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The circuit topology of a folded linear polymer is the arrangement of its intra-molecular contacts. Examples of linear polymers with intra-molecular contacts are nucleic acids and proteins. Proteins fold via the formation of contacts of various natures, including hydrogen bonds, disulfide bonds, and beta-beta interactions.^[1] RNA molecules fold by forming hydrogen bonds between nucleotides, forming nested or non-nested structures. Contacts in the genome are established via protein bridges including CTCF and cohesins and are measured by technologies including Hi-C.^[2] Circuit topology categorizes the topological arrangement of these physical contacts, that are referred to as hard contacts (or h-contacts). Furthermore, chains can fold via knotting (or the formation of “soft” contacts (s-contacts)). Circuit topology uses a similar language to categorize both “soft” and “hard” contacts, and provides a full description of a folded linear chain. In this framework, a “circuit” refers to a segment of the chain where each contact site within the segment forms connections with other contact sites within the same segment, and thus is not left unpaired. A folded chain can thus be studied based on its constituting circuits.

A simple example of a folded chain is a chain with two hard contacts. For a chain with two binary contacts, three arrangements are available: parallel (P), series (S), and crossed (X). For a chain with n contacts, the topology can be described by an n by n matrix in which each element illustrates the relation between a pair of contacts and may take one of the three states, P, S and X. Multivalent contacts can also be categorised in full or via decomposition into several binary contacts. Similarly, circuit topology allows for the classification of the pairwise arrangements of chain crossings and tangles, thus providing a complete 3D description of folded chains. Furthermore, one can apply circuit topology operations to soft and hard contacts to generate complex folds, using a bottom-up engineering approach.

Both knot theory and circuit topology aim to describe chain entanglement, making it important to understand their relationship. Knot theory considers any entangled chain as a connected sum of prime knots, which are themselves undecomposable. Circuit topology splits any entangled chains (including prime knots) into basic structural units called soft contacts, and lists simple rules on how soft contacts can be put together.^[3][4] An advantage of circuit topology is that it can be applied to open linear chains with intra-chain interactions, so-called hard contacts.^[5] This enabled topological analysis of proteins and genomes, which are often described as unknot in knot theory.^[6][7] Finally, circuit topology enables studying interactions between hard contacts and entanglements and can identify slip knots, while knot theory typically overlooks hard contacts and split knots. Thus, circuit topology serves as a complementary approach to knot theory.

Circuit topology has implications for folding kinetics and molecular evolution and has been applied to engineer polymers including molecular origami.^[8][9] Circuit topology along with contact order and size are determinants of the folding rate of linear polymers.^[10] It has also been applied to quantify the conformational organisation of intrinsically disordered proteins.^[11][12][13] In protein structure prediction, coarse-grained generative approaches recover global topological features of protein folds and enable rapid contact map prediction.^[14] Circuit topology can be applied to characterise the topology of multi-chain systems as well, including biomolecular condensates and aggregates.^[15][16] For example, it has been used to characterise coil–globule transitions and aggregation in polymers, revealing multichain topological motifs during collective structural transitions.^[17] In addition, circuit topology has been applied to classify conformational substates in amyloid polypeptides, providing insight into aggregation mechanisms and their modulation by small molecules.^[18] Finally, the topology approach can also be used for medical applications including disease analysis,^[19] and drug response predictions.^[20][21]

• Topology (chemistry)

• Scalvini, Barbara; Sheikhhassani, Vahid; Mashaghi, Alireza (2021). “Topological principles of protein folding”. Physical Chemistry Chemical Physics. 23 (37): 21316–21328. Bibcode:2021PCCP…2321316S. doi:10.1039/D1CP03390E. hdl:1887/3277889. PMID 34545868. S2CID 237583577.
• Golovnev, Anatoly; Mashaghi, Alireza (September 2020). “Generalized Circuit Topology of Folded Linear Chains”. iScience. 23 (9) 101492. Bibcode:2020iSci…23j1492G. doi:10.1016/j.isci.2020.101492. PMC 7481252. PMID 32896769.
• Heidari, Maziar; Schiessel, Helmut; Mashaghi, Alireza (24 June 2020). “Circuit Topology Analysis of Polymer Folding Reactions”. ACS Central Science. 6 (6): 839–847. doi:10.1021/acscentsci.0c00308. PMC 7318069. PMID 32607431.

