|
Twist Knot
In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. (That is, a twist knot is any Whitehead double of an unknot.) The twist knots are an infinite family of knots, and are considered the simplest type of knots after the torus knots. Construction A twist knot is obtained by linking together the two ends of a twisted loop. Any number of half-twists may be introduced into the loop before linking, resulting in an infinite family of possibilities. The following figures show the first few twist knots: Image:One-Twist Trefoil.png, One half-twist (trefoil knot, 31) Image:Blue Figure-Eight Knot.png, Two half-twists (figure-eight knot, 41) Image:Blue Three-Twist Knot.png, Three half-twists ( 52 knot) Image:Blue Stevedore Knot.png, Four half-twists (stevedore knot, 61) Image:Blue 7_2 Knot.png, Five half-twists (72 knot) Image:Blue 8_1 Knot.png, Six half-twists (81 knot) Properties All ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blue 8 1 Knot
Blue is one of the three primary colours in the RYB colour model (traditional colour theory), as well as in the RGB (additive) colour model. It lies between violet and cyan on the spectrum of visible light. The term ''blue'' generally describes colours perceived by humans observing light with a dominant wavelength that's between approximately 450 and 495 nanometres. Most blues contain a slight mixture of other colours; azure contains some green, while ultramarine contains some violet. The clear daytime sky and the deep sea appear blue because of an optical effect known as Rayleigh scattering. An optical effect called the Tyndall effect explains blue eyes. Distant objects appear more blue because of another optical effect called aerial perspective. Blue has been an important colour in art and decoration since ancient times. The semi-precious stone lapis lazuli was used in ancient Egypt for jewellery and ornament and later, in the Renaissance, to make the pigment ultram ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unknotting Number
In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself (crossing switch) to untie it. If a knot has unknotting number n, then there exists a diagram of the knot which can be changed to unknot by switching n crossings. The unknotting number of a knot is always less than half of its crossing number. This invariant was first defined by Hilmar Wendt in 1936. Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. The following table show the unknotting numbers for the first few knots: Image:Blue Trefoil Knot.png, Trefoil knot unknotting number 1 Image:Blue Figure-Eight Knot.png, Figure-eight knot unknotting number 1 Image:Blue Cinquefoil Knot.png, Cinquefoil knot unknotting number 2 Image:Blue Three-Twist Knot.png, Three-twist knot unknotting number 1 Image:Blue Stevedore Knot.png, Stevedore knot unknotting number 1 Ima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jones Polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t^ with integer coefficients. Definition by the bracket Suppose we have an oriented link L, given as a knot diagram. We will define the Jones polynomial V(L) by using Louis Kauffman's bracket polynomial, which we denote by \langle~\rangle. Here the bracket polynomial is a Laurent polynomial in the variable A with integer coefficients. First, we define the auxiliary polynomial (also known as the normalized bracket polynomial) :X(L) = (-A^3)^\langle L \rangle, where w(L) denotes the writhe of L in its given diagram. The writhe of a diagram is the number of positive crossings (L_ in the figure below) minus the number of negative crossings (L_). The writhe is not a knot invariant. X(L) is a knot invariant si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial. Definition Let ''K'' be a knot in the 3-sphere. Let ''X'' be the infinite cyclic cover of the knot complement of ''K''. This covering can be obtained by cutting the knot complement along a Seifert surface of ''K'' and gluing together infinitely many copies of the resulting manifold with boundary in a cyclic manner. There is a covering transformation ''t'' acting on ''X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figure-eight Knot (mathematics)
In knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number (knot theory), crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure-eight knot is a prime knot. Origin of name 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. Description A simple parametric representation of the figure-eight knot is as the set of all points (''x'',''y'',''z'') where : \begin x & = \left(2 + \cos \right) \cos \\ y & = \left(2 + \cos \right) \sin \\ z & = \sin \end for ''t'' varying over the real numbers (see 2D visual realization at bottom right). The figure-eight knot is Prime knot, prime, alternating knot, alternating, rational knot, rational with an associated value of 5/3, and is Chiral kn ... [...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 superposed (not to be confused with 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-superposable mirror ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible Knot
