spar into a loose strangle knot tied around another spar will illustrate this point. In relation to the upper spar, the crossings of the knot come to very closely resemble those of a constrictor knot.

Perhaps because of this Clifford Ashley described the transom knot as both “a modification of”^[3] and “closely related to”^[4] the constrictor knot. Despite these descriptions the transom knot is consistently illustrated in The Ashley Book of Knots as being based on a strangle knot.

Suggested for binding kite sticks by Ashley,^[5] it is useful generally as a light-duty or temporary square lashing. To reinforce, a second transom knot can be made on the opposite side and at a right-angle to the first.^[2][5]

Fiador knot
Names	Fiador knot, Ole fiador knot, Theodore knot, Hackamore diamond knot
Category	Loop
Related	Bottle sling, Diamond knot
Typical use	rope halters, hackamores, and hobbles
ABoK	#1110, #2569

The fiador knot (also Theodore knot) is a decorative, symmetrical knot used in equine applications to create items such as rope halters, hobbles, and components of the fiador on some hackamore designs. As traditionally described, it is a four strand diamond knot in which six of the eight ends loop back into the knot, thus allowing it to be tied with a single line.^[1] While a specific knot is discussed in this article, the fiador knot has also been treated as an entire class of multi-strand knots similarly made with a single line.^[2][3]

The origin of the variant name “Theodore knot”, used in the United States, is a corruption of the Spanish fiador. American cowboys likewise corrupted a number of other closely related terms, substituting “hackamore” for jaquima and “McCarty” for mecate.^[4]

Knotting authority Clifford Ashley relates Philip Ashton Rollins’s suggestion that, “When Theodore Roosevelt, ‘the hero of San Juan Hill,’ visited the Southwest, shortly after the Spanish–American War, it was a foregone conclusion that the Spanish name ‘Fiador’ would be corrupted to ‘Theodore’ in his honor.”^[1]

Considered a difficult knot to tie, cowboys were said to have been able to collect a fee for tying it.^[5] Ashley went so far as to include it in a chapter covering trick knots in The Ashley Book of Knots stating archly, “the trick is to succeed in tying it.”^[6]

Many methods have been devised to tie the fiador knot,^[7] including fixtures used to hold the parts in shape while tying.^[4][8] More recent sources have shown a simpler method of forming the fiador knot using a flat precursor knot.^[9][10][11]

The following images show a method for tying the fiador knot:

There are several ways the fiador knot is used with certain types of horse tack:

• The knot is used on, and shares its name with, the fiador of a hackamore.^[16] The fiador knot holds the four strands of the fiador together under the horse’s jaw, while a doubled bottle sling—sometimes called a “hackamore knot” in this context—is used to attach the fiador to the heel knot of the bosal, or noseband, of the hackamore. A becket hitch is used to secure the fiador around the throatlatch of the horse.^[3] In North America, again according to Ashley, “…the method originated in the South American pampas and worked its way, via Mexico, to the Southwestern cow country, arriving there soon after the conclusion of the Spanish–American War.”^[1]
• On knotted rope halters, the knot often is used under the jaw both as a decorative knot, and also to fashion the lower loop onto which a lead rope is attached. On a rope halter, the fiador knot is made from one continuous piece of rope, and is, along with a series of double overhand knots, one of two types of knots that comprise most rope halters.
• For one style of rope hobbles, a brass ring may be attached to the double loops on one side of the knot to join the hobble for the horse’s other front foot. On the other side, a diamond knot terminates the two loose ends and the single loop is placed over this to encircle the horse’s fetlock. A small rope slide (melted with a solder iron) on this single loop is pushed against the diamond knot to prevent the loop from slipping off the foot.

• List of knots

• Tying a Fiador Knot
• Rope Halter with a Fiador Knot

In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R³. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q are not coprime (in which case the number of components is gcd(p, q)). A torus knot is trivial (equivalent to the unknot) if and only if either p or q is equal to 1 or −1.

The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot.

A torus knot can be rendered geometrically in multiple ways which are topologically equivalent (see Properties below) but geometrically distinct. The convention used in this article and its figures is the following.

The (p,q)-torus knot winds q times around a circle in the interior of the torus, and p times around its axis of rotational symmetry.^{[note 1]}. If p and q are not relatively prime, then we have a torus link with more than one component.

The direction in which the strands of the knot wrap around the torus is also subject to differing conventions. The most common is to have the strands form a right-handed screw for p q > 0.^[3][4][5]

The (p,q)-torus knot can be given by the parametrization

${\displaystyle {\begin{aligned}x&=r\cos(p\phi )\\y&=r\sin(p\phi )\\z&=-\sin(q\phi )\end{aligned}}}$

where ${\displaystyle r=\cos(q\phi )+2}$ and ${\displaystyle 0<\phi <2\pi }$. This lies on the surface of the torus given by ${\displaystyle (r-2)^{2}+z^{2}=1}$ (in cylindrical coordinates).