Cinquefoil
Common name	Double overhand knot
Arf invariant	1
Braid length	5
Braid no.	2
Bridge no.	2
Crosscap no.	1
Crossing no.	5
Genus	2
Hyperbolic volume	0
Stick no.	8
Unknotting no.	2
Conway notation	[5]
A–B notation	51
Dowker notation	6, 8, 10, 2, 4
Last / Next	41 / 52
Other
alternating, torus, fibered, prime, reversible

In knot theory, the cinquefoil knot, also known as Solomon’s seal knot or the pentafoil knot, is one of two knots with crossing number five, the other being the three-twist knot. It is listed as the 5₁ knot in the Alexander-Briggs notation, and can also be described as the (5,2)-torus knot. The cinquefoil is the closed version of the double overhand knot.

The cinquefoil is a prime knot. Its writhe is 5, and it is invertible but not amphichiral.^[1] Its Alexander polynomial is

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

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

${\displaystyle \nabla (z)=z^{4}+3z^{2}+1}$,

and its Jones polynomial is

${\displaystyle V(q)=q^{-2}+q^{-4}-q^{-5}+q^{-6}-q^{-7}.}$

These are the same as the Alexander, Conway, and Jones polynomials of the knot 10₁₃₂. However, the Kauffman polynomial can be used to distinguish between these two knots.

The name “cinquefoil” comes from the five-petaled flowers of plants in the genus Potentilla.

• Pentagram
• Trefoil knot
• 7₁ knot
• Skein relation

• A Pentafoil Knot at the Wayback Machine (archived June 4, 2004)

Simple Simon Under
Category	Bend
Efficiency	high
Origin	Harry Asher, published in 1989
Related	simple Simon over bend, simple Simon symmetric bend, simple Simon double bend
Releasing	Fair
Typical use	suitable for dissimilar ropes, works well with synthetic ropes.

The simple Simon under bend is a knot belonging to the category bend. It was invented by Harry Asher. It is more secure than the similar Simple Simon over and more effective with quite large differences in thickness of the two ropes.^[1]

The simple Simon under holds well even with different sized ropes, or slippery synthetic ropes.^[2]

• 
  The Sheet bend was the starting point of developing the Simple Simon over bend.^[3]
• 
  Simple Simon Over.
  The working part passes over the standing (loaded) part of the rope.
• 
  Simple Simon Under.
  The working part passes under the standing (loaded) part of the rope.

Tie as shown in the images. Note that, as in the sheet bend, the two running ends should emerge on the same side of the knot.^[4]

• 
  Form a bight with the left rope.
• 
  Pass the right rope down through the bight.
• 
  Go underneath of the working end over the bight.
• 
  Pass underneath the bight.
• 
  The working part passes beneath the standing (loaded) part of the right,^[5] you got an X.
• 
  Bring your working end down up through the bight
• 
  Tighten the bend.

• Knot
• List of bend knots
• List of knots

In mathematics, a chord diagram consists of a cyclic order on a set of objects, together with a one-to-one pairing (perfect matching) of those objects. Chord diagrams are conventionally visualized by arranging the objects in their order around a circle, and drawing the pairs of the matching as chords of the circle.

The number of different chord diagrams that may be given for a set of ${\displaystyle 2n}$ cyclically ordered objects is the double factorial ${\displaystyle (2n-1)!!}$.^[1] There is a Catalan number of chord diagrams on a given ordered set in which no two chords cross each other.^[2] The crossing pattern of chords in a chord diagram may be described by a circle graph, the intersection graph of the chords: it has a vertex for each chord and an edge for each two chords that cross.^[3]

