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In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement.^[1] Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include composite knots, cable knots, and Whitehead doubles. A satellite link is one that orbits a companion knot K in the sense that it lies inside a regular neighborhood of the companion.^[2]: 217

A satellite knot ${\displaystyle K}$ can be picturesquely described as follows: start by taking a nontrivial knot ${\displaystyle K’}$ lying inside an unknotted solid torus ${\displaystyle V}$. Here “nontrivial” means that the knot ${\displaystyle K’}$ is not allowed to sit inside of a 3-ball in ${\displaystyle V}$ and ${\displaystyle K’}$ is not allowed to be isotopic to the central core curve of the solid torus. Then tie up the solid torus into a nontrivial knot.

This means there is a non-trivial embedding ${\displaystyle f\colon V\to S^{3}}$ and ${\displaystyle K=f\left(K’\right)}$. The central core curve of the solid torus ${\displaystyle V}$ is sent to a knot ${\displaystyle H}$, which is called the “companion knot” and is thought of as the planet around which the “satellite knot” ${\displaystyle K}$ orbits. The construction ensures that ${\displaystyle f(\partial V)}$ is a non-boundary parallel incompressible torus in the complement of ${\displaystyle K}$. Composite knots contain a certain kind of incompressible torus called a swallow-follow torus, which can be visualized as swallowing one summand and following another summand.

Since ${\displaystyle V}$ is an unknotted solid torus, ${\displaystyle S^{3}\setminus V}$ is a tubular neighbourhood of an unknot ${\displaystyle J}$. The 2-component link ${\displaystyle K’\cup J}$ together with the embedding ${\displaystyle f}$ is called the pattern associated to the satellite operation.

A convention: people usually demand that the embedding ${\displaystyle f\colon V\to S^{3}}$ is untwisted in the sense that ${\displaystyle f}$ must send the standard longitude of ${\displaystyle V}$ to the standard longitude of ${\displaystyle f(V)}$. Said another way, given any two disjoint curves ${\displaystyle c_{1},c_{2}\subset V}$, ${\displaystyle f}$ preserves their linking numbers i.e.: ${\displaystyle \operatorname {lk} (f(c_{1}),f(c_{2}))=\operatorname {lk} (c_{1},c_{2})}$.

When ${\displaystyle K’\subset \partial V}$ is a torus knot, then ${\displaystyle K}$ is called a cable knot. Examples 3 and 4 are cable knots. The cable constructed with given winding numbers (m,n) from another knot K, is often called the (m,n) cable of K.

If ${\displaystyle K’}$ is a non-trivial knot in ${\displaystyle S^{3}}$ and if a compressing disc for ${\displaystyle V}$ intersects ${\displaystyle K’}$ in precisely one point, then ${\displaystyle K}$ is called a connect-sum. Another way to say this is that the pattern ${\displaystyle K’\cup J}$ is the connect-sum of a non-trivial knot ${\displaystyle K’}$ with a Hopf link.

If the link ${\displaystyle K’\cup J}$ is the Whitehead link, ${\displaystyle K}$ is called a Whitehead double. If ${\displaystyle f}$ is untwisted, ${\displaystyle K}$ is called an untwisted Whitehead double.

• 
  Example 1: A connect-sum of a trefoil and figure-8 knot.
• 
  Example 2: The Whitehead double of the figure-8.
• 
  Example 3: A cable of a connect-sum.
• 
  Example 4: A cable of a trefoil.
• 
  Example 5: A knot which is a 2-fold satellite i.e.: it has non-parallel swallow-follow tori.
• 
  Example 6: A knot which is a 2-fold satellite i.e.: it has non-parallel swallow-follow tori.

Examples 5 and 6 are variants on the same construction. They both have two non-parallel, non-boundary-parallel incompressible tori in their complements, splitting the complement into the union of three manifolds. In 5, those manifolds are: the Borromean rings complement, trefoil complement, and figure-8 complement. In 6, the figure-8 complement is replaced by another trefoil complement.