Other parameterizations are also possible, because knots are defined up to continuous deformation. The illustrations for the (2,3)- and (3,8)-torus knots can be obtained by taking ${\displaystyle r=\cos(q\phi )+4}$, and in the case of the (2,3)-torus knot by furthermore subtracting respectively ${\displaystyle 3\cos((p-q)\phi )}$ and ${\displaystyle 3\sin((p-q)\phi )}$ from the above parameterizations of x and y. The latter generalizes smoothly to any coprime p,q satisfying ${\displaystyle p<q<2p}$.

A torus knot is trivial if and only if either p or q is equal to 1 or −1.^[4][5]

Each nontrivial torus knot is prime^[6] and chiral.^[4]

The (p,q) torus knot is equivalent to the (q,p) torus knot.^[3][5][7] The (p,−q) torus knot is the obverse (mirror image) of the (p,q) torus knot.^[5] The (−p,−q) torus knot is equivalent to the (p,q) torus knot except for the reversed orientation.

Any (p,q)-torus knot can be made from a closed braid with p strands. The appropriate braid word is ^[8]

${\displaystyle (\sigma _{1}\sigma _{2}\cdots \sigma _{p-1})^{q}.}$

(This formula assumes the common convention that braid generators are right twists,^{[4][8][9][10]} which is not followed by the Wikipedia page on braids.)

The crossing number of a (p,q) torus knot with p,q > 0 is given by

c = min((p−1)q, (q−1)p).

The genus of a torus knot with p,q > 0 is

${\displaystyle g={\frac {1}{2}}(p-1)(q-1).}$

The Alexander polynomial of a torus knot is ^[3][8]

${\displaystyle t^{k}{\frac {(t^{pq}-1)(t-1)}{(t^{p}-1)(t^{q}-1)}},}$ where ${\displaystyle k=-{\frac {(p-1)(q-1)}{2}}.}$

The Jones polynomial of a (right-handed) torus knot is given by

${\displaystyle t^{(p-1)(q-1)/2}{\frac {1-t^{p+1}-t^{q+1}+t^{p+q}}{1-t^{2}}}.}$

The complement of a torus knot in the 3-sphere is a Seifert-fibered manifold, fibred over the disc with two singular fibres.

Let Y be the p-fold dunce cap with a disk removed from the interior, Z be the q-fold dunce cap with a disk removed from its interior, and X be the quotient space obtained by identifying Y and Z along their boundary circle. The knot complement of the (p, q) -torus knot deformation retracts to the space X. Therefore, the knot group of a torus knot has the presentation

${\displaystyle \langle x,y\mid x^{p}=y^{q}\rangle .}$

Torus knots are the only knots whose knot groups have nontrivial center (which is infinite cyclic, generated by the element ${\displaystyle x^{p}=y^{q}}$ in the presentation above).

The stretch factor of the (p,q) torus knot, as a curve in Euclidean space, is Ω(min(p,q)), so torus knots have unbounded stretch factors. Undergraduate researcher John Pardon won the 2012 Morgan Prize for his research proving this result, which solved a problem originally posed by Mikhail Gromov.^[11][12]

The (p,q)−torus knots arise when considering the link of an isolated complex hypersurface singularity. One intersects the complex hypersurface with a hypersphere, centred at the isolated singular point, and with sufficiently small radius so that it does not enclose, nor encounter, any other singular points. The intersection gives a submanifold of the hypersphere.

Let p and q be coprime integers, greater than or equal to two. Consider the holomorphic function ${\displaystyle f:\mathbb {C} ^{2}\to \mathbb {C} }$ given by ${\displaystyle f(w,z):=w^{p}+z^{q}.}$ Let ${\displaystyle V_{f}\subset \mathbb {C} ^{2}}$ be the set of ${\displaystyle (w,z)\in \mathbb {C} ^{2}}$ such that ${\displaystyle f(w,z)=0.}$ Given a real number ${\displaystyle 0<\varepsilon \ll 1,}$ we define the real three-sphere ${\displaystyle \mathbb {S} _{\varepsilon }^{3}\subset \mathbb {R} ^{4}\hookrightarrow \mathbb {C} ^{2}}$ as given by ${\displaystyle |w|^{2}+|z|^{2}=\varepsilon ^{2}.}$ The function ${\displaystyle f}$ has an isolated critical point at ${\displaystyle (0,0)\in \mathbb {C} ^{2}}$ since ${\displaystyle \partial f/\partial w=\partial f/\partial z=0}$ if and only if ${\displaystyle w=z=0.}$ Thus, we consider the structure of ${\displaystyle V_{f}}$ close to ${\displaystyle (0,0)\in \mathbb {C} ^{2}.}$ In order to do this, we consider the intersection ${\displaystyle V_{f}\cap \mathbb {S} _{\varepsilon }^{3}\subset \mathbb {S} _{\varepsilon }^{3}.}$ This intersection is the so-called link of the singularity ${\displaystyle f(w,z)=w^{p}+z^{q}.}$ The link of ${\displaystyle f(w,z)=w^{p}+z^{q}}$, where p and q are coprime, and both greater than or equal to two, is exactly the (p,q)−torus knot.^[13]