In knot theory, a chord diagram can be used to describe the sequence of crossings along the planar projection of a knot, with each point at which a crossing occurs paired with the point that crosses it. To fully describe the knot, the diagram should be annotated with an extra bit of information for each pair, indicating which point crosses over and which crosses under at that crossing. With this extra information, the chord diagram of a knot is called a Gauss diagram.^[4] In the Gauss diagram of a knot, every chord crosses an even number of other chords, or equivalently each pair in the diagram connects a point in an even position of the cyclic order with a point in an odd position, and sometimes this is used as a defining condition of Gauss diagrams.^[5]

In algebraic geometry, chord diagrams can be used to represent the singularities of algebraic plane curves.^[6]

• Arrangement of lines
• Kontsevich invariant

Simple Simon Over
Category	Bend
Efficiency	high
Origin	Harry Asher, published in 1989
Related	simple Simon under bend, simple Simon symmetric bend, simple Simon double bend, sheet bend
Releasing	Fair
Typical use	suitable for dissimilar ropes, works well with synthetic ropes.

The simple Simon over bend is a knot belonging to the category bend. The simple Simon under holds well even with slippery synthetic ropes,^[1] but is less secure than the similar simple Simon under.^[2]

The difference is just whether the green working end goes over the green standing (loaded) end (Simple Simon over) or under the green standing (loaded) end (simple Simon under).

It was invented by Dr. Harry Asher^[3] and published in 1989.^[4]

When I had decided that the way to try for new bends was to think of the two halves separately, and then decide how to put them together. There seemed to be no better way than to start with the two halfs that make up the famous Sheet bend … an open loop and a single hitch.

• 
  The Sheet bend was the starting point of developing the Simple Simon Over bend.^[6]
• 
  Simple Simon Over.
  The working part passes over the standing (loaded) part of the rope.
• 
  Simple Simon Under.
  The working part passes under the standing (loaded) part of the rope.

Tie as shown in the images. In the sheet bend, the two running ends should emerge on the same side of the knot.^[7]

• 
  Form a bight with the left rope.
• 
  Pass the right rope down through the bight.
• 
  Go above and over the bight.
• 
  Pass underneath the bight.
• 
  Go over the top, you got an X
• 
  Bring your working end down up through the bight
• 
  Tighten the bend.

• Knot
• List of bend knots
• List of knots

In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral.

A chiral object and its mirror image are said to be enantiomorphs. The word chirality is derived from the Greek χείρ (cheir), the hand, the most familiar chiral object; the word enantiomorph stems from the Greek ἐναντίος (enantios) ‘opposite’ + μορφή (morphe) ‘form’.

The tetrominos S and Z are enantiomorphs in 2-dimensions

S	Z

Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule.

Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral if you can wear them inside-out.^[1]

The J-, L-, S- and Z-shaped tetrominoes of the popular video game Tetris also exhibit chirality, but only in a two-dimensional space. Individually they contain no mirror symmetry in the plane.

A figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry. In Euclidean geometry any isometry can be written as ${\displaystyle v\mapsto Av+b}$ with an orthogonal matrix ${\displaystyle A}$ and a vector ${\displaystyle b}$. The determinant of ${\displaystyle A}$ is either 1 or −1 then. If it is −1 the isometry is orientation-reversing, otherwise it is orientation-preserving.

A general definition of chirality based on group theory exists.^[2] It does not refer to any orientation concept: an isometry is direct if and only if it is a product of squares of isometries, and if not, it is an indirect isometry. The resulting chirality definition works in spacetime.^[3][4]

In two dimensions, every figure which possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry. (An axis of symmetry of a figure ${\displaystyle F}$ is a line ${\displaystyle L}$, such that ${\displaystyle F}$ is invariant under the mapping ${\displaystyle (x,y)\mapsto (x,-y)}$, when ${\displaystyle L}$ is chosen to be the ${\displaystyle x}$-axis of the coordinate system.) For that reason, a triangle is achiral if it is equilateral or isosceles, and is chiral if it is scalene.

Consider the following pattern:

This figure is chiral, as it is not identical to its mirror image:

But if one prolongs the pattern in both directions to infinity, one receives an (unbounded) achiral figure which has no axis of symmetry. Its symmetry group is a frieze group generated by a single glide reflection.

