Category: Knots

Blackwall hitch
Category	Hitch
Related	Half hitch
Releasing	Non-jamming
Typical use	To temporarily attach a rope to a hook when they both are of equal size.
Caveat	Likely to slip if subjected to more than ordinary strain
ABoK	#1875

The blackwall hitch is a temporary means of attaching a rope to a hook. Made of a simple half hitch over the hook, it will only hold when subjected to constant tension. It is used when the rope and hook are of equal size, but it is likely to slip if subjected to more than ordinary tension. Human life should never be trusted to it.^[1]

• List of knots

P(5,3,-2) = T(5,3) = 10₁₂₄

P(3,3,-2) = T(4,3) = 8₁₉

Only two knots are both torus and pretzel^[1]

In the mathematical theory of knots, a pretzel link is a special kind of link. It consists of a finite number of tangles made of two intertwined circular helices. The tangles are connected cyclicly,^[2] and the first component of the first tangle is connected to the second component of the second tangle, the first component of the second tangle is connected to the second component of the third tangle, and so on. Finally, the first component of the last tangle is connected to the second component of the first. A pretzel link which is also a knot (that is, a link with one component) is a pretzel knot.

Each tangle is characterized by its number of twists: positive if they are counter-clockwise or left-handed, negative if clockwise or right-handed. In the standard projection of the ${\displaystyle (p_{1},\,p_{2},\dots ,\,p_{n})}$ pretzel link, there are ${\displaystyle p_{1}}$left-handed crossings in the first tangle, ${\displaystyle p_{2}}$in the second, and, in general, ${\displaystyle p_{n}}$in the nth.

A pretzel link can also be described as a Montesinos link with integer tangles.

The ${\displaystyle (p_{1},p_{2},\dots ,p_{n})}$ pretzel link is a knot iff both ${\displaystyle n}$ and all the ${\displaystyle p_{i}}$ are odd or exactly one of the ${\displaystyle p_{i}}$ is even.^[3]

The ${\displaystyle (p_{1},\,p_{2},\dots ,\,p_{n})}$ pretzel link is split if at least two of the ${\displaystyle p_{i}}$ are zero; but the converse is false.

The ${\displaystyle (-p_{1},-p_{2},\dots ,-p_{n})}$ pretzel link is the mirror image of the ${\displaystyle (p_{1},\,p_{2},\dots ,\,p_{n})}$ pretzel link.

The ${\displaystyle (p_{1},\,p_{2},\dots ,\,p_{n})}$ pretzel link is isotopic to the ${\displaystyle (p_{2},\,p_{3},\dots ,\,p_{n},\,p_{1})}$ pretzel link. Thus, too, the ${\displaystyle (p_{1},\,p_{2},\dots ,\,p_{n})}$ pretzel link is isotopic to the ${\displaystyle (p_{k},\,p_{k+1},\dots ,\,p_{n},\,p_{1},\,p_{2},\dots ,\,p_{k-1})}$ pretzel link.^[3]

The ${\displaystyle (p_{1},\,p_{2},\,\dots ,\,p_{n})}$ pretzel link is isotopic to the ${\displaystyle (p_{n},\,p_{n-1},\dots ,\,p_{2},\,p_{1})}$ pretzel link. However, if one orients the links in a canonical way, then these two links have opposite orientations.

The (1, 1, 1) pretzel knot is the (right-handed) trefoil; the (−1, −1, −1) pretzel knot is its mirror image.

The (5, −1, −1) pretzel knot is the stevedore knot (6₁).

If p, q, r are distinct odd integers greater than 1, then the (p, q, r) pretzel knot is a non-invertible knot.

The (2p, 2q, 2r) pretzel link is a link formed by three linked unknots.

The (−3, 0, −3) pretzel knot (square knot (mathematics)) is the connected sum of two trefoil knots.

The (0, q, 0) pretzel link is the split union of an unknot and another knot.

