1.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2.
Trefoil knot
–
In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of knot theory. The trefoil knot is named after the three-leaf clover plant, specifically, any curve isotopic to a trefoil knot is also considered to be a trefoil. In addition, the 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 a parametric equation. In algebraic geometry, the trefoil can also be obtained as the intersection in C2 of the unit 3-sphere S3 with the plane curve of zeroes of the complex polynomial z2 + w3. If one end of a tape or belt is turned over three times and then pasted to the other, the forms a trefoil knot. The trefoil knot is chiral, in the sense that a knot can be distinguished from its own mirror image. The two resulting variants are known as the trefoil and the right-handed trefoil. It is not possible to deform a left-handed trefoil continuously into a right-handed trefoil, though the trefoil knot is chiral, it is also invertible, meaning that there is no distinction between a counterclockwise-oriented trefoil and a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the over and under crossings, the trefoil knot is nontrivial, meaning that it is not possible to untie a trefoil knot in three dimensions without cutting it. From a mathematical point of view, this means that a knot is not isotopic to the unknot. In particular, there is no sequence of Reidemeister moves that will untie 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, 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 it is a prime knot, and is listed as 31 in the Alexander-Briggs notation. The Dowker notation for the trefoil is 462, the trefoil can be described as the -torus knot. It is also the knot obtained by closing the braid σ13, the trefoil is an alternating knot

3.
Solomon's knot
–
Solomons knot is the most common name for a traditional decorative motif used since ancient times, and found in many cultures. Despite the name, it is classified as a link, and is not a true knot according to the definitions of mathematical knot theory, the Solomons knot consists of two closed loops, which are doubly interlinked in an interlaced manner. If laid flat, the Solomons knot is seen to have four crossings where the two loops interweave under and over each other and this contrasts with two crossings in the simpler Hopf link. In most artistic representations, the parts of the loops that alternately cross over and under each other become the sides of a central square, while four loopings extend outward in four directions. The four extending loopings may have oval, square, or triangular endings, or may terminate with free-form shapes such as leaves, lobes, blades, the Solomons knot often occurs in ancient Roman mosaics, usually represented as two interlaced ovals. Tzippori National Park, Israel, has Solomons Knots in stone mosaics at the site of an ancient synagogue, across the Middle East, historical Islamic sites show Solomons knot as part of Muslim tradition. It appears over the doorway of a twentieth century CE mosque/madrasa in Cairo. Two versions of Solomons knot are included in the recently excavated Yattir Mosaic in Jordan, to the east, it is woven into an antique Central Asian prayer rug. To the west, Solomons knot appeared in Moorish Spain, the British Museum, London, England has a fourteenth-century CE Egyptian Quran with a Solomons Knot as its frontispiece. Home of Peace Mausoleum, a Jewish Cemetery, Los Angeles, California, USA has multiple images of Solomons knot in stone, saint Sophias Greek Orthodox Cathedral, Byzantine District of Los Angeles, California, USA has an olive wood Epitaphios with Solomons knots carved at each corner. The Epitaphios is used in the Greek Easter services, powell Library University of California at Los Angeles, USA has ceiling beams in the Main Reading Room covered with Solomons Knots. Built in 1926 CE, the room also features a central Dome of Wisdom bordered by Solomons knots. In Latin, this configuration was sometimes known as sigillum Salomonis and it was associated with the Biblical monarch Solomon because of his reputation for wisdom and knowledge. This phrase is rendered into English as Solomons knot, since seal of Solomon has other conflicting meanings. Imbolo describes the design on the textiles of the Kuba people of Congo. Nodo di Salomone is the Italian term for Solomons Knot, and is used to name the Solomons Knot mosaic found at the ruins of a synagogue at Ostia, the ancient seaport for Rome. Since the knot has been used across a number of cultures and historical eras, because there is no visible beginning or ending, it may represent immortality and eternity—as does the more complicated Buddhist Endless Knot. Because the knot seems to be two entwined figures, it is interpreted as a Lovers Knot, although that name may indicate another knot