In three dimensions, every figure that possesses a mirror plane of symmetry S₁, an inversion center of symmetry S₂, or a higher improper rotation (rotoreflection) S_n axis of symmetry^[5] is achiral. (A plane of symmetry of a figure ${\displaystyle F}$ is a plane ${\displaystyle P}$, such that ${\displaystyle F}$ is invariant under the mapping ${\displaystyle (x,y,z)\mapsto (x,y,-z)}$, when ${\displaystyle P}$ is chosen to be the ${\displaystyle x}$–${\displaystyle y}$-plane of the coordinate system. A center of symmetry of a figure ${\displaystyle F}$ is a point ${\displaystyle C}$, such that ${\displaystyle F}$ is invariant under the mapping ${\displaystyle (x,y,z)\mapsto (-x,-y,-z)}$, when ${\displaystyle C}$ is chosen to be the origin of the coordinate system.) Note, however, that there are achiral figures lacking both plane and center of symmetry. An example is the figure

${\displaystyle F_{0}=\left\{(1,0,0),(0,1,0),(-1,0,0),(0,-1,0),(2,1,1),(-1,2,-1),(-2,-1,1),(1,-2,-1)\right\}}$

which is invariant under the orientation reversing isometry ${\displaystyle (x,y,z)\mapsto (-y,x,-z)}$ and thus achiral, but it has neither plane nor center of symmetry. The figure

${\displaystyle F_{1}=\left\{(1,0,0),(-1,0,0),(0,2,0),(0,-2,0),(1,1,1),(-1,-1,-1)\right\}}$

also is achiral as the origin is a center of symmetry, but it lacks a plane of symmetry.

Achiral figures can have a center axis.

A knot is called achiral if it can be continuously deformed into its mirror image, otherwise it is called a chiral knot. For example, the unknot and the figure-eight knot are achiral, whereas the trefoil knot is chiral.

• Asymmetry
• Chiral polytope
• Chirality (chemistry)
• Chirality (physics)
• Parity (physics)
• Skewness
• Vertex algebra

• Flapan, Erica (2000). When Topology Meets Chemistry. Outlook. Cambridge University Press and Mathematical Association of America. ISBN 0-521-66254-0.

• Symmetry, Chirality, Symmetry Measures and Chirality Measures: General Definitions
• Chiral Polyhedra by Eric W. Weisstein, The Wolfram Demonstrations Project.
• Chiral manifold at the Manifold Atlas.

The signature of a knot is a topological invariant in knot theory. It may be computed from the Seifert surface.

Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K. The Seifert form of S is the pairing ${\displaystyle \phi :H_{1}(S)\times H_{1}(S)\to \mathbb {Z} }$ given by taking the linking number ${\displaystyle \operatorname {lk} (a^{+},b^{-})}$ where ${\displaystyle a,b\in H_{1}(S)}$ and ${\displaystyle a^{+},b^{-}}$ indicate the translates of a and b respectively in the positive and negative directions of the normal bundle to S.

Given a basis ${\displaystyle b_{1},…,b_{2g}}$ for ${\displaystyle H_{1}(S)}$ (where g is the genus of the surface) the Seifert form can be represented as a 2g-by-2g Seifert matrix V, ${\displaystyle V_{ij}=\phi (b_{i},b_{j})}$. The signature of the matrix ${\displaystyle V+V^{t}}$, thought of as a symmetric bilinear form, is the signature of the knot K.

Slice knots are known to have zero signature.