(−2, 3, 2n + 1) pretzel links are especially useful in the study of 3-manifolds. Many results have been stated about the manifolds that result from Dehn surgery on the (−2,3,7) pretzel knot in particular.

The hyperbolic volume of the complement of the (−2,3,8) pretzel link is 4 times Catalan’s constant, approximately 3.66. This pretzel link complement is one of two two-cusped hyperbolic manifolds with the minimum possible volume, the other being the complement of the Whitehead link.^[6]

A Montesinos link. In this example, ${\displaystyle e=-3}$ , ${\displaystyle \alpha _{1}/\beta _{1}=-3/2}$ and ${\displaystyle \alpha _{2}/\beta _{2}=5/2}$.

Edible (−2,3,7) pretzel knot

Another edible (–2,3,7) pretzel knot, glazed to near perfection

• Trotter, Hale F.: Non-invertible knots exist, Topology, 2 (1963), 272–280.
• Burde, Gerhard; Zieschang, Heiner (2003). Knots. De Gruyter studies in mathematics. Vol. 5 (2nd revised and extended ed.). Walter de Gruyter. ISBN 3110170051. ISSN 0179-0986. Zbl 1009.57003.

In mathematics, the Birman–Murakami–Wenzl (BMW) algebra, introduced by Joan Birman and Hans Wenzl (1989) and Jun Murakami (1987), is a two-parameter family of algebras ${\displaystyle \mathrm {C} _{n}(\ell ,m)}$ of dimension ${\displaystyle 1\cdot 3\cdot 5\cdots (2n-1)}$ having the Hecke algebra of the symmetric group as a quotient. It is related to the Kauffman polynomial of a link. It is a deformation of the Brauer algebra in much the same way that Hecke algebras are deformations of the group algebra of the symmetric group.

For each natural number n, the BMW algebra ${\displaystyle \mathrm {C} _{n}(\ell ,m)}$ is generated by ${\displaystyle G_{1}^{\pm 1},G_{2}^{\pm 1},\dots ,G_{n-1}^{\pm 1},E_{1},E_{2},\dots ,E_{n-1}}$ and relations:

${\displaystyle G_{i}G_{j}=G_{j}G_{i},\mathrm {if} \left\vert i-j\right\vert \geq 2,}$

${\displaystyle G_{i}G_{i+1}G_{i}=G_{i+1}G_{i}G_{i+1},}$         ${\displaystyle E_{i}E_{i\pm 1}E_{i}=E_{i},}$

${\displaystyle G_{i}+{G_{i}}^{-1}=m(1+E_{i}),}$

${\displaystyle G_{i\pm 1}G_{i}E_{i\pm 1}=E_{i}G_{i\pm 1}G_{i}=E_{i}E_{i\pm 1},}$      ${\displaystyle G_{i\pm 1}E_{i}G_{i\pm 1}={G_{i}}^{-1}E_{i\pm 1}{G_{i}}^{-1},}$

${\displaystyle G_{i\pm 1}E_{i}E_{i\pm 1}={G_{i}}^{-1}E_{i\pm 1},}$      ${\displaystyle E_{i\pm 1}E_{i}G_{i\pm 1}=E_{i\pm 1}{G_{i}}^{-1},}$

${\displaystyle G_{i}E_{i}=E_{i}G_{i}=l^{-1}E_{i},}$     ${\displaystyle E_{i}G_{i\pm 1}E_{i}=lE_{i}.}$

These relations imply the further relations:

${\displaystyle E_{i}E_{j}=E_{j}E_{i},\mathrm {if} \left\vert i-j\right\vert \geq 2,}$

${\displaystyle (E_{i})^{2}=(m^{-1}(l+l^{-1})-1)E_{i},}$

${\displaystyle {G_{i}}^{2}=m(G_{i}+l^{-1}E_{i})-1.}$

This is the original definition given by Birman and Wenzl. However a slight change by the introduction of some minus signs is sometimes made, in accordance with Kauffman’s ‘Dubrovnik’ version of his link invariant. In that way, the fourth relation in Birman & Wenzl’s original version is changed to

