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47-398: The valknut is a symbol consisting of three interlocked triangles . It appears on a variety of objects from the archaeological record of the ancient Germanic peoples . The term valknut is a modern development; it is not known what term or terms were used to refer to the symbol historically. Scholars have proposed a variety of explanations for the symbol, sometimes associating it with

94-717: A trefoil knot also seen in the triquetra . This unicursal form is found, for example, on the Tängelgårda stone. The symbol also appears in tricursal form, consisting of three linked triangles, topologically equivalent to the Borromean rings . This tricursal form can be seen on one of the Stora Hammars stones , as well as upon the Nene River Ring, and on the Oseberg ship bed post. Although other forms are topologically possible, these are

141-407: A "symbol-using, symbol making, and symbol misusing animal" to suggest that a person creates symbols as well as misuses them. One example he uses to indicate what he means by the misuse of the symbol is the story of a man who, when told that a particular food item was whale blubber, could barely keep from throwing it up. Later, his friend discovered it was actually just a dumpling. But the man's reaction

188-603: A carving of the symbol on an ornately stylized bedpost and the Oseberg tapestry fragments , a partially preserved tapestry found within the ship burial, also features the symbol. Additionally, the valknut appears prominently on two picture stones from Gotland , Sweden : the Stora Hammars I stone and the Tängelgårda stone . The historically attested instances of the symbol appear in two traditional, topologically distinct forms. The symbol appears in unicursal form, topologically

235-463: A compact knot complement S 3 ∖ int ⁡ ( N ε ( K ) {\displaystyle S^{3}\setminus \operatorname {int} (\mathrm {N} _{\varepsilon }(\mathbf {K} )} .) The Alexander polynomial of the trefoil knot is Δ ( t ) = t − 1 + t − 1 , {\displaystyle \Delta (t)=t-1+t^{-1},} and

282-686: A means of complex communication that often can have multiple levels of meaning. Symbols are the basis of all human understanding and serve as vehicles of conception for all human knowledge. Symbols facilitate understanding of the world in which we live, thus serving as the grounds upon which we make judgments. In this way, people use symbols not only to make sense of the world around them but also to identify and cooperate in society through constitutive rhetoric . Human cultures use symbols to express specific ideologies and social structures and to represent aspects of their specific culture. Thus, symbols carry meanings that depend upon one's cultural background. As

329-552: A particular symbol's apparent meaning. Consequently, symbols with emotive power carry problems analogous to false etymologies . The context of a symbol may change its meaning. Similar five-pointed stars might signify a law enforcement officer or a member of the armed services , depending upon the uniform . Symbols are used in cartography to communicate geographical information (generally as point, line, or area features). As with other symbols, visual variables such as size, shape, orientation, texture, and pattern provide meaning to

376-481: A result, the meaning of a symbol is not inherent in the symbol itself but is culturally learned. Heinrich Zimmer gives a concise overview of the nature, and perennial relevance, of symbols. Concepts and words are symbols, just as visions, rituals, and images are; so too are the manners and customs of daily life. Through all of these, a transcendent reality is mirrored. There are so many metaphors reflecting and implying something which, though thus variously expressed,

423-531: A smooth 2-dimensional disk in the 4-dimensional ball; one way to prove this is to note that its signature is not zero. Another proof is that its Alexander polynomial does not satisfy the Fox-Milnor condition . The trefoil is a fibered knot , meaning that its complement in S 3 {\displaystyle S^{3}} is a fiber bundle over the circle S 1 {\displaystyle S^{1}} . The trefoil K may be viewed as

470-438: A symbol always "points beyond itself" to something that is unquantifiable and mysterious; symbols open up the "depth dimension of reality itself". Symbols are complex, and their meanings can evolve as the individual or culture evolves. When a symbol loses its meaning and power for an individual or culture, it becomes a dead symbol. When a symbol becomes identified with the deeper reality to which it refers, it becomes idolatrous as

517-806: A theological sense signifying a formula used in the Roman Catholic Church as a sort of synonym for 'the credo'; by extension in the early Renaissance it came to mean 'a maxim' or 'the external sign of a sacrament'; these meanings were lost in secular contexts. It was during the Renaissance in the mid-16th century that the word took on the meaning that is dominant today, that of 'a natural fact or object evoking by its form or its nature an association of ideas with something abstract or absent'; this appears, for example, in François Rabelais , Le Quart Livre , in 1552. This French word derives from Latin, where both