Knot signatures can also be defined in terms of the Alexander module of the knot complement. Let ${\displaystyle X}$ be the universal abelian cover of the knot complement. Consider the Alexander module to be the first homology group of the universal abelian cover of the knot complement: ${\displaystyle H_{1}(X;\mathbb {Q} )}$. Given a ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$-module ${\displaystyle V}$, let ${\displaystyle {\overline {V}}}$ denote the ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$-module whose underlying ${\displaystyle \mathbb {Q} }$-module is ${\displaystyle V}$ but where ${\displaystyle \mathbb {Z} }$ acts by the inverse covering transformation. Blanchfield’s formulation of Poincaré duality for ${\displaystyle X}$ gives a canonical isomorphism ${\displaystyle H_{1}(X;\mathbb {Q} )\simeq {\overline {H^{2}(X;\mathbb {Q} )}}}$ where ${\displaystyle H^{2}(X;\mathbb {Q} )}$ denotes the 2nd cohomology group of ${\displaystyle X}$ with compact supports and coefficients in ${\displaystyle \mathbb {Q} }$. The universal coefficient theorem for ${\displaystyle H^{2}(X;\mathbb {Q} )}$ gives a canonical isomorphism with ${\displaystyle \operatorname {Ext} _{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),\mathbb {Q} [\mathbb {Z} ])}$ (because the Alexander module is ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$-torsion). Moreover, just like in the quadratic form formulation of Poincaré duality, there is a canonical isomorphism of ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$-modules ${\displaystyle \operatorname {Ext} _{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),\mathbb {Q} [\mathbb {Z} ])\simeq \operatorname {Hom} _{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),[\mathbb {Q} [\mathbb {Z} ]]/\mathbb {Q} [\mathbb {Z} ])}$, where ${\displaystyle [\mathbb {Q} [\mathbb {Z} ]]}$ denotes the field of fractions of ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$. This isomorphism can be thought of as a sesquilinear duality pairing ${\displaystyle H_{1}(X;\mathbb {Q} )\times H_{1}(X;\mathbb {Q} )\to [\mathbb {Q} [\mathbb {Z} ]]/\mathbb {Q} [\mathbb {Z} ]}$ where ${\displaystyle [\mathbb {Q} [\mathbb {Z} ]]}$ denotes the field of fractions of ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$. This form takes value in the rational polynomials whose denominators are the Alexander polynomial of the knot, which as a ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$-module is isomorphic to ${\displaystyle \mathbb {Q} [\mathbb {Z} ]/\Delta K}$. Let ${\displaystyle tr:\mathbb {Q} [\mathbb {Z} ]/\Delta K\to \mathbb {Q} }$ be any linear function which is invariant under the involution ${\displaystyle t\longmapsto t^{-1}}$, then composing it with the sesquilinear duality pairing gives a symmetric bilinear form on ${\displaystyle H_{1}(X;\mathbb {Q} )}$ whose signature is an invariant of the knot.

All such signatures are concordance invariants, so all signatures of slice knots are zero. The sesquilinear duality pairing respects the prime-power decomposition of ${\displaystyle H_{1}(X;\mathbb {Q} )}$—i.e.: the prime power decomposition gives an orthogonal decomposition of ${\displaystyle H_{1}(X;\mathbb {R} )}$. Cherry Kearton has shown how to compute the Milnor signature invariants from this pairing, which are equivalent to the Tristram-Levine invariant.

• Link concordance

• C.Gordon, Some aspects of classical knot theory. Springer Lecture Notes in Mathematics 685. Proceedings Plans-sur-Bex Switzerland 1977.
• J.Hillman, Algebraic invariants of links. Series on Knots and everything. Vol 32. World Scientific.
• C.Kearton, Signatures of knots and the free differential calculus, Quart. J. Math. Oxford (2), 30 (1979).
• J.Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44, 229-244 (1969)
• J.Milnor, Infinite cyclic coverings, J.G. Hocking, ed. Conf. on the Topology of Manifolds, Prindle, Weber and Schmidt, Boston, Mass, 1968 pp. 115–133.
• K.Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117, 387-482 (1965)
• A.Ranicki On signatures of knots Slides of lecture given in Durham on 20 June 2010.
• H.Trotter, Homology of group systems with applications to knot theory, Ann. of Math. (2) 76, 464-498 (1962)

In the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image (when identical while reversed). An oriented knot that is equivalent to its mirror image is an amphichiral knot, also called an achiral knot. The chirality of a knot is a knot invariant. A knot’s chirality can be further classified depending on whether or not it is invertible.