• Unknot, 3₁ knot (3,2), 5₁ knot (5,2), 7₁ knot (7,2), 8₁₉ knot (4,3), 9₁ knot (9,2), 10₁₂₄ knot (5,3)

Table #	A-B	Image	P	Q	Cross #
0	01		1	0	0
3a1	31		2	3	3
5a2	51		2	5	5
7a7	71		2	7	7
8n3	819		3	4	8
9a41	91		2	9	9
10n21	10124		3	5	10
11a367			2	11	11
13a4878			2	13	13
14n21881			3	7	14
15n41185			4	5	15
15a85263			2	15	15
16n783154			3	8	16
			2	17	17
			2	19	19
			3	10	20
			4	7	21
			2	21	21
			3	11	22
			2	23	23
			5	6	24
			2	25	25
			3	13	26
			4	9	27
			2	27	27
			5	7	28
			3	14	28
			2	29	29
			2	31	31
			5	8	32
			3	16	32
			4	11	33
			2	33	33
			3	17	34
			6	7	35
			2	35	35
			5	9	36
			7	8	48
			7	9	54
			8	9	63

A g-torus knot is a closed curve drawn on a g-torus. More technically, it is the homeomorphic image of a circle in S³ which can be realized as a subset of a genus g handlebody in S³ (whose complement is also a genus g handlebody). If a link is a subset of a genus two handlebody, it is a double torus link.^[14]

For genus two, the simplest example of a double torus knot that is not a torus knot is the figure-eight knot.^[15][16]

• Alternating knot
• Hyperbolic knot
• Irrational winding of a torus
• Satellite knot

• “36 Torus Knots“, The Knot Atlas.
• Weisstein, Eric W. “Torus Knot”. MathWorld.
• Torus knot renderer in Actionscript
• Fun with the PQ-Torus Knot

Farrimond Friction Hitch
Category	Hitch
Origin	Barry Farrimond MBE 2008
Related	Taut-line hitch, Prusik knot, Siberian hitch
Releasing	Quick release
Typical use	Camping, Adjusting line tension

The Farrimond friction hitch is a quick release adjustable friction hitch for use on lines under tension. It is useful when the length of a line will need to be periodically adjusted in order to maintain or adjust tension whilst remaining quick and easy to untie; such as when hanging the ridge line for a Basha. It can be used in very effective conjunction with the Siberian hitch for this purpose. It can also be used as a mooring knot.

The first known presentation of this knot was made by British actor Barry Farrimond MBE in 2008 during a demonstration at the Yellow Wood Bush Camp, Wales.

The diagram below is shown from a birds eye perspective with the green dot representing a fixed point such as a tree or post which a ridge line might be attached to. After passing the working end around the back of this fixed point, create a loop that is then placed on top of the ridge line as in fig 1. Once this has been done take the loop and wrap it around the ridge line (following the directions shown by the red arrows) until you reach fig 4. Next take the working end of the rope and create a bight in it. Follow the red arrow in fig 4 which shows the bight being passed under and through the hoop of rope to form the knot’s quick release mechanism. Once the knot has been tightened up it should be able to resist considerable load on the ridge line whilst remaining easy to adjust and quick to release.

• List of friction hitch knots
• List of knots
• Prusik
• Siberian hitch

• The Farrimond at Animated Knots
• Barry Farrimond’s webpage

Tom fool’s knot
Names	Tom fool’s knot, Tom fool knot, Conjurer’s knot, Bow knot
Category	Trick
Related	Handcuff knot, Sheepshank, Fireman’s chair knot
ABoK	#1141, #2290, #2291, #2534

The Tom fool’s knot, also called the conjurer’s knot or bow knot, is sometimes considered a handcuff knot but is somewhat inferior for this purpose to the knot which usually bears that name.^[1]: 208 It is a good knot with which to commence a slightly fancy sheepshank.^[1]: 210 It is also used as a trick knot due to the speed with which it can be made.^[1]: 406

Tom fool’s knot is believed to be the knot epankylotos brokhos described by the 1st Century Greek physician Heraklas.^[2]

It is formed by making two loops, not exactly overlaying each other. The inner half of each hitch or loop is pulled under and through the outer side of the opposite loop.