1. (Kauffman skein relation)

   ${\displaystyle G_{i}-{G_{i}}^{-1}=m(1-E_{i}),}$

Given invertibility of m, the rest of the relations in Birman & Wenzl’s original version can be reduced to

2. (Idempotent relation)

   ${\displaystyle (E_{i})^{2}=(m^{-1}(l-l^{-1})+1)E_{i},}$

3. (Braid relations)

   ${\displaystyle G_{i}G_{j}=G_{j}G_{i},{\text{if }}\left\vert i-j\right\vert \geqslant 2,{\text{ and }}G_{i}G_{i+1}G_{i}=G_{i+1}G_{i}G_{i+1},}$

4. (Tangle relations)

   ${\displaystyle E_{i}E_{i\pm 1}E_{i}=E_{i}{\text{ and }}G_{i}G_{i\pm 1}E_{i}=E_{i\pm 1}E_{i},}$

5. (Delooping relations)

   ${\displaystyle G_{i}E_{i}=E_{i}G_{i}=l^{-1}E_{i}{\text{ and }}E_{i}G_{i\pm 1}E_{i}=lE_{i}.}$

• The dimension of ${\displaystyle \mathrm {C} _{n}(\ell ,m)}$ is ${\displaystyle (2n)!/(2^{n}n!)}$.
• The Iwahori–Hecke algebra associated with the symmetric group ${\displaystyle S_{n}}$ is a quotient of the Birman–Murakami–Wenzl algebra ${\displaystyle \mathrm {C} _{n}}$.
• The Artin braid group embeds in the BMW algebra: ${\displaystyle B_{n}\hookrightarrow \mathrm {C} _{n}}$.

It is proved by Morton & Wassermann (1989) that the BMW algebra ${\displaystyle \mathrm {C} _{n}(\ell ,m)}$ is isomorphic to the Kauffman’s tangle algebra ${\displaystyle \mathrm {KT} _{n}}$. The isomorphism ${\displaystyle \phi \colon \mathrm {C} _{n}\to \mathrm {KT} _{n}}$ is defined by
and

Define the face operator as

${\displaystyle U_{i}(u)=1-{\frac {i\sin u}{\sin \lambda \sin \mu }}(e^{i(u-\lambda )}G_{i}-e^{-i(u-\lambda )}{G_{i}}^{-1})}$,

where ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ are determined by

${\displaystyle 2\cos \lambda =1+(l-l^{-1})/m}$

and

${\displaystyle 2\cos \lambda =1+(l-l^{-1})/(\lambda \sin \mu )}$.

Then the face operator satisfies the Yang–Baxter equation.

${\displaystyle U_{i+1}(v)U_{i}(u+v)U_{i+1}(u)=U_{i}(u)U_{i+1}(u+v)U_{i}(v)}$

Now ${\displaystyle E_{i}=U_{i}(\lambda )}$ with

${\displaystyle \rho (u)={\frac {\sin(\lambda -u)\sin(\mu +u)}{\sin \lambda \sin \mu }}}$.

In the limits ${\displaystyle u\to \pm i\infty }$, the braids ${\displaystyle {G_{j}}^{\pm }}$ can be recovered up to a scale factor.

In 1984, Vaughan Jones introduced a new polynomial invariant of link isotopy types which is called the Jones polynomial. The invariants are related to the traces of irreducible representations of Hecke algebras associated with the symmetric groups. Murakami (1987) showed that the Kauffman polynomial can also be interpreted as a function ${\displaystyle F}$ on a certain associative algebra. In 1989, Birman & Wenzl (1989) constructed a two-parameter family of algebras ${\displaystyle \mathrm {C} _{n}(\ell ,m)}$ with the Kauffman polynomial ${\displaystyle K_{n}(\ell ,m)}$ as trace after appropriate renormalization.