There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, invertible, positively amphichiral noninvertible, negatively amphichiral noninvertible, and fully amphichiral invertible.^[1]

The possible chirality of certain knots was suspected since 1847 when Johann Listing asserted that the trefoil was chiral,^[2] and this was proven by Max Dehn in 1914. P. G. Tait found all amphichiral knots up to 10 crossings and conjectured that all amphichiral knots had even crossing number. Mary Gertrude Haseman found all 12-crossing and many 14-crossing amphichiral knots in the late 1910s.^[3][4] But a counterexample to Tait’s conjecture, a 15-crossing amphichiral knot, was found by Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks in 1998.^[5] However, Tait’s conjecture was proven true for prime, alternating knots.^[6]

• Both possible trefoil knots.
• 
  The left-handed trefoil knot.
• 
  The right-handed trefoil knot.

The simplest chiral knot is the trefoil knot, which was shown to be chiral by Max Dehn. All nontrivial torus knots are chiral. The Alexander polynomial cannot distinguish a knot from its mirror image, but the Jones polynomial can in some cases; if V_k(q) ≠ V_k(q⁻¹), then the knot is chiral, however the converse is not true. The HOMFLY polynomial is even better at detecting chirality, but there is no known polynomial knot invariant that can fully detect chirality.^[7]

An amphichiral knot is one which has an orientation-reversing self-homeomorphism of the 3-sphere, α, fixing the knot set-wise.
All amphichiral alternating knots have even crossing number. The first amphichiral knot with odd crossing number is a 15-crossing knot discovered by Hoste et al.^[6]

Negative amphichiral

If the self-homeomorphism, α, reverses the orientation of the knot, it is said to be negative amphichiral. This is equivalent to the knot being isotopic to the reverse of its mirror image. The noninvertible knot with this property that has the fewest crossings is the knot 8₁₇.^[8]

Siberian Hitch
Names	Siberian Hitch, Evenk knot, Evenk Slippery Figure of Eight Hitch
Category	Hitch
Related	Halter hitch, Slipped buntline hitch, Highwayman’s hitch, Packer’s knot, Figure-eight knot, Farrimond friction hitch
Releasing	Quick release
Typical use	Bushcraft

The Siberian hitch (or Evenk knot) is a hitch knot used to attach a rope to an object. It is a type of slipped figure-eight noose. The hitch is known for having a tying method suitable even while wearing heavy gloves or mittens in cold climates. As a slipped knot it can be released simply by pulling the working end of the rope.

The hitch and its associated tying method were recorded in use among the Nenets people of northern Russia in the early 1990s. The knot’s ease of tying and releasing while wearing cold weather gear was cited as a primary advantage.^[1][2]

It was also used by Ray Mears during his bushcraft television series.^[3]

While it can be tied by other methods, it is associated with the one demonstrated in the following video.^[1][2]

• Discussion of Siberian hitch, including tying diagram

Chinese knotting
Example of Chinese knotwork
Chinese name
Traditional Chinese	中國結
Simplified Chinese	中国结
Literal meaning	Chinese knot
Transcriptions Standard Mandarin Hanyu Pinyin Zhōngguó jié
English name
English	Chinese knotting/ Chinese knots/ Decorative knots

Chinese knotting, also known as zhongguo jie (Chinese: 中國結; pinyin: Zhōngguó jié), is a Chinese folk art with ties to Buddhism and Taoism.^[1] A Chinese knot is made from a single length of cord that is woven into different shapes, with each shape having a symbolic meaning.^[2] The most common color used in Chinese knotting is red, a color associated with luck in Chinese culture, although any color can be used. Charms, beads, and jade are sometimes incorporated into a Chinese knot. It is believed that Chinese knotting originated for recording information and exchanging messages before writing was commonplace. Traditionally, Chinese knots acted as good-luck charms to ward off evil spirits. Chinese knots are used today to decorate homes during festivities and are also commonly seen in traditional jade jewellery and traditional Chinese clothing.^[1]

Chinese knots come in a variety of shapes and sizes. They are made from a single cord and are often double-layered and symmetrical in all directions.^[3][4][5] Satin cording is the most widely used material, especially when the knotting is done for clothing and jewellery; however, cotton, parachute cord, and other materials are frequently used as well. Knots are often paired with tassels, which are created separately and then incorporated into the main work.^[1]

Chinese knots are created in a variety of colors such as gold, green, blue, or black, though the most commonly used color is red, which symbolizes good luck and prosperity.