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564-423: A trefoil knot is also considered to be a trefoil. In addition, the mirror image of a trefoil knot is also considered to be a trefoil. In topology and knot theory, the trefoil is usually defined using a knot diagram instead of an explicit parametric equation. In algebraic geometry , the trefoil can also be obtained as the intersection in C of the unit 3-sphere S with the complex plane curve of zeroes of

611-423: A trefoil. Proving this requires the construction of a knot invariant that distinguishes the trefoil from the unknot. The simplest such invariant is tricolorability : the trefoil is tricolorable, but the unknot is not. In addition, virtually every major knot polynomial distinguishes the trefoil from an unknot, as do most other strong knot invariants. In knot theory, the trefoil is the first nontrivial knot, and

658-401: Is ineffable, though thus rendered multiform, remains inscrutable. Symbols hold the mind to truth but are not themselves the truth, hence it is delusory to borrow them. Each civilisation, every age, must bring forth its own." In the book Signs and Symbols , it is stated that A symbol   ... is a visual image or sign representing an idea – a deeper indicator of universal truth. Semiotics

705-409: Is not possible to deform a left-handed trefoil continuously into a right-handed trefoil, or vice versa. (That is, the two trefoils are not ambient isotopic .) Though chiral, the trefoil knot is also invertible, meaning that there is no distinction between a counterclockwise -oriented and a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the over and under crossings, not

752-597: Is substituted for another in order to change the meaning. In other words, if one person does not understand a certain word or phrase, another person may substitute a synonym or symbol in order to get the meaning across. However, upon learning the new way of interpreting a specific symbol, the person may change his or her already-formed ideas to incorporate the new information. Jean Dalby Clift says that people not only add their own interpretations to symbols, but they also create personal symbols that represent their own understanding of their lives: what she calls "core images" of

799-546: Is the only knot with crossing number three. It is a prime knot , and is listed as 3 1 in the Alexander-Briggs notation . The Dowker notation for the trefoil is 4 6 2, and the Conway notation is [3]. The trefoil can be described as the (2,3)- torus knot . It is also the knot obtained by closing the braid σ 1 . The trefoil is an alternating knot . However, it is not a slice knot , meaning it does not bound

846-513: Is the study of signs, symbols, and signification as communicative behavior. Semiotics studies focus on the relationship of the signifier and the signified, also taking into account the interpretation of visual cues, body language, sound, and other contextual clues. Semiotics is linked with linguistics and psychology. Semioticians not only study what a symbol implies but also how it got its meaning and how it functions to make meaning in society. For example, symbols can cause confusion in translation when

893-451: Is used for a variety of purposes in modern popular culture. The valknut symbol is used as a religious symbol by some adherents of Heathenry , a new religious movement inspired by historic Germanic paganism . In Europe, the Swedish forest products company Svenska Cellulosa Aktiebolaget uses a triquetra valknut as their logo, which can be commonly seen on many products produced by the company;

940-462: The Conway polynomial is ∇ ( z ) = z 2 + 1. {\displaystyle \nabla (z)=z^{2}+1.} The Jones polynomial is V ( q ) = q − 1 + q − 3 − q − 4 , {\displaystyle V(q)=q^{-1}+q^{-3}-q^{-4},} and the Kauffman polynomial of

987-519: The DFB has used a logo inspired by the unicursal form of the valknut for the Germany national football team since 1991. The symbol appears as the fretboard inlay on some of Arch Enemy / Carcass guitarist Michael Amott 's signature Dean Guitars "Tyrant" models, and it is also used as a logo by American engineering firm RedViking. In Civilization VI , the valknut is the national symbol of Norway , which in

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1034-468: The Milnor map ϕ ( z , w ) = ( z 2 + w 3 ) / | z 2 + w 3 | {\displaystyle \phi (z,w)=(z^{2}+w^{3})/|z^{2}+w^{3}|} as the fibre bundle projection of the knot complement S 3 ∖ K {\displaystyle S^{3}\setminus \mathbf {K} } to

1081-412: The jötunn Hrungnir : "Hrungnir had a heart that was famous. It was made of hard stone with three sharp-pointed corners just like the carved symbol hrungnishjarta [Hrungnir's heart]." Comparisons have been made between this symbol description and the symbol known as the valknut. Hilda Ellis Davidson theorizes a connection between the valknut, the god Odin , and "mental binds": For instance, beside

1128-416: The trefoil knot is the simplest example of a nontrivial knot . The trefoil can be obtained by joining the two loose ends of a common overhand knot , resulting in a knotted loop . As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory. The trefoil knot is named after the three-leaf clover (or trefoil) plant. The trefoil knot can be defined as the curve obtained from