• Birman, Joan S.; Wenzl, Hans (1989), “Braids, link polynomials and a new algebra”, Transactions of the American Mathematical Society, 313 (1), American Mathematical Society: 249–273, doi:10.1090/S0002-9947-1989-0992598-X, ISSN 0002-9947, JSTOR 2001074, MR 0992598
• Murakami, Jun (1987), “The Kauffman polynomial of links and representation theory”, Osaka Journal of Mathematics, 24 (4): 745–758, ISSN 0030-6126, MR 0927059
• Morton, Hugh R.; Wassermann, Antony J. (1989). “A basis for the Birman–Wenzl algebra”. arXiv:1012.3116 [math.QA].

The Pratt knot is a method of tying a necktie. It is also known as the Shelby knot.^[1][2][3] The knot was created in the late 1950s by Jerry Pratt, an employee of the US Chamber of Commerce.^[4] It was popularized as the Shelby knot after then 92-year-old Pratt taught it in 1986 to television reporter Don Shelby, who he felt had been tying his tie poorly on the air.^[5] Shelby then refined the Pratt knot with local clothier Kingford Bavender and wore it on the air with a spread collar where it stood out and attracted attention for its symmetry and trim precision.^[6]

The knot is a variation on the Nicky knot. Both the Pratt and Nicky knots are tied inside out, though only the Nicky knot is self-releasing. Before its popularization in a 1989 New York Times article, the knot was unknown within the fashion world and not recorded in the tie industry’s standard reference guide of the time, Getting Knotted – 188 Knots for Necks by Davide Mosconi and Riccardo Villarosa in Milan, Italy.^[6][7]

The Pratt knot uses less length than the half-Windsor or Windsor knots, and so is well suited to shorter ties or taller men. Unlike the four-in-hand knot, the Pratt method produces a symmetrical knot. It is of medium thickness.

Using notation from and according to The 85 Ways to Tie a Tie, the knot is tied

• Lo Ci Lo Ri Co T (knot 5).

• List of knots

• Encyclopedia of tie knots at Thomas Fink’s homepage
• Grog. “Pratt (Shelby) tying guide”. Animated Knots.
• Pratt (Shelby) tying guide at TieGuide.com

In mathematics, biquandles and biracks are sets with binary operations that generalize quandles and racks. In the theory of virtual knots, biquandles are analagous to quandles in the theory of classical knots. Biracks and racks have the same relation, while a biquandle is a birack which satisfies some additional conditions.

A birack is a set ${\displaystyle X}$, two right-invertible operations ${\displaystyle {\underline {\triangleright }}}$${\displaystyle ,{\overline {\triangleright }}}$ and a bijection ${\displaystyle \pi \colon X\to X}$ such that for all ${\displaystyle a,b,c\in X}$,

1. ${\textstyle \pi (a{\overline {\triangleright }}a)=a{\underline {\triangleright }}a}$ and ${\displaystyle \pi (a){\overline {\triangleright }}a=a{\underline {\triangleright }}\pi (a)}$.

2. The map ${\displaystyle H\colon X\times X\to X\times X}$ defined by ${\displaystyle H(a,b)=(b{\overline {\triangleright }}a,a{\underline {\triangleright }}b)}$ is invertible.

3. The exchange laws

• ${\displaystyle (a{\underline {\triangleright }}b){\underline {\triangleright }}(c{\underline {\triangleright }}b)=(a{\underline {\triangleright }}c){\underline {\triangleright }}(b{\overline {\triangleright }}c)}$

• ${\displaystyle (a{\underline {\triangleright }}b){\overline {\triangleright }}(c{\underline {\triangleright }}b)=(a{\overline {\triangleright }}c){\underline {\triangleright }}(b{\overline {\triangleright }}c)}$
• ${\displaystyle (a{\overline {\triangleright }}b){\overline {\triangleright }}(c{\overline {\triangleright }}b)=(a{\overline {\triangleright }}c){\overline {\triangleright }}(b{\underline {\triangleright }}c)}$.