In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot (mathematics), knot that can be ambient isotopy, 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 (knot theory), link equivalent of an invertible knot. There are only five knot symmetry types, indicated by Chiral_knot, 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 (mathematics), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crossing Number (knot Theory)
In the mathematics, mathematical area of knot theory, the crossing number of a knot (mathematics), knot is the smallest number of crossings of any diagram of the knot. It is a knot invariant. Examples By way of example, the unknot has crossing number 0 (number), zero, the trefoil knot three and the figure-eight knot (mathematics), figure-eight knot four. There are no other knots with a crossing number this low, and just two knots have crossing number five, but the number of knots with a particular crossing number increases rapidly as the crossing number increases. Tabulation Tables of prime knots are traditionally indexed by crossing number, with a subscript to indicate which particular knot out of those with this many crossings is meant (this sub-ordering is not based on anything in particular, except that torus knots then twist knots are listed first). The listing goes 31 (the trefoil knot), 41 (the figure-eight knot), 51, 52, 61, etc. This order has not changed significa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slice Knot
A slice knot is a knot (mathematics), mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space. Definition A knot K \subset S^3 is said to be a topologically slice knot or a smoothly slice knot, if it is the boundary of an Embedding, embedded disk in the 4-ball B^4, which is Local flatness, locally flat or Smoothness, smooth, respectively. Here we use S^3 = \partial B^4: the 3-sphere S^3 = \ is the boundary (topology), boundary of the four-dimensional ball (mathematics), ball B^4 = \. Every smoothly slice knot is topologically slice because a smoothly embedded disk is locally flat. Usually, smoothly slice knots are also just called slice. Both types of slice knots are important in 3- and 4-dimensional topology. Smoothly slice knots are often illustrated using knots diagrams of ribbon knots and it is an open question whether there are any smoothly slice knots which are not ribbon knots (′Slice-ribbon conjecture′). Cone construction The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stevedore Knot (mathematics)
A dockworker (also called a longshoreman, stevedore, docker, wharfman, lumper or wharfie) is a waterfront manual laborer who loads and unloads ships. As a result of the intermodal shipping container revolution, the required number of dockworkers has declined by over 90% since the 1960s. Etymology The word ''stevedore'' () originated in Portugal or Spain, and entered the English language through its use by sailors. It started as a phonetic spelling of ''estivador'' ( Portuguese) or ''estibador'' (Spanish), meaning ''a man who loads ships and stows cargo'', which was the original meaning of ''stevedore'' (though there is a secondary meaning of "a man who stuffs" in Spanish); compare Latin ''stīpāre'' meaning ''to stuff'', as in ''to fill with stuffing''. In Ancient and Modern Greek, the verb στοιβάζω (stivazo) means pile up. In Great Britain and Ireland, people who load and unload ships are usually called ''dockers''; in Australia, they are called ''stevedores'', ''dock ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-bridge Knot
In the mathematical field of knot theory, a 2-bridge knot is a knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ... which can be regular isotoped so that the natural height function given by the ''z''-coordinate has only two maxima and two minima as critical points. 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, and . 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. Schubert normal form The names rational knot and rational link were coined by John Conway who defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twist Knot Stevedore Steps Horizontal
Twist may refer to: In arts and entertainment Film, television, and stage * ''Twist'' (2003 film), a 2003 independent film loosely based on Charles Dickens's novel ''Oliver Twist'' * ''Twist'' (2021 film), a 2021 modern rendition of ''Oliver Twist'' starring Rafferty Law * ''The Twist'' (1976 film), a 1976 film co-written and directed by Claude Chabrol * ''The Twist'' (1992 film), a 1992 documentary film directed by Ron Mann * ''Twist'' (stage play), a 1995 stage thriller by Miles Tredinnick * Twist, the main character on television series '' The Fresh Beat Band'' and its spin-off ''Fresh Beat Band of Spies'' * Oliver Twist (other), name of several film, television, and musical adaptations based on Charles Dickens's novel ''Oliver Twist'' * "Twist" (''Only Murders in the Building''), a 2021 episode of the TV series ''Only Murders in the Building'' * Jack Twist, a character in the 2005 film ''Brokeback Mountain'' * Twist Morgan, a character in the television seri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