4.
Borromean rings
–
In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link. In other words, no two of the three rings are linked with other as a Hopf link, but nonetheless all three are linked. Although the typical picture of the Borromean rings may lead one to think the link can be formed from geometrically ideal circular rings, freedman and Skora prove that a certain class of links, including the Borromean links, cannot be exactly circular. Alternatively, this can be seen from considering the link diagram, in either case, the third circle must pass through this plane or sphere four times, without lying in it, which is impossible, see. It is, however, true that one can use ellipses and these may be taken to be of arbitrarily small eccentricity, i. e. Apart from indicating which strand crosses over the other, link diagrams use the notation to show two strands crossing, as graph diagrams use to show four edges meeting at a common vertex. The result has three loops, linked together as Borromean rings. In knot theory, the Borromean rings are an example of a Brunnian link, although each pair of rings is unlinked. There are a number of ways of seeing this and this is non-trivial in the fundamental group, and thus the Borromean rings are linked. Another way is that the cohomology of the complement supports a non-trivial Massey product, in arithmetic topology, there is an analogy between knots and prime numbers in which one considers links between primes. The triple of primes are linked modulo 2 but are pairwise unlinked modulo 2, therefore, these primes have been called a proper Borromean triple modulo 2 or mod 2 Borromean primes. The Borromean rings are a link, the complement of the Borromean rings in the 3-sphere admits a complete hyperbolic metric of finite volume. The canonical polyhedral decomposition of the complement consists of two regular ideal octahedra, the volume is 16Л =7. 32772… where Л is the Lobachevsky function. If one cuts the Borromean rings, one obtains one iteration of the braid, conversely, if one ties together the ends of a standard braid. Just as removing one Borromean ring unlinks the remaining two, removing one strand of the standard braid unbraids the other two, they are the basic Brunnian link and Brunnian braid, respectively. In the standard link diagram, the Borromean rings are ordered non-transitively, in a cyclic order. Using the colors above, these are red over green, green over blue, blue over red – and thus removing any one ring. Similarly, in the standard braid, each strand is above one of the others, the name Borromean rings comes from their use in the coat of arms of the aristocratic Borromeo family in Northern Italy

5.
Knot theory
–
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, in mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Knots can be described in various ways, given a method of description, however, there may be more than one description that represents the same knot. For example, a method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in different ways using a knot diagram. Therefore, a problem in knot theory is determining when two descriptions represent the same knot. A complete algorithmic solution to this exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, important invariants include knot polynomials, knot groups, and hyperbolic invariants. The original motivation for the founders of theory was to create a table of knots and links. Over six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century, to gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other spaces and objects other than circles can be used. Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space, archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics, Knots appear in various forms of Chinese artwork dating from several centuries BC. The endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures, the Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork. Mathematical studies of knots began in the 19th century with Gauss, in the 1860s, Lord Kelvins theory that atoms were knots in the aether led to Peter Guthrie Taits creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings and this record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology. This would be the approach to knot theory until a series of breakthroughs transformed the subject. In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem, many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants

6.
Knot (mathematics)
–
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations. A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, the term knot is also applied to embeddings of S j in S n, especially in the case j = n −2. The branch of mathematics that studies knots is known as knot theory, a knot is an embedding of the circle into three-dimensional Euclidean space. Or the 3-sphere, S3, since the 3-sphere is compact, two knots are defined to be equivalent if there is an ambient isotopy between them. A knot in R3, can be projected onto a plane R2, in this case, by choosing a projection side, one can completely encode the isotopy class of the knot by its regular projection by recording a simple over/under information at these crossings. In graph theory terms, a projection of a knot. The local modifications of this graph which allow to go from one diagram to any other diagram of the knot are called Reidemeister moves. The simplest knot, called the unknot or trivial knot, is a circle embedded in R3. In the ordinary sense of the word, the unknot is not knotted at all, the simplest nontrivial knots are the trefoil knot, the figure-eight knot and the cinquefoil knot. Several knots, linked or tangled together, are called links, Knots are links with a single component. A polygonal knot is a knot whose image in R3 is the union of a set of line segments. A tame knot is any knot equivalent to a polygonal knot, Knots which are not tame are called wild, and can have pathological behavior. In knot theory and 3-manifold theory, often the adjective tame is omitted, smooth knots, for example, are always tame. A framed knot is the extension of a knot to an embedding of the solid torus D2 × S1 in S3. The framing of the knot is the number of the image of the ribbon I × S1 with the knot. A framed knot can be seen as the ribbon and the framing is the number of twists. This definition generalizes to a one for framed links