If ${\textstyle \pi =id}$ is the identity map, ${\textstyle X}$ is called a biquandle.^[1]

Biracks and biquandles were first introduced by Roger Fenn, Mercedes Jordan-Santana and Louis Kauffman in 2004.^[2]

Note that the three conditions above correspond directly to the three Reidemeister moves in knot therory, showing the close connections between knots and biracks.^[3]

Let ${\textstyle X}$ be a set with two bijection ${\displaystyle \sigma ,\tau \colon X\to X}$ that commute. Then the constant action birack is defined by ${\displaystyle a{\underline {\triangleright }}b=\sigma (a)}$ and ${\displaystyle a{\overline {\triangleright }}b=\tau (x)}$ and ${\displaystyle \pi (a)=\tau ^{-1}(\sigma (a))}$.

Any rack ${\displaystyle (X,\triangleright )}$ is a birack with ${\displaystyle a{\underline {\triangleright }}b=a\triangleright b}$ and ${\displaystyle a{\overline {\triangleright }}b=a}$ for all ${\displaystyle a,b\in X}$. Note that if we insert these operations into the conditions in the definition above, we regain the exact definition of a rack.

Let ${\displaystyle R}$ be a commutative ring with identity and ${\displaystyle X}$ an ${\displaystyle R[s^{\pm 1},t^{\pm 1}]}$-module. Then the Alexander biquandle is defined as ${\displaystyle a{\underline {\triangleright }}b=ta+(s-t)b}$ and ${\displaystyle a{\overline {\triangleright }}b=sa}$. For ${\displaystyle s=1}$ this is a quandle.

• Fenn, Roger; Jordan-Santana, Mercedes; Kauffman, Louis (2004). “Biquandles and Virtual Links”. Topology and its Applications. 145 (1–3): 157–175. doi:10.1016/j.topol.2004.06.008.
• Fenn, Roger; Rourke, Colin; Sanderson, Brian (1993). “An Introduction to Species and the Rack Space”. Topics in Knot Theory. NATO ASI Series. Vol. 399. Springer. pp. 33–55. doi:10.1007/978-94-011-1695-4_4.
• Kauffman, Louis H. (1999). “Virtual Knot Theory”. European Journal of Combinatorics. 20 (7): 663–690. doi:10.1006/eujc.1999.0314.

Portuguese bowline
Category	Loop
Related	Spanish bowline
Typical use	Boat tow, Bosun’s chair, Double anchor, Litter bridle
ABoK	#1072, #1848
Instructions	see external links

The Portuguese bowline (Portuguese: Nó volta do calafate), also known as the Bowline on a coil, Caulker’s bowline, or Lisbon surprise, is a variant of the bowline with two loops. The two loops are adjustable in size. Rope can be pulled from one loop into the other, even after tightening. Among other applications, the knot can be used as an equalising anchor, a litter bridle or a makeshift Bosun’s chair.^[1] It is often regarded as one of the more important bowline variations.^[2][3]

It is sometimes known as the French bowline, not to be confused with the separate French bowline, thanks to its description under that name in the 1922 book Standard Seamanship for the Merchant Service by Felix Riesenberg,^[4] as Riesenberg had been taught the knot by a French sailor in the voyage he described in Under Sail.^[5] The traditional name of this knot is Portuguese bowline.^[6]

It is tied in a way that is similar to regular bowlines.

• Click on a picture to see its full-size version.
• 
  Make an overhand loop (loop with the working end on top).
• 
  Put the working end through the loop.
• 
  Put it through again. From here, the steps are the same as the regular bowline.
• 
  Put it around the standing end and back through the loop.

here is a practical way to tie it after letting the rope go through the two anchor points counter clockwise. It consists of a half hitch of the main part where the main part is at the bottom, going under the middle of the part between the two anchor points, then a bight from the main part going over the middle part and through that half hitch, which then itself gets the end part go through it, as a last step the end part is collapsed to be the bight into the final (flag hitch of the) bowline where the original half hitch squeezes it.

