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Invertible Knot
In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed. A non-invertible knot is any knot which does not have this property. The invertibility of a knot is a knot invariant. An invertible link is the link equivalent of an invertible knot. There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, reversible, positively amphichiral noninvertible, negatively amphichiral noninvertible, and fully amphichiral invertible.. Background It has long been known that most of the simple knots, such as the trefoil knot and the figure-eight knot are invertible. In 1962 Ralph Fox conjectured that some knots were non-invertible, but it was not proved that non-invertible knots exist until Hale Trotter discovered an infinite family of pretzel knots that were non-invertible in 1963.. It is now known almost all knots are non-invertible. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ralph Fox
Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the ''Golden Age of differential topology'', and he played an important role in the modernization and main-streaming of knot theory. Biography Ralph Fox attended Swarthmore College for two years, while studying piano at the Leefson Conservatory of Music in Philadelphia. He earned a master's degree from Johns Hopkins University, and a PhD degree from Princeton University in 1939. His doctoral dissertation, ''On the Lusternick-Schnirelmann Category'', was directed by Solomon Lefschetz. (In later years he disclaimed all knowledge of the Lusternik–Schnirelmann category, and certainly never published on the subject again.) He directed 21 doctoral dissertations, including those of John Milnor, John Stallings, Francisco González-Acuña, Guillermo Torres-Diaz and Barry Mazur, and supervised Ken ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conway Notation (knot Theory)
In knot theory, Conway notation, invented by John Horton Conway, is a way of describing knots that makes many of their properties clear. It composes a knot using certain operations on tangles to construct it. Basic concepts Tangles In Conway notation, the tangles are generally algebraic 2-tangles. This means their tangle diagrams consist of 2 arcs and 4 points on the edge of the diagram; furthermore, they are built up from rational tangles using the Conway operations. he following seems to be attempting to describe only integer or 1/n rational tanglesTangles consisting only of positive crossings are denoted by the number of crossings, or if there are only negative crossings it is denoted by a negative number. If the arcs are not crossed, or can be transformed into an uncrossed position with the Reidemeister moves, it is called the 0 or ∞ tangle, depending on the orientation of the tangle. Operations on tangles If a tangle, ''a'', is reflected on the NW-SE line, it is den ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8 17 Knot
8 (eight) is the natural number following 7 and preceding 9. In mathematics 8 is: * a composite number, its proper divisors being , , and . It is twice 4 or four times 2. * a power of two, being 2 (two cubed), and is the first number of the form , being an integer greater than 1. * the first number which is neither prime nor semiprime. * the base of the octal number system, which is mostly used with computers. In octal, one digit represents three bits. In modern computers, a byte is a grouping of eight bits, also called an octet. * a Fibonacci number, being plus . The next Fibonacci number is . 8 is the only positive Fibonacci number, aside from 1, that is a perfect cube. * the only nonzero perfect power that is one less than another perfect power, by Mihăilescu's Theorem. * the order of the smallest non-abelian group all of whose subgroups are normal. * the dimension of the octonions and is the highest possible dimension of a normed division algebra. * the first number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tunnel Number
In mathematics, the tunnel number of a knot, as first defined by Bradd Clark, is a knot invariant, given by the minimal number of arcs (called tunnels) that must be added to the knot so that the complement becomes a handlebody. The tunnel number can equally be defined for links. The boundary of a regular neighbourhood of the union of the link and its tunnels forms a Heegaard splitting of the link exterior. Examples * The unknot is the only knot with tunnel number 0. * The trefoil knot In knot theory, 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 loose ends of a common overhand knot, resulting in a knotted loop. As the simplest kno ... has tunnel number 1. In general, any nontrivial torus knot has tunnel number 1. Every link ''L'' has a tunnel number. This can be seen, for example, by adding a 'vertical' tunnel at every crossing in a diagram of ''L''. It follows from this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x \mapsto -x), reciprocation (x \mapsto 1/x), and complex conjugation (z \mapsto \bar z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition of two involutions ''f'' and ''g'' is an involution if and only if they commute: . Involutions on finite sets The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800: :a_0 = a_1 = 1 and a_n = a_ + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amphichiral Knot