In 1949^[3] Horst Schubert proved that every oriented knot in ${\displaystyle S^{3}}$ decomposes as a connect-sum of prime knots in a unique way, up to reordering, making the monoid of oriented isotopy-classes of knots in ${\displaystyle S^{3}}$ a free commutative monoid on countably-infinite many generators. Shortly after, he realized he could give a new proof of his theorem by a close analysis of the incompressible tori present in the complement of a connect-sum. This led him to study general incompressible tori in knot complements in his epic work Knoten und Vollringe,^[4] where he defined satellite and companion knots.

Schubert’s demonstration that incompressible tori play a major role in knot theory was one several early insights leading to the unification of 3-manifold theory and knot theory. It attracted Waldhausen’s attention, who later used incompressible surfaces to show that a large class of 3-manifolds are homeomorphic if and only if their fundamental groups are isomorphic.^[5] Waldhausen conjectured what is now the Jaco–Shalen–Johannson-decomposition of 3-manifolds, which is a decomposition of 3-manifolds along spheres and incompressible tori. This later became a major ingredient in the development of geometrization, which can be seen as a partial-classification of 3-dimensional manifolds. The ramifications for knot theory were first described in the long-unpublished manuscript of Bonahon and Siebenmann.^[6]

In Knoten und Vollringe, Schubert proved that in some cases, there is essentially a unique way to express a knot as a satellite. But there are also many known examples where the decomposition is not unique.^[7] With a suitably enhanced notion of satellite operation called splicing, the JSJ decomposition gives a proper uniqueness theorem for satellite knots.^[8][9]

• Hyperbolic knot
• Torus knot
• Bing double

Butterfly loop
A butterfly loop with a carabiner.
Names	Butterfly loop, alpine butterfly knot, butterfly knot, lineman’s loop, lineman’s rider
Category	Loop
Related	Alpine butterfly bend, farmer’s loop, artillery loop, span loop
Releasing	Non-jamming
Typical use	Fixed loop on the bight. Isolating a worn section of rope.
ABoK	#331, #532,[1] #1053
Instructions

The butterfly loop, also known as lineman’s loop, butterfly knot, alpine butterfly knot and lineman’s rider, is a knot used to form a fixed loop in the middle of a rope. Tied in the bight, it can be made in a rope without access to either of the ends; this is a distinct advantage when working with long climbing ropes. The butterfly loop is an excellent mid-line rigging knot; it handles multi-directional loading well^[2] and has a symmetrical shape that makes it easy to inspect.^[2] In a climbing context it is also useful for traverse lines, some anchors, shortening rope slings, and for isolating damaged sections of rope.^[3]

The earliest known presentation of the knot was in A.A. Burger’s 1914 work Rope and Its Uses, included in an agricultural extension bulletin from what is now Iowa State University.^[4] Burger called the knot a lineman’s rider stating it was often used by “linemen and especially telephone men”. The knot’s security and ability to withstand tension in any direction are both discussed.^[5]

The knot’s association with mountaineering—and with butterflies—originates from a 1928 article in Alpine Journal by C.E.I. Wright and J.E. Magowan.^[6] The authors claim to have developed the butterfly noose themselves while attempting to improve the selection of knots available to climbers. The name is “so styled on the basis of a more or less fanciful resemblance imagined in the form of the knot.” In the second part of the article they express dissatisfaction regarding their earlier use of the word “noose,” since the knot is non-collapsing, and refer to the knot as butterfly loop or simply butterfly.^[7] Wright and Magowan call the butterfly loop “new,” along with several other of their knots, in the sense they were unable to identify any earlier record of them. However, they prudently added that it “might be rash to claim they have never been used before.”^[8]

Clifford Ashley presented the knot in 1944 (text & image #1053), calling it the lineman’s loop; he attributed its first publication to J.M. Drew, but made no specific reference as to the source of this claim.^[9] A 1912 article called “Some Knots and Splices” by Drew appears in the bibliography of The Ashley Book of Knots.^[10] A 1913 reprint of this Drew article does not mention the butterfly loop.^[11] Nor does Drew’s 1942 book Ropework : Knots, Hitches, Splices, Halters –and presumably earlier edition 1936 (but which has 66pp vs. 58 for 1942?!)– own book on knots present this knot. But in his contributed “Chapter 12 Rope Work”, pp.202 .. 252 to Lester Griswold’s Handicraft does present the knot. (Curiously, Ashley gives no hint that this book which he twice praises contains a full knots chapter written by Drew!)