• Handcuff knot, a similar knot sometimes incorrectly identified as a Tom fool’s knot
• List of knots

• Ian Knot, shoelace knot based on Tom fool’s knot

Farmer’s loop
Names	Farmer’s loop, Wireman’s knot[1]
Category	Loop
Related	Alpine butterfly knot, Artillery loop, Span loop
Releasing	Non-jamming
Typical use	Climbing, agriculture
ABoK	#1054, #1056, #2565

The farmer’s loop is a knot which forms a fixed loop.^[2] As a midline loop knot made with a bight, it is related to several other similar knots, including the alpine butterfly knot and artillery loop.

If pulled with one hand holding one end, the other hand holding the start side of the loop that is the continuation of the same end, before tightening the knot of the loop, it may capsize to a slip knot with a complicated and heavy knot.

It is tied on one hand to make a loop about twice the size of that hand (use fingers for a smaller one, thumb-hook-to-elbow for a large one), as follows:^[3][4][5]

1. start with the rope 3 times around the palm of one hand, let the ends hang down,
2. then pull the initial middle turn up from the top edge and place it over to the right (of the right loops top edge)
3. then pull the now new middle turn up from the top edge and place it over to the left
4. then pull the now new middle turn up from the top edge and place it over to the right
5. then pull the now new middle turn up to form the loop, dress and tighten before use

The knot is a good one on all three counts—lead, security, and strength. Moreover, the method of tying is both ingenious and distinctive, and once mastered, it is not apt to be forgotten.

To tie: Take three turns around the left arm or hand, according to the size of the material being used. Move the center turn to the outside three times, as indicated by the arrows, first right, then left, and finally right again. Finally, pull out (extend) the center turn, and the knot is ready for use.

Cornell University professor Howard W. Riley published this knot in an agricultural extension pamphlet devoted to farming knots in 1912.^[2] He was shown the knot by a farmer at the 1910 Genesee County Fair in Batavia, New York. Riley noted that he had never seen the knot described in any reference book.^[6]

• List of knots

• Farmer’s Loop

Falconer’s knot
Category	Hitch
Related	Halter hitch
Releasing	Non-jamming
Typical use	Falconry

The falconer’s knot is a knot used in falconry to tether a bird of prey to a perch. Some sources show this knot to be identical to the halter hitch,^[1] but with a specific method of single-handed tying needed when the other hand is occupied holding the bird, which makes this knot very useful.

The falconer’s knot is usually tied one handed with the right hand (using two fingers to hold the end, and the thumb to hook behind the end) as follows:

1. The rope is passed around the perch, with the tail end to the further side extending to the left.
2. The right hand, is placed palm up, underneath both pieces of rope and a middle point of the tail piece is pinch/gripped between the index and middle fingers, as if one were cutting the rope with scissors.
3. The thumb reaches over the main part, and over the “scissoring fingers”, points first down to the right under the tail side, then upwards to hook the tail side rope with the back of the thumbnail,
4. Keeping the thumb in the same position hooking the tail side, the wrist with the pinched tail is then rotated to the right as if signalising “GO AWAY!” so that the back of the hand ends up facing up at the near side of both ropes while the scissoring fingers still hold the tail, and pulled now under the main part to the right.
5. Due to the rotation, the thumb (still over the main part) will have a loop of the tail side wrapped around. The “scissored” rope (still under the main part) is then to be put through the loop around the thumb, pushing with the fingers. The thumb may also help it through the loop.
6. The result is an Overhand knot of the tail, where the knot is around the main part, with a slip i.e. a Halter hitch. This is achieved without involving the left hand which usually is busy holding the animal attached to the main part. The knot is then tightened towards the perch, then the free tail end passed through the new slip loop, to secure (just in case the animal has learned to untie the slip knot by pulling the end).

To untie, one simply pulls out the free end, tugs hard, and it should untie. When securing birds of prey, two knots will often be used as birds can learn to untie them.

• 
  Falconer’s knot 1 : pinching fingers from below, hooking thumb from above
• 
  Falconer’s knot 2 : hand rotated counterclockwise
• 
  Falconer’s knot 3 : Loop around the thumb, end between fingers
• 
  Falconer’s knot 4 : End bight slipped through loop around thumb
• 
  Falconer’s knot 5 : Tightened
• 
  Falconer’s knot 6 : Locked

• List of knots

• Instructional video on tying the Falconer’s knot at youtube.com
• How to Tie a Falconer’s Knot at themodernapprentice.com