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from its mirror image; that is, it cannot be superimposed onto it. Conversely, a mirror image of an ''achiral'' object, such as a sphere, cannot be distinguished from the object. A chiral object and its mirror image are called ''enantiomorphs'' (Greek, "opposite forms") or, when referring to molecules, '' enantiomers''. A non-chiral object is called ''achiral'' (sometimes also ''amphichiral'') and can be superposed on its mirror image. The term was first used by Lord Kelvin in 1893 in the second Robert Boyle Lecture at the Oxford University Junior Scientific Club which was published in 1894: Human hands are perhaps the most recognized example of chirality. The left hand is a non-superimposable mirror image of the right hand; no matter how ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost All
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null. In contrast, "almost no" means "a negligible amount"; that is, "almost no elements of X" means "a negligible amount of elements of X". Meanings in different areas of mathematics Prevalent meaning Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) but finitely many". This use occurs in philosophy as well. Similarly, "almost all" can mean "all (elements of an uncountable set) but countably many". Examples: * Almost all positive integers are greater than 1012. * Almost all prime numbers are odd (2 is the only exception). * Almost all polyhedra are irregular (as there are only nine exceptions: the five platonic solids and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pretzel Link
In the mathematical theory of knots, a pretzel link is a special kind of link. It consists of a finite number tangles made of two intertwined circular helices. The tangles are connected cyclicly, the first component of the first tangle is connected to the second component of the second tangle, etc., with the first component of the last tangle connected to the second component of the first. A pretzel link which is also a knot (i.e. 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 (p_1,\,p_2,\dots,\,p_n) 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. Some basic results The (p_1,p_2,\dots,p_n) pretzel link is a knot iff both n and all the p_i ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hale Trotter
Hale Freeman Trotter (30 May 1931 – 17 January 2022)biographical information from ''American Men and Women of Science'', Thomson Gale 2004 was a Canadian-American mathematician, known for the Lie–Trotter product formula, the Steinhaus–Johnson–Trotter algorithm, and the Lang–Trotter conjecture. He was born in Kingston, Ontario. He died in Princeton, New Jersey on January 17, 2022. Biography The son of historian Reginald George Trotter, Hale Trotter studied at Queen's University in Kingston with bachelor's degree in 1952 and master's degree in 1953. He received in 1956 his PhD from Princeton University under William Feller with thesis ''Convergence of semigroups of operators''. Trotter was from 1956 to 1958 at Princeton University the ''Fine Instructor'' for mathematics and from 1958 to 1960 an assistant professor at Queen's University. He was from 1962 to 1963 a visiting associate professor, from 1963 to 1969 an associate professor, and from 1969 until his retirement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figure-eight Knot (mathematics)
Figure 8 (figure of 8 in British English) may refer to: * 8 (number), in Arabic numerals Entertainment * ''Figure 8'' (album), a 2000 album by Elliott Smith * "Figure of Eight" (song), a 1989 song by Paul McCartney * '' Figure Eight EP'', a 2008 EP by This Et Al * "Figure 8" (song), a 2012 song by Ellie Goulding from ''Halcyon'' * "Figure Eight", an episode and song from the children's educational series ''Schoolhouse Rock!'' * "Figure of Eight", song by Status Quo from ''In Search of the Fourth Chord'' * "Figure 8", a song by FKA Twigs from the EP ''M3LL155X'' Geography * Figure Eight Island, North Carolina, United States * Figure Eight Lake, Alberta, Canada * Figure-Eight Loops, feature of the Historic Columbia River Highway in Guy W. Talbot State Park Mathematics and sciences * Figure-eight knot (mathematics), in knot theory * ∞, symbol meaning infinity * Lemniscate, various types of mathematical curve that resembles a figure 8 * Figure 8, a two-lobed Lissajous c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