1175-411: The "symbol is taken for reality." The symbol itself is substituted for the deeper meaning it intends to convey. The unique nature of a symbol is that it gives access to deeper layers of reality that are otherwise inaccessible. A symbol's meaning may be modified by various factors including popular usage, history , and contextual intent . The history of a symbol is one of many factors in determining

1222-418: The circle S 1 {\displaystyle S^{1}} . The fibre is a once-punctured torus . Since the knot complement is also a Seifert fibred with boundary, it has a horizontal incompressible surface—this is also the fiber of the Milnor map . (This assumes the knot has been thickened to become a solid torus N ε ( K ), and that the interior of this solid torus has been removed to create

1269-402: The complex polynomial z  +  w (a cuspidal cubic ). If one end of a tape or belt is turned over three times and then pasted to the other, the edge forms a trefoil knot. The trefoil knot is chiral , in the sense that a trefoil knot can be distinguished from its own mirror image. The two resulting variants are known as the left-handed trefoil and the right-handed trefoil . It

1316-682: The figure of Odin on his horse shown on several memorial stones there is a kind of knot depicted, called the valknut , related to the triskele . This is thought to symbolize the power of the god to bind and unbind, mentioned in the poems and elsewhere. Odin had the power to lay bonds upon the mind, so that men became helpless in battle, and he could also loosen the tensions of fear and strain by his gifts of battle-madness, intoxication, and inspiration. Davidson says that similar symbols are found beside figures of wolves and ravens on "certain cremation urns" from Anglo-Saxon cemeteries in East Anglia . Because

1363-420: The following parametric equations : The (2,3)- torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus ( r − 2 ) 2 + z 2 = 1 {\displaystyle (r-2)^{2}+z^{2}=1} : Any continuous deformation of the curve above is also considered a trefoil knot. Specifically, any curve isotopic to

1410-456: The game is led by Harald Hardrada and mostly representative of Vikings rather than the modern country. The valknut has seen some use by White supremacists . The Anti-Defamation League notes that "nonracist pagans may also use this symbol, so one should carefully examine it in context rather than assume that a particular use of the symbol is racist." Bogd Bank of Mongolia uses the same symbol as their main corporate logo. In modern Norway,

1457-641: The god Odin , and it has been compared to the three-horned symbol found on the 9th-century Snoldelev Stone , to which it may be related. The valknut appears on a wide variety of objects found in areas inhabited by the Germanic peoples. The symbol is prominently featured on the Nene River Ring , an Anglo-Saxon gold finger ring dated to around the 8th to 9th centuries. A wooden bed in the Viking Age Oseberg Ship buried near Tønsberg , Norway , features

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1504-486: The masculine noun symbolus and the neuter noun symbolum refer to "a mark or sign as a means of recognition." The Latin word derives from Ancient Greek : σύμβολον symbolon , from a verb meaning 'put together', 'compare', alluding to the Classical practice of breaking a piece of ceramic in two and giving one half to the person who would receive a future message, and one half to the person who would send it: when

1551-460: The only attested forms found so far. In Norwegian Bokmål , the term valknute is used for a polygon with a loop on each of its corners. In the English language, the looped, four-cornered symbol is called Saint John's Arms . Several explanations for the symbol have been proposed: Chapter 17 of the 13th century Prose Edda book Skáldskaparmál contains the following description of the heart of

1598-434: The orientation of the curve. But the knot has rotational symmetry. The axis is about a line perpendicular to the page for the 3-coloured image. The trefoil knot is nontrivial, meaning that it is not possible to "untie" a trefoil knot in three dimensions without cutting it. Mathematically, this means that a trefoil knot is not isotopic to the unknot . In particular, there is no sequence of Reidemeister moves that will untie

1645-852: The person. Clift argues that symbolic work with these personal symbols or core images can be as useful as working with dream symbols in psychoanalysis or counseling. William Indick suggests that the symbols that are commonly found in myth, legend, and fantasy fulfill psychological functions and hence are why archetypes such as "the hero", "the princess" and "the witch" have remained popular for centuries. Symbols can carry symbolic value in three primary forms: Ideological, comparative, and isomorphic. Ideological symbols such as religious and state symbols convey complex sets of beliefs and ideas that indicate "the right thing to do". Comparative symbols such as prestigious office addresses, fine art, and prominent awards indicate answers to questions of "better or worse" and "superior or inferior". Isomorphic symbols blend in with