The loop is typically attached to a climbing harness by 2 carabiners together with gates to opposite sides from each other.

It can also be used to isolate a worn section of rope, where the knot is tied such that the worn section is isolated in the loop (which of course does not receive a carabiner nor bear any loads in this case).^[3] The loop portion is isolated when the other two legs are loaded, and in fact the butterfly can be tied as a bend with the ends emerging where the loop would be.^[12][13]

Errors in tying the butterfly loop can produce a similar looking but inferior knot, the so-called “false butterfly”, which is prone to slipping. However, some sources suggest this behavior can be exploited purposely for shock absorption.^[3] Wright and Magowan called this less secure loop knot the “half-hitch noose”.^[14]

• Forms stable, secure loop after initial setting
• Allows for the knot to be loaded three ways; by each end of the main line and the loop
• Relatively easy to untie after loading (more difficult if wet)
• Size of loop can be adjusted more easily than with bulkier or more complex loop knots
• Easy to inspect^[2]
• Can easily be tied with gloves on
• Can easily be tied one-handed

• Difficult to tie around a solid ring or similar object, as when a rethreaded figure eight is needed
• Improper tying can result in similar looking but inferior “false butterfly” knot
• Works best with softer ropes^[2]

The double butterfly loop has two non-collapsing loops, allowing for two clip-in points, both of which have the same advantages and disadvantages of a single-loop butterfly.^[12][15]

• List of knots

• Alpine Butterfly Knot
• Butterfly Loop animation

The San Diego jam knot is a common fishing knot used to tie a line to the hook, swivel, clip, or artificial fly. This knot is also known as the San Diego knot, reverse clinch knot or Heiliger knot.

This is a common knot used by fishermen^[1] because it is simple to tie, is strong^[2] and can be used with many kinds of line including mono-filament, fluorocarbon, and braided fishing line.^[3] It is an alternative to another fishing knot, the clinch knot.^[4]

The San Diego jam knot is intended to be tied to a ring or a hook with an eye on the back end. It is tied by first passing the main line through the eye, and then doubling the free end back over the main line. Next the fisherman wraps the free end around the doubled main line five times (more turns may be recommended for light line or fewer for heavy line)^[5] working towards the eye. The free end is then passed through the loop that has formed at the eye, and subsequently through the loop around the main line that was formed by the first wrap. The dampened main line and free end are pulled to snug the knot tight.^[5][6] A variation tied using line that is doubled prior to passing it through the eye is known as the doubled San Diego jam knot.^[7][8]

This knot is thought to have originated as a quick and reliable way to tie the heavy “iron” jigs by fishermen chasing tuna on long-range boats, such as those that fished in Mexican waters.^[3][9]

• NorthAmericanFishing (15 April 2011). “Knot Wars Excerpt: How To Tie The Legendary San Diego Jam Fishing Knot”. YouTube. North American Fishing Club. Retrieved 17 November 2022.

• List of knots

Sailor’s hitch
Category	Hitch
Releasing	Non-jamming

The sailor’s hitch is a secure, jam-proof hitch knot.^[1] A hitch knot is a type of knot that has the ability to fit to the size and shape of an object that it is being tied to.^[2]

The sailor’s hitch is similar in function and appearance to the swing hitch.

The sailor’s hitch can be used in such a way that allows a smaller rope to be attached to a large rope. The smaller rope should be pulled to the left while the bight should go through the final tuck to form the final product of a sailor’s hitch.

This knot can also serve the purpose of a cleat hitch.

There is another variation of the knot with several more turns that is called the gripping sailor’s hitch.^[3] The gripping sailor’s hitch is commonly confused with the icicle hitch, but it has distinctions with the last tuck of the knot that allows them to be different.^[4]

The sailor’s knot is used in the following circumstances:

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

• Video: Tying the Gripping Sailor’s Hitch

Butterfly Bend
Names	Butterfly Bend, Strait bend[1]
Category	Bend
Related	butterfly loop, Hunter’s bend, Zeppelin bend, Ashley’s bend
Releasing	Non-jamming
Caveat	Errors in tying can result in a similar looking but insecure bend
Instructions

The butterfly bend is a knot used to join the ends of two ropes together. It is the analogous bend form of the butterfly loop,^[1] in that it is the butterfly loop with the loop cut.^[2] The observation that the butterfly loop is secure enough to isolate a worn or damaged section of rope within the loop indicated that the bend form of the knot would be similarly secure.^[3]