The knot is often tied in the bight by sailors,^[7] rescuers,^[8][9] and others.^[10][11] ABoK #1083, earlier depicted by Hjalmar Öhrvall,^[12] is an example of a Portuguese bowline on a bight.^[13]

Other times it is tied with a follow through,^[14] that is, the knot is retraced.^[15]

A No-twist variant is sometimes tied for specific applications.^[16][17] Other variants involve method of tying.^[18]

Among arborists,^[19] in rescue,^[20][21] and other vertical professions, it is frequently used alongside knots like the equalising eight in two-point anchors.^[22]

It originated as a sailing knot and is still frequently used in sailing.^[23] Its most common application in sailing was to substitute for a Bosun’s chair, but at the risk of suspension trauma.^[5] It is still used to tow boats.^[24] Similar horizontal load applications exist beyond sailing.^[25]

It is used in search and rescue both to form a litter tag.^[26] and in some cases the litter bridle itself.^[27][28][29]

It can be used to make a hanging bike rack.^[30]

In a 2019 slop pull test, the Portuguese bowline on a bight with Yosemite finish^[31] failed on an 11 mm high tenacity polyester rope at a minimum of 36.9 kN when it was pulled from its anchors into a forward facing bight, and on an 8 mm nylon cord at a minimum of 9.6 kN (only 9.1 kN without the Yosemite finish).^[32]

• List of knots

• Root, Dave (2002-10-12). “French Bowline or Portuguese Bowline or ABOK #1072”. Dave.Root.
• Day, Cyrus L.; Jarman, Colin (2012) [2006]. “Portuguese Bowline”. Knots and Splices (2nd ed.). London: Adlard Coles Nautical. p. 19. ISBN 978-0-7136-7748-5.
• Shaw, George Russell (2012) [1933]. “Portuguese Bowline”. Practical and Ornamental Knots (2nd ed.). p. 28. ISBN 978-0-486-46020-8.
• Sherry, John E. (2018). “Portuguese Bowline”. NetKnots.com.
• Appleton, Joe (2023-09-09). “How To Tie A French Bowline Knot”. Boat Safe.
• Nyinx (n.d.). “Portuguese Bowline”. Knots 3D.
• Brumelis, Oscar; Achouch, Sahar (2025-01-02). “Portuguese Bowline Knot”. The Knots Manual.

In knot theory, a field of mathematics, the Bing double of a knot is a link with two components which follow the pattern of the knot and “hook together”. Bing doubles were introduced in Bing (1952) by their namesake, the American mathematician R. H. Bing.^[1] The Bing double of a slice knot is a slice link, though it is unknown whether the converse is true.^[2] The components of a Bing double bound disjoint Seifert surfaces.^[2]

The Bing double of a knot K is defined by placing the Bing double of the unknot in the solid torus surrounding it, as shown in the figure, and then twisting that solid torus into the shape of K.^[2] This definition is similar to that for Whitehead doubles. The Bing double of the unknot is also called the Bing link.^[3]

• Slice knot
• Seifert surface
• Satellite knot
• Whitehead double

• Cochran, Tim; Harvey, Shelly; Leidy, Constance (2008), “Link concordance and generalized doubling operators”, Algebraic & Geometric Topology, 8 (3): 1593–1646, arXiv:0801.3677, doi:10.2140/agt.2008.8.1593.
• Cha, Jae Choon; Livingston, Charles; Ruberman, Daniel (2008), “Algebraic and Heegaard–Floer invariants of knots with slice Bing doubles”, Mathematical Proceedings of the Cambridge Philosophical Society, 144 (2): 403–410, arXiv:math/0612419, doi:10.1017/S0305004107000795.

Ply-split braiding is a technique where one twisted cord (“splitter”) passes through another twisted cord or cords splitting the plies of the latter cords (“splittee” cords). This is unlike weaving or many forms of braiding where cloth is formed by threads interlacing in an over-under sequence. Pattern is formed by cord color, and splitting order.

Ply-split braiding is an ancient art that is practiced for making elaborate camel girths and other animal regalia of hand-spun goat hair, wool or sometimes cotton in northwestern India.