7.
Carrick mat
–
The carrick mat is a flat woven decorative knot which can be used as a mat or pad. Its name is based on the mats decorative-type carrick bend with the ends connected together, a larger form, called the prolong knot, is made by expanding the basic carrick mat by extending, twisting, and overlapping its outer bights, then weaving the free ends through them. This process may be repeated to produce an arbitrarily long mat, in its basic form it is the same as a 3-lead, 4-bight Turks head knot. The basic carrick mat, made two passes of rope, also forms the central motif in the logo of the International Guild of Knot Tyers. When tied to form a cylinder around the opening, instead of lying flat. List of knots 8_18, The Knot Atlas

8.
2-bridge knot
–
Equivalently, these are the knots with bridge number 2, the smallest possible bridge number for a nontrivial knot. Other names for 2-bridge knots are rational knots, 4-plats, 2-bridge links are defined similarly as above, but each component will have one min and max. 2-bridge knots were classified by Horst Schubert, using the fact that the 2-sheeted branched cover of the 3-sphere over the knot is a lens space, the names rational knot and rational link were coined by John Conway who defined them as arising from numerator closures of rational tangles. Horst Schubert, Über Knoten mit zwei Brücken, Mathematische Zeitschrift 65, louis H. Kauffman, Sofia Lambropoulou, On the classification of rational knots, L Enseignement Mathématique,49, 357–410. Adams, The Knot Book, An elementary introduction to the theory of knots. American Mathematical Society, Providence, RI,2004, xiv+307 pp. ISBN 0-8218-3678-1 Table and invariants of rational knots with up to 16 crossings

9.
Pretzel link
–
In the mathematical theory of knots, a pretzel link is a special kind of link. A pretzel link which is also a knot is a pretzel knot, in the standard projection of the pretzel link, there are p 1 left-handed crossings in the first tangle, p 2 in the second, and, in general, p n in the nth. A pretzel link can also be described as a Montesinos link with integer tangles, the pretzel link is a knot iff both n and all the p i are odd or exactly one of the p i is even. The pretzel link is split if at least two of the p i are zero, but the converse is false, the pretzel link is the mirror image of the pretzel link. The pretzel link is link-equivalent to the pretzel link, thus, too, the pretzel link is link-equivalent to the pretzel link. The pretzel link is link-equivalent to the pretzel link, however, if one orients the links in a canonical way, then these two links have opposite orientations. The pretzel knot is the trefoil, the knot is its mirror image. The pretzel knot is the stevedore knot, if p, q, r are distinct odd integers greater than 1, then the pretzel knot is a non-invertible knot. The pretzel link is a formed by three linked unknots. The pretzel knot is the sum of two trefoil knots. The pretzel link is the union of an unknot and another knot. A Montesinos link is a kind of link that generalizes pretzel links. A Montesinos link which is also a knot is a Montesinos knot, a Montesinos link is composed of several rational tangles. One notation for a Montesinos link is K, in this notation, e and all the α i and β i are integers. Many results have been stated about the manifolds that result from Dehn surgery on the knot in particular. The hyperbolic volume of the complement of the link is 4 times Catalans constant. This pretzel link complement is one of two two-cusped hyperbolic manifolds with the possible volume, the other being the complement of the Whitehead link. Trotter, Hale F. Non-invertible knots exist, Topology,2, 272–280