When Phil D. Smith made the first known presentation of the Hunter’s bend in 1953 (under the name “rigger’s bend”),^[4] he described it as a modification to the butterfly bend.^[3] While the bend form had been known to mountaineers, nautical rigger Brion Toss brought the knot to a wider audience when he published it in 1975. Unaware of the earlier publication, Toss called the butterfly bend the strait bend after the Strait of Juan de Fuca.^[1][5]

The butterfly bend can be tied using a subset of the methods used for tying the loop form by holding the two rope ends together and treating them as if they were a single bight. However, specific methods have been developed for tying the bend form directly, including the one shown below and characterizable using the mnemonic device “A d through a b; ‘twixt the two and toward me”:

A properly tied butterfly bend should be as secure as the equivalent loop form.^[1] However, subtle positioning errors during the above shown tying method can result in a similar looking but insecure bend knot.^[6]

• Knot
• List of bend knots
• List of knots

Sailmaker’s whipping
Category	Whipping
Related	West Country whipping
Typical use	Whipping
ABoK	#3446

The sailmaker’s whipping is one of the most durable and stable of rope whippings known. According to The Ashley Book of Knots, “palm-and-needle whipping, or sailmaker’s whipping, is the most satisfactory of all.”^[1]

The West Country whipping is a quick practical method using twine, having 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.

• List of knots

• Grog. “Sailmaker’s Whipping”. Animated Knots. Retrieved May 5, 2013.

In mathematics the Burau representation is a representation of the braid groups, named after and originally studied by the German mathematician Werner Burau^[1] during the 1930s. The Burau representation has two common and near-equivalent formulations, the reduced and unreduced Burau representations.

Consider the braid group B_n to be the mapping class group of a disc with n marked points D_n. The homology group H₁(D_n) is free abelian of rank n. Moreover, the invariant subspace of H₁(D_n) (under the action of B_n) is primitive and infinite cyclic. Let π : H₁(D_n) → Z be the projection onto this invariant subspace. Then there is a covering space C_n corresponding to this projection map. Much like in the construction of the Alexander polynomial, consider H₁(C_n) as a module over the group-ring of covering transformations Z[Z], which is isomorphic to the ring of Laurent polynomials Z[t, t⁻¹]. As a Z[t, t⁻¹]-module, H₁(C_n) is free of rank n − 1. By the basic theory of covering spaces, B_n acts on H₁(C_n), and this representation is called the reduced Burau representation.

The unreduced Burau representation has a similar definition, namely one replaces D_n with its (real, oriented) blow-up at the marked points. Then instead of considering H₁(C_n) one considers the relative homology H₁(C_n, Γ) where γ ⊂ D_n is the part of the boundary of D_n corresponding to the blow-up operation together with one point on the disc’s boundary. Γ denotes the lift of γ to C_n. As a Z[t, t⁻¹]-module this is free of rank n.

By the homology long exact sequence of a pair, the Burau representations fit into a short exact sequence

0 → V_r → V_u → D ⊕ Z[t, t⁻¹] → 0,

where V_r (resp. V_u) is the reduced (resp. unreduced) Burau B_n-module and D ⊂ Zⁿ is the complement to the diagonal subspace, in other words:

${\displaystyle D=\left\{\left(x_{1},\cdots ,x_{n}\right)\in \mathbf {Z} ^{n}:x_{1}+\cdots +x_{n}=0\right\},}$

and B_n acts on Zⁿ by the permutation representation.

Let σ_i denote the standard generators of the braid group B_n. Then the unreduced Burau representation may be given explicitly by mapping

${\displaystyle \sigma _{i}\mapsto \left({\begin{array}{c|cc|c}I_{i-1}&0&0&0\\\hline 0&1-t&t&0\\0&1&0&0\\\hline 0&0&0&I_{n-i-1}\end{array}}\right),}$

for 1 ≤ i ≤ n − 1, where I_k denotes the k × k identity matrix. Likewise, for n ≥ 3 the reduced Burau representation is given by

${\displaystyle \sigma _{1}\mapsto \left({\begin{array}{cc|c}-t&1&0\\0&1&0\\\hline 0&0&I_{n-3}\end{array}}\right),}$