The first written description of the technique appeared in 1976 with Virginia Harvey’s “Split-Ply Twining”.^[1] In the introduction, she describes seeing two camel girths at Convergence 1974, and says that Peter Collingwood “suspected the pieces were produced by pulling one yarn through the ply of another”. The ply-split girths examined for this publication were created with only one technique, now known as single course oblique twining (SCOT).

In 1974 and 1975, two graduate students from UCLA conducted field research in Rajasthan and Gujarat in north-western India. They were introduced to Ishwar Singh, a master girth maker, who spent several weeks with them demonstrating the construction of the girths. Their 1982 publication, “The Ply-Split Camel Girths of West India” ^[2] was the first to document the entire traditional process, including spinning, plying, cordmaking, and three structures of ply-splitting. These are SCOT, mentioned above, as well as plain oblique twining (POT) and two-layered oblique interlacing (TLOI).

The designs showed prominence in the market and were even used to decorate camels for wedding processions.^[3][4]

Today, the ply-split braiding technique is used by fiber artists to create handmade decorative items including neckwear, bags, household décor, garments and three-dimensional structures such as baskets and sculptures.^[5][6][7][8]

Contemporary braid makers use a variety of yarns such as cotton, linen, hemp, silk, paper, or rayon. The ply-splitting process requires minimal equipment: A four-hook cord maker^[9][10] to make the cords, and a gripfid for splitting the plies of one or more cords and drawing a cord back through the split cords.

The premier reference is Peter Collingwood’s The Techniques of Ply-split Braiding.^[11] which presents a comprehensive view of the history and techniques. A more easily accessible history is found in David Fraser’s paper “View From The Shoulders Of Thar Masters: New Space For Ply-Split Braiding”.^[12] Other publications include Julie Hedges, Ply-split Braiding, An Introduction^[13] and Ply-split Braiding, Further Techniques.^[14] Linda Hendrickson’s work includes “Great SCOT! A Beginner’s Guide to Ply-Split Braids in Single-Course Oblique Twining”,^[15] “How to Make Ply-Split Baskets”,^[16] “How to Make Ply-Split Braids & Bands”,^[17] and several instructional videos on YouTube.^{[18][19][20][21][22][23][24]} There have been many publications in weaving journals.^{[25][26][27][28][29][30][31][32]}

In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II₁ subfactor.^[1] They also provide an appropriate algebraic framework for many knot invariants (in particular the Jones polynomial), and have been used in describing the properties of Khovanov homology with respect to tangle composition.^[2][3] Any subfactor planar algebra provides a family of unitary representations of Thompson groups.^[4]
Any finite group (and quantum generalization) can be encoded as a planar algebra.^[1]

The idea of the planar algebra is to be a diagrammatic axiomatization of the standard invariant.^[1][5][6]

Bimini twist

The Bimini twist ^[1] is a fishing knot used for offshore trolling and sportsfishing and the creation of double-line leaders.

A Bimini twist creates a loop at the end of the line in which it is tied. The loop is secured at the top with a long barrel of coiled line created by the tying process. A Bimini twist loop is stronger than the line itself. It is one of the rare knots that does not weaken the line in which it is tied. It is a simple method of doubling your fishing line in order to prevent chafing or to create the necessary loop in order to attach a wind-on leader without using strength in the mainline. For use in fishing applications, the old standby is 20-30 initial twists in nylon monofilament and 60 or more initial-twists in Spectra-type braided line.

An article in Sportfishing Magazine in February 2007 made the claim that fewer twists created greater strength. However, the holding mechanism in a Bimini Twist is the friction created by the twists. It was quickly and has since been often demonstrated that the 12-twist knot (proposed in the article) in Spectra-braid slipped before breaking. It is not known what testing errors led to the erroneous conclusion that fewer twists made a stronger knot.

• 
  How to tie a Bimini twist
• 
  How to tie a Bimini twist
• 
  Tie with help

• Video instructions on how to tie a Bimini twist
• fisha.com