1692-540: The same symbol means different things in the source and target languages. A potential error documented in survey translation is the symbol of "x" used to denote "yes" when marking a response in the English language surveys, but "x" usually means "no" in the Chinese convention. Symbols allow the human brain continuously to create meaning using sensory input and decode symbols through both denotation and connotation . An alternative definition of symbol , distinguishing it from

1739-399: The set of pairs ( z , w ) {\displaystyle (z,w)} of complex numbers such that | z | 2 + | w | 2 = 1 {\displaystyle |z|^{2}+|w|^{2}=1} and z 2 + w 3 = 0 {\displaystyle z^{2}+w^{3}=0} . Then this fiber bundle has

1786-821: The surrounding cultural environment such that they enable individuals and organizations to conform to their surroundings and evade social and political scrutiny. Examples of symbols with isomorphic value include wearing a professional dress during business meetings, shaking hands to greet others in the West, or bowing to greet others in the East. A single symbol can carry multiple distinct meanings such that it provides multiple types of symbolic value. Paul Tillich argued that, while signs are invented and forgotten, symbols are born and die. There are, therefore, dead and living symbols. A living symbol can reveal to an individual hidden levels of meaning and transcendent or religious realities. For Tillich

1833-408: The symbol appears on picture stones with Odin and on burial gifts in the Oseberg ship burial, Rudolf Simek says that the symbol may have been associated with religious practices surrounding death. The valknut is topologically equivalent to either the Borromean rings , the trefoil knot , or (in modern use only) a closed three-link chain, depending on the particular artistic depiction: The symbol

1880-454: The symbol. According to semiotics , map symbols are "read" by map users when they make a connection between the graphic mark on the map (the sign ), a general concept (the interpretant ), and a particular feature of the real world (the referent ). Map symbols can thus be categorized by how they suggest this connection: A symbolic action is an action that symbolizes or signals what the actor wants or believes. The action conveys meaning to

1927-448: The term sign was proposed by Swiss psychoanalyst Carl Jung . In his studies on what is now called Jungian archetypes , a sign stands for something known, as a word stands for its referent. He contrasted a sign with a symbol : something that is unknown and that cannot be made clear or precise. An example of a symbol in this sense is Christ as a symbol of the archetype called self . Kenneth Burke described Homo sapiens as

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1974-678: The trefoil is L ( a , z ) = z a 5 + z 2 a 4 − a 4 + z a 3 + z 2 a 2 − 2 a 2 . {\displaystyle L(a,z)=za^{5}+z^{2}a^{4}-a^{4}+za^{3}+z^{2}a^{2}-2a^{2}.} The HOMFLY polynomial of the trefoil is L ( α , z ) = − α 4 + α 2 z 2 + 2 α 2 . {\displaystyle L(\alpha ,z)=-\alpha ^{4}+\alpha ^{2}z^{2}+2\alpha ^{2}.} The knot group of

2021-402: The trefoil is given by the presentation ⟨ x , y ∣ x 2 = y 3 ⟩ {\displaystyle \langle x,y\mid x^{2}=y^{3}\rangle } or equivalently ⟨ x , y ∣ x y x = y x y ⟩ . {\displaystyle \langle x,y\mid xyx=yxy\rangle .} This group

2068-518: The two fit perfectly together, the receiver could be sure that the messenger bearing it did indeed also carry a genuine message from the intended person. A literary or artistic symbol as an "outward sign" of something else is a metaphorical extension of this notion of a message from a sender to a recipient. In English, the meaning "something which stands for something else" was first recorded in 1590, in Edmund Spenser 's Faerie Queene . Symbols are

2115-421: The viewers. Symbolic action may overlap with symbolic speech , such as the use of flag burning to express hostility or saluting the flag to express patriotism. In response to intense public criticism, businesses, organizations, and governments may take symbolic actions rather than, or in addition to, directly addressing the identified problems. Trefoil knot In knot theory , a branch of mathematics ,

2162-453: The word "valknut" means "knot of those fallen in battle", connecting the symbol to the god Odin and representing the glory of death in battle. Symbol In the arts, symbolism is the use of a concrete element to represent a more abstract idea. In cartography , an organized collection of symbols forms a legend for a map. The word symbol derives from the late Middle French masculine noun symbole , which appeared around 1380 in

2209-514: Was a direct consequence of the symbol of "blubber" representing something inedible in his mind. In addition, the symbol of "blubber" was created by the man through various kinds of learning . Burke goes on to describe symbols as also being derived from Sigmund Freud 's work on condensation and displacement , further stating that symbols are not just relevant to the theory of dreams but also to "normal symbol systems". He says they are related through "substitution", where one word, phrase, or symbol

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