${\displaystyle \sigma _{i}\mapsto \left({\begin{array}{c|ccc|c}I_{i-2}&0&0&0&0\\\hline 0&1&0&0&0\\0&t&-t&1&0\\0&0&0&1&0\\\hline 0&0&0&0&I_{n-i-2}\end{array}}\right),\quad 2\leq i\leq n-2,}$

${\displaystyle \sigma _{n-1}\mapsto \left({\begin{array}{c|cc}I_{n-3}&0&0\\\hline 0&1&0\\0&t&-t\end{array}}\right),}$

while for n = 2, it maps

${\displaystyle \sigma _{1}\mapsto \left(-t\right).}$

If a knot K is the closure of a braid f in B_n, then, up to multiplication by a unit in Z[t, t⁻¹], the Alexander polynomial Δ_K(t) of K is given by

${\displaystyle {\frac {1-t}{1-t^{n}}}\det(I-f_{*}),}$

where f_∗ is the reduced Burau representation of the braid f.

For example, if f = σ₁σ₂ in B₃, one finds by using the explicit matrices above that

${\displaystyle {\frac {1-t}{1-t^{n}}}\det(I-f_{*})=1,}$

and the closure of f_* is the unknot whose Alexander polynomial is 1.

The first nonfaithful Burau representations were found by John A. Moody without the use of computer, using a notion of winding number or contour integration.^[3] A more conceptual understanding, due to Darren D. Long and Mark Paton^[4] interprets the linking or winding as coming from Poincaré duality in first homology relative to the basepoint of a covering space, and uses the intersection form (traditionally called Squier’s Form as Craig Squier was the first to explore its properties).^[5] Stephen Bigelow combined computer techniques and the Long–Paton theorem to show that the Burau representation is not faithful for n ≥ 5.^[6][7][8] Bigelow moreover provides an explicit non-trivial element in the kernel as a word in the standard generators of the braid group: let

${\displaystyle \psi _{1}=\sigma _{3}^{-1}\sigma _{2}\sigma _{1}^{2}\sigma _{2}\sigma _{4}^{3}\sigma _{3}\sigma _{2},\quad \psi _{2}=\sigma _{4}^{-1}\sigma _{3}\sigma _{2}\sigma _{1}^{-2}\sigma _{2}\sigma _{1}^{2}\sigma _{2}^{2}\sigma _{1}\sigma _{4}^{5}.}$

Then an element of the kernel is given by the commutator

${\displaystyle [\psi _{1}^{-1}\sigma _{4}\psi _{1},\psi _{2}^{-1}\sigma _{4}\sigma _{3}\sigma _{2}\sigma _{1}^{2}\sigma _{2}\sigma _{3}\sigma _{4}\psi _{2}].}$

The Burau representation for n = 2, 3 has been known to be faithful for some time. The faithfulness of the Burau representation when n = 4 is an open problem. The Burau representation appears as a summand of the Jones representation, and for n = 4, the faithfulness of the Burau representation is equivalent to that of the Jones representation, which on the other hand is related to the question of whether or not the Jones polynomial is an unknot detector.^[9]

Craig Squier showed that the Burau representation preserves a sesquilinear form.^[5] Moreover, when the variable t is chosen to be a transcendental unit complex number near 1, it is a positive-definite Hermitian pairing. Thus the Burau representation of the braid group B_n can be thought of as a map into the unitary group U(n).

• “Burau’s Theorem“, The Knot Atlas.

The Russian braid (Russian: Русская коса, Russkaya kosa) is a national traditional hairstyle in Russia. It has an ancient history since the times of the Rus’. In modern Russia the hairstyle is still widespread among girls and women, while the symbolism behind the Russian braid is no longer so strong in modern Russia. It also plays an important role in Russian folk dance and Russian folk song ensembles.