10.
Figure-eight knot (mathematics)
–
In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot, the figure-eight knot is a prime knot. The name is given because tying a normal figure-eight knot in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot. A simple parametric representation of the knot is as the set of all points where x = cos y = sin z = sin for t varying over the real numbers. The figure-eight knot is prime, alternating, rational with a value of 5/2. The figure-eight knot is also a fibered knot and this follows from other, less simple representations of the knot, It is a homogeneous closed braid, and a theorem of John Stallings shows that any closed homogeneous braid is fibered. It is the link at of a critical point of a real-polynomial map F, R4→R2. Bernard Perron found the first such F for this knot, namely, F = G, the figure-eight knot has played an important role historically in the theory of 3-manifolds. Sometime in the mid-to-late 1970s, William Thurston showed that the figure-eight was hyperbolic and this construction, new at the time, led him to many powerful results and methods. For example, he was able to show all but ten Dehn surgeries on the figure-eight knot resulted in non-Haken, non-Seifert-fibered irreducible 3-manifolds. Many more have been discovered by generalizing Thurstons construction to other knots, the figure-eight knot is also the hyperbolic knot whose complement has the smallest possible volume,2.02988. According to the work of Chun Cao and Robert Meyerhoff, from this perspective, the figure-eight knot can be considered the simplest hyperbolic knot. The figure eight knot complement is a double-cover of the Gieseking manifold, however, it is not currently known whether the figure-eight knot is the only one that achieves the bound of 10. A well-known conjecture is that the bound is 6, the symmetry between q and q −1 in the Jones polynomial reflects the fact that the figure-eight knot is achiral. Ian Agol, Bounds on exceptional Dehn filling, Geometry & Topology 4, mR1799796 Chun Cao and Robert Meyerhoff, The orientable cusped hyperbolic 3-manifolds of minimum volume, Inventiones Mathematicae,146, no. MR1869847 Marc Lackenby, Word hyperbolic Dehn surgery, Inventiones Mathematicae 140, no

11.
Link (knot theory)
–
In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component, links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a reference link, usually called the unlink. For example, a co-dimension two link in 3-dimensional space is a subspace of 3-dimensional Euclidean space whose connected components are homeomorphic to circles, the simplest nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles linked together once. The circles in the Borromean rings are collectively linked despite the fact that no two of them are directly linked, the Borromean rings thus form a Brunnian link and in fact constitute the simplest such link. The notion of a link can be generalized in a number of ways, frequently the word link is used to describe any submanifold of the sphere S n diffeomorphic to a disjoint union of a finite number of spheres, S j. If M is disconnected, the embedding is called a link, if M is connected, it is called a knot. While links are defined as embeddings of circles, it is interesting and especially technically useful to consider embedded intervals. The type of a tangle is the manifold X, together with an embedding of ∂ X. The condition that the boundary of X lies in R × says that intervals either connect two lines or connect two points on one of the lines, but imposes no conditions on the circles. Tangles include links, braids, and others besides – for example, in this context, a braid is defined as a tangle which is always going down – whose derivative always has a non-zero component in the vertical direction. In particular, it must consist solely of intervals, and not double back on itself, however, a string link is a tangle consisting of only intervals, with the ends of each strand required to lie at. – i. e. connecting the integers, and ending in the order that they began, if this has ℓ components. A string link need not be a braid – it may double back on itself, a braid that is also a string link is called a pure braid, and corresponds with the usual such notion. The key technical value of tangles and string links is that they have algebraic structure, the tensor structure is given by juxtaposition of tangles – putting one tangle to the right of the other. For a fixed ℓ, isotopy classes of ℓ-component string links form a monoid, however, concordance classes of string links do have inverses, where inverse is given by flipping the string link upside down, and thus form a group. Linking number Hyperbolic link Unlink Link group

12.
Whitehead link
–
In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links, Whitehead spent much of the 1930s looking for a proof of the Poincaré conjecture. In 1934, the Whitehead link was used as part of his construction of the now-named Whitehead manifold, the link is created with two projections of the unknot, one circular loop and one figure eight-shaped loop intertwined such that they are inseparable and neither loses its form. Excluding the instance where the figure eight thread intersects itself, the Whitehead link has four crossings, because each underhand crossing has a paired upperhand crossing, its linking number is 0. It is not isotopic to the unlink, but it is homotopic to the unlink. In braid theory notation, the link is written σ12 σ22 σ1 −1 σ2 −2 and its Jones polynomial is V = t −32. This polynomial and V are the two factors of the Jones polynomial of the L10a140 link, notably, V is the Jones polynomial for the mirror image of a link having Jones polynomial V. The hyperbolic volume of the complement of the Whitehead link is 4 times Catalans constant, the Whitehead link complement is one of two two-cusped hyperbolic manifolds with the minimum possible volume, the other being the complement of the pretzel link with parameters. Solomons knot Weeks manifold Whitehead double L5a1 knot-theoretic link, The Knot Atlas