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5₁ Knot
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. Properties The cinquefoil is a prime knot. Its writhe is 5, and it is invertible but not amphichiral. Its Alexander polynomial is :\Delta(t) = t^2 - t + 1 - t^ + t^, since \begin1 & -1 & 0&0\\ 0 & 1 &-1 &0 \\ 0& 0& 1&-1 \\ 0& 0& 0&1\end is a possible Seifert matrix, or because of its Conway polynomial, which is :\nabla(z) = z^4 + 3z^2 + 1, and its Jones polynomial is :V(q) = q^ + q^ - q^ + q^ - q^. 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. History The name "cinquefoil" comes from the five-pet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, \mathbb^3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself. Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways. Therefore, a fundamental p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seifert Surface
In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface. Seifert surfaces are also interesting in their own right, and the subject of considerable research. Specifically, let ''L'' be a tame oriented knot or link in Euclidean 3-space (or in the 3-sphere). A Seifert surface is a compact, connected, oriented surface ''S'' embedded in 3-space whose boundary is ''L'' such that the orientation on ''L'' is just the induced orientation from ''S''. Note that any compact, connected, oriented surface with nonempty boundary in Euclidean 3-space is the Seifert surface associated to its boundary link. A single knot or link can have many different inequivalent Seifert surfaces. A Seifert surface must be oriented. It is possi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trefoil Knot
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot (mathematics), knot. The trefoil can be obtained by joining the two loose ends of a common overhand knot, resulting in a knotted loop (topology), 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. Descriptions The trefoil knot can be defined as the curve obtained from the following parametric equations: :\begin x &= \sin t + 2 \sin 2t \\ y &= \cos t - 2 \cos 2t \\ z &= -\sin 3t \end 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: :\begin x &= (2+\cos 3t) \cos 2t \\ y &= (2+\cos 3t )\sin 2t \\ z &= \sin 3t \end Any continuous deformation of the curve above is also considered a trefoil knot. Specifically, any curve Homotopy#Isotopy, isotopic to a trefoil knot is also co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around the five points creates a similar symbol referred to as the pentacle, which is used widely by Wiccans and in paganism, or as a sign of life and connections. The word ''pentagram'' comes from the Greek language, Greek word πεντάγραμμον (''pentagrammon''), from πέντε (''pente''), "five" + γραμμή (''grammē''), "line". The word pentagram refers to just the star and the word pentacle refers to the star within a circle, although there is some overlap in usage. The word ''pentalpha'' is a 17th-century revival of a post-classical Greek name of the shape. History Early history Early pentagrams have been found on Sumerian pottery from Ur c. 3500 Common Era, BCE, and the five-pointed star was at various times the symbol of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cinquefoil Knot
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. Properties The cinquefoil is a prime knot. Its writhe is 5, and it is invertible but not amphichiral. Its Alexander polynomial is :\Delta(t) = t^2 - t + 1 - t^ + t^, since \begin1 & -1 & 0&0\\ 0 & 1 &-1 &0 \\ 0& 0& 1&-1 \\ 0& 0& 0&1\end is a possible Seifert matrix, or because of its Conway polynomial, which is :\nabla(z) = z^4 + 3z^2 + 1, and its Jones polynomial is :V(q) = q^ + q^ - q^ + q^ - q^. 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. History The name "cinquefoil" comes from the five-pet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potentilla
''Potentilla'' is a genus containing over 500 species of Annual plant, annual, Biennial plant, biennial and Perennial plant, perennial herbaceous plant, herbaceous flowering plants in the rose family (biology), family, Rosaceae. Potentillas may also be called cinquefoils in English, but they have also been called five fingers and silverweeds. Some species are called tormentils, though this is often used specifically for Common Tormentil, common tormentil (''P. erecta''). Others are referred to as barren strawberries, which may also refer to ''Potentilla sterilis, P. sterilis'' in particular, or to the closely related ''Waldsteinia fragarioides''. Several other cinquefoils formerly included here are now separated in distinct genera – notably the popular garden shrub ''P. fruticosa'', now ''Dasiphora fruticosa''. Potentillas are generally found throughout the northern continents of the world (holarctic), though some occur in montane biomes of the New Guinea Highlands. Descrip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kauffman Polynomial
In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as :F(K)(a,z)=a^L(K)\,, where w(K) is the writhe of the link diagram and L(K) is a polynomial in ''a'' and ''z'' defined on link diagrams by the following properties: *L(O) = 1 (O is the unknot). *L(s_r)=aL(s), \qquad L(s_\ell)=a^L(s). *''L'' is unchanged under type II and III Reidemeister moves. Here s is a strand and s_r (resp. s_\ell) is the same strand with a right-handed (resp. left-handed) curl added (using a type I Reidemeister move). Additionally ''L'' must satisfy Kauffman's skein relation: : The pictures represent the ''L'' polynomial of the diagrams which differ inside a disc as shown but are identical outside. Kauffman showed that ''L'' exists and is a regular isotopy invariant of unoriented links. It follows easily that ''F'' is an ambient isotopy invariant of oriented links. The Jones polynomial is a special case of the Kauf ... [...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] |
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] |
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] |