In Rus it was uncommon for women to cut their hair, so they would grow it for a long period of time and braid their long hair. The Russian braid symbolized honor and pride and had several meanings in Old Russia. One large and long braid was worn by girls in active search for a groom, while two braids which were tied around the head meant the girl was in marriage. If a colourful ribbon was woven into a braid, it meant that the girl is for marriage, when two ribbons appeared, it meant that the official groom was found. The end of a long braid was often adorned by a kosnik, a piece of jewelry that was made of birch bark.^{[1][2][3][4][5]}

Buntline hitch
Left: Buntline hitch Right: Slipped buntline hitch
Names	Buntline hitch, Stunsail tack bend,[1] Studding sail tack bend, Inside clove hitch
Category	Hitch
Related	Clove hitch, Two half-hitches, Lobster buoy hitch, Corned beef knot
Releasing	Jamming
ABoK	#55, #397, #1229, #1711, #1712, #1807, #1838, #1847, #1918, #2408

The buntline hitch is a knot used for attaching a rope to an object. It is formed by passing the working end around an object, then making a clove hitch around the rope’s standing part and taking care that the turns of the clove hitch progress towards the object rather than away from it. Secure and easily tied, the buntline hitch will jam when subjected to extreme loads. Given the knot’s propensity to jam, it is often made in slipped form.

The buntline hitch, when bent to a yard, makes a more secure knot than two half hitches, but is more liable to jam. It differs from two half hitches in that the second half hitch is inside instead of outside the first one.

Simple and effective, the buntline hitch dates to the Age of Sail, when it was used to secure buntlines to the foot of the sails^[3][4] on square-rigged ships. That the buntline hitch was the preferred knot speaks to its security and reliability.^[3][5] Once set, repeated jerking and slatting tend to tighten it further rather than loosen it.^[6] Its compact size allowed the foot of the sail to be drawn up as closely as possible to the buntline deadeyes on top of the yard.

It has gained in popularity in recent years due to its performance in slippery modern synthetic lines.^[3][6]

The buntline hitch is useful for attaching lines to rings, eyes, posts, rods, and railings where a compact and secure knot is required. The non-slipped form is appropriate for moderate loads or where the knot will not need to be untied often.^[7] If heavily loaded it can be difficult or impossible to untie without the aid of a marlinspike.^[3]

The slipped form is more versatile and convenient when a secure temporary hitch is needed. For example, the slipped buntline hitch is an excellent choice to fasten a line to one side of a vehicle’s luggage rack, with a trucker’s hitch being used on the other side to tension the line over a load placed between them.

The buntline hitch is the same knot as the four-in-hand knot used for neckties.^[6] When it is made in flat material in the manner used to fasten a necktie, the working end is brought more parallel to the standing part during tightening than generally seen when made in cylindrical cordage for load-bearing purposes.

The buntline hitch is simply a clove hitch tied around the standing part, with the turns of the clove hitch progressing towards the object.

Slipped variation

The slipped variation is made by passing a bight through on the final step instead of the end. The knot may be released by pulling at the free end of the rope.

While the buntline hitch is considered a secure knot, the turns of the clove hitch must progress towards the object, otherwise the much less secure two half-hitches will result.

Although not generally required, a round turn can first be made around the object causing the buntline hitch to be even less prone to slipping.^[8]

A buntline hitch may also be tied with an extra half turn for security: see highpoint hitch.

• List of hitch knots

• Buntline Hitch
• Slipped Buntline Hitch Variations
• Stunsail Tack Bend

Running bowline
Category	Running
Origin	Ancient
Related	Bowline, noose
Releasing	Non-jamming
Typical use	Fishing out floating objects that have fallen overboard. Tightening the squaresail to the yard in high winds.
Caveat	None.
ABoK	#1117, #2071

The running bowline is a knot consisting of a bowline looped around its own standing end to create a noose.

The running bowline is strong and secure. It slides easily and can be undone just as simply.

1117. The RUNNING BOWLINE KNOT is referred to by name, in A Four Years’ Voyage by G. Roberts (1726), as the “RUNNING BOWLING KNOT.” It is the knot universally used at sea when a NOOSE is called for. According to an old nautical authority it “is used for throwing over anything out of reach, or anything under water.” Any lumber that has dropped overboard or any rigging that has gone adrift is recovered by its means. ^[1]

Tie a bowline in the end of a line with a small loop, and by appearance one then passes the standing part through the loop to form the noose. However, this method of forming the noose is practicable only for a short piece of line. Alternatively, one can tie the bowline tied directly around the standing part or, having tied the bowline first, one would form a bight in the standing part and pull it through the loop of the bowline.

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