Types and shapes

Chinese knot scholar Lydia Chen lists eleven basic types of Chinese decorative knotwork. Complex knots are constructed from repeating or combining basic knots.^[5][6]

Types of Chinese knots^[6]

Name	Chinese name	Alternate names	Images
Chinese button knot	中國鈕扣結(traditional) 中国纽扣结(simplified)	Knife lanyard knot, Bosun whistle knot
Cloverleaf knot	三葉草結 (traditional) 三叶草结 (simplified)	Four-flower knot, dragonfly knot; ginger knot (생쪽매듭, Korean)
Cross knot	十字結 (traditional) 十字结 (simplified)	Square knot, friendship knot, Japanese crown knot	Front view Back view
Double connection knot	雙結 (traditional) 双结 (simplified)	Matthew Walker knot; dorae knot (도래매듭, Korean)
Double coin knot	雙錢結 (traditional) 双钱结 (simplified)	Carrick bend, Josephine knot, Awaji musubi (淡路結び, あわじ結び, abalone knot); wing knot (날개매듭, Korean)
Good luck knot	好運結 (traditional) 好运结 (simplified)	lovers knot (동심결매듭, Korean)
Pan Chang knot	盤長結 (traditional) 盘长结 (simplified)	Coil knot, temple knot, endless knot, 2×2 mystic knot; chrysanthemum knot (국화매듭, Korean)	Pan Chang knots A 4-row Pan Chang knot with cross knots An 8-row Pan Chang knot with overlapping ears A 3D structure of a Pan Chang knot 3D structure of a Pan Chang knot (top view) 3D structure of a Pan Chang knot (side view)
Plafond knot	平結 (traditional) 平结 (simplified)	spectacle/glasses knot (안경매듭, Korean); caisson ceiling knot
Round brocade knot	圓錦結 (traditional) 圆锦结(simplified)	six-flower knot; apricot/plum blossom knot (매화매듭, Korean)
Sauvastika knot	萬字結(traditional) 万字结 (simplified)	Agemaki (Japanese), Sailor’s cross; dragonfly wing knot (잠자리날개매듭, Korean)

Archaeological studies indicate that the art of tying knots dates back to prehistoric times. Discoveries include 100,000-year-old bone needles used for sewing and bodkins used to untie knots. Due to the delicate nature of the medium, little evidence of prehistoric Chinese knotting exists today. Some of the earliest evidence of knotting has been preserved on bronze vessels from the Warring States period (481–221 BCE), Buddhist carvings from the Northern dynasties period (317–581), and on silk paintings from the Western Han period (206 BCE – 9 CE).

Ancient totem

In addition to their use in recording, knots became a totem and belief motif.^[15] A double coin knot pattern painting on a silk banner was discovered by archaeologists in the Mawangdui tombs (206 BCE – CE 9).^[16] The pattern is of intertwined dragons forming a double coin knot in the middle of the fabric painting. The upper part of the fabric painting depicts the ancient deities Fuxi and Nüwa, the initiators of marriage in China, from whom many ancient poems derive “love” as a meaning of the double coin knot.^[5]: 10 There is evidence from the 3,000-year-old Yinxu oracle bone script that knots were recognized as symbols rather than for functional use.^[17]

Decorative art

According to Lydia Chen, the earliest tangible evidence of knots as a decorative motif is on a small high-stemmed square pot from the Spring and Autumn period (770–476 BCE), which is now displayed in the Shanxi Museum.^[18][5]: 5 However, archaeology research has found that the earliest decorative knot artifact in China can be traced back to 4000 years ago, when a three-row rattan knotting of a double coin knot was excavated from Liangzhu ruins.^[17][19]

Knots gradually evolved into a distinct decorative art in China, beginning with the use of ribbon knotting and decorative knots on clothing during the Spring and Autumn period. This is attested in the Zuo Zhuan, where it is written that:^[20]

“The collar has an intersection, and the belt is tied as knots.” [衣有襘．帶有結]

Chinese knotting was thus derived from the Lào zi culture. The Chinese word Lào is an ancient Chinese term for knots, and it was customary to tie a knot at the waist with silk or cotton ribbon.^[7]

Sui to Ming dynasties

The Sui and Tang dynasties (581–906 CE) saw the first peak of the Lào zi culture when basic knots, such as the Swastika knot and the round brocade knot, became popular adornments on garments, both among the nobility and the commoners.^[5]: 12 Knots were cherished not only as symbols and tools, but also as an essential part of everyday life to decorate and express thoughts and feelings.^[7]

In the Tang and Song dynasty (960–1279 CE), the love-based knot became an important symbol, as evidenced in many of the poems, novels, and paintings of the era. In the memoir Dongjing Meng Hua Lu (東京夢華錄) written by Meng Yuanlao, it is observed that in the traditional wedding custom, a Concentric knot needed to be held by the bride and groom.^[21] Other ancient poems used the Concentric knot to portray love, such as Luo Binwang’s poem:^[22]

“Knot the ribbon as the Concentric knot, interlock the love as the clothes.” [同心结缕带，连理织成衣].

It was also mentioned in a poem written by Huang Tingjian:

“We had a time knotting together, loving as the ribbon tied.” [曾共结，合欢罗带].

The most famous poem about the Love knot was written by Meng Jiao in Jie Ai (结爱 – lit. ‘Bond of Love‘).^[23]

The phenomenon of knot-tying continued to steadily evolve over thousands of years with the development of more sophisticated techniques and increasingly intricate woven patterns. During the Song and Yuan dynasties (960–1368), the Pan Chang knot, today’s most recognizable Chinese knot, became popular. Much artwork evidence has also shown the knots as clothing decoration during the Ming dynasty (1368–1644); for instance, in Tang Yin’s artwork, a knotting ribbon is clearly shown.

Chinese knots in paintings

Painting by Tang Yin, 1520.

Making the Bride’s gown, between 1700 and 1825, Qing dynasty

Qing dynasty

During the Qing dynasty (1644–1911), Chinese knotting evolved from folklore to an acceptable art form in Chinese society. The Lào zi culture again became popular during the Qing dynasty. During that time, basic knots were widely used to embellish everyday objects such as ruyi, sachets, purses, fan tassels, spectacle cases, and rosaries, and the single knot technique was extended into complicated knots.^[5]: 14

Chinese knots in daily items

Mirror and needle case

Mirror

Toy

Brisé Fan

Objects decorated with Chinese knots dating from the Qing dynasty, 19th century

According to the Chinese classical novel Dream of the Red Chamber, the Lào zi was developed and spread between the middle and upper nobility, who used Lào zi as a way to express love and luck between family members, lovers, and friends.^[24] It was also a form of honorable craftsmanship studied and created by maids in the Imperial Palace. As written in the Gongnü Tan Wang lu (宫女谈往录), when knotting, the maids of Ci Xi were able to quickly produce many different knots.^[25]: 29

Republic of China

There was little development of knotting during the Republic of China (1912–1949). Simpler knots were popular, for example the pan kou, which had been developed before the Qing dynasty,^[26] used knot button ornaments designed particularly for the cheongsam in this period.^[27]

20th and 21st centuries

Knowledge and interest in Chinese knotting had declined considerably by the 1970s,^[28]: 64 when Lydia Chen helped bring about a renewal of interest in the art form through the Chinese Knotting Promotion Center.^[29] Chinese knotting has since become a popular symbol and souvenir in festivals and commodity markets.^{[7][28]: 64}

The use of pan kou on clothing and knots as a folk craft remains alive in China.^[30]: 98

• Chinese art
• Chinese folk art
• Chinese paper cutting
• Chinese paper folding
• Endless knot
  • China: Pan kou; Lào zi
  • Japan: Mizuhiki
  • Korea: Norigae
• List of Japanese tea ceremony equipment#Shifuku
• Macrame

• Media related to Chinese knots at Wikimedia Commons