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Stick Number
In the mathematical theory of knots, the stick number is a knot invariant that intuitively gives the smallest number of straight "sticks" stuck end to end needed to form a knot. Specifically, given any knot K, the stick number of K, denoted by \operatorname(K), is the smallest number of edges of a polygonal path equivalent Known values Six is the lowest stick number for any nontrivial knot. There are few knots whose stick number can be determined exactly. Gyo Taek Jin determined the stick number of a (p,q)-torus knot T(p,q) in case the parameters p and q are not too far from each other: The same result was found independently around the same time by a research group around Colin Adams, but for a smaller range of parameters. Bounds The stick number of a knot sum can be upper bounded by the stick numbers of the summands: \text(K_1\#K_2)\le \text(K_1)+ \text(K_2)-3 \, Related invariants The stick number of a knot K is related to its crossing number c(K) by the following ineq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trefoil Valknut Cropped
A trefoil () is a graphic form composed of the outline of three overlapping rings, used in architecture and Christian symbolism, among other areas. The term is also applied to other symbols with a threefold shape. A similar shape with four rings is called a quatrefoil. Architecture Ornamentation 'Trefoil' is a term in Gothic architecture given to the ornamental foliation or cusping introduced in the heads of window-lights, tracery, and panellings, in which the centre takes the form of a three-lobed leaf (formed from three partially overlapping circles). One of the earliest examples is in the plate tracery at Winchester Cathedral (1222–1235). The fourfold version of an architectural trefoil is a quatrefoil. A simple trefoil shape in itself can be symbolic of the Trinity, while a trefoil combined with an equilateral triangle was also a moderately common symbol of the Christian Trinity during the late Middle Ages in some parts of Europe, similar to a barbed quatrefoil. Two form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
In the mathematical field of 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, Unknot, 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 (in topology, a circle is not bound to the classical geometric concept, but to all of its homeomorphisms). 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 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 descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Invariant
In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers (algebraic), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory (for example, "a ''knot invariant'' is a rule that assigns to any knot a quantity such that if and are equivalent then ."). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification,Purcell, Jessica (2020). ''Hyperbolic Knot Theory'', p.7. American Mathematical Society. "A ''knot invariant'' is a function from the set of knots to some other set whose value depends only on the equiva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygonal Path
In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments connecting the consecutive vertices. Name A polygonal chain may also be called a polygonal curve, polygonal path, polyline,. piecewise linear curve, broken line or, in geographic information systems, a linestring or linear ring. Variations A simple polygonal chain is one in which only consecutive (or the first and the last) segments intersect and only at their endpoints. A closed polygonal chain is one in which the first vertex coincides with the last one, or, alternatively, the first and the last vertices are also connected by a line segment. A simple closed polygonal chain in the plane is the boundary of a simple polygon. Often the term "polygon" is used in the meaning of "closed polygonal chain", but in some cases it is important to dr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus Knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers ''p'' and ''q''. A torus link arises if ''p'' and ''q'' are not coprime (in which case the number of components is gcd(''p, q'')). A torus knot is trivial (equivalent to the unknot) if and only if either ''p'' or ''q'' is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot. Geometrical representation A torus knot can be rendered geometrically in multiple ways which are topologically equivalent (see Properties below) but geometrically distinct. The convention used in this article and its figures is the following. The (''p'',''q'')-torus knot winds ''q'' times around a circle in the interior of the torus, and ''p'' times around its axis of rotational symmetry.. If ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colin Adams (mathematician)
Colin Conrad Adams (born October 13, 1956) is a mathematician primarily working in the areas of hyperbolic 3-manifolds and knot theory. His book, ''The Knot Book'', has been praised for its accessible approach to advanced topics in knot theory. He is currently Francis Christopher Oakley Third Century Professor of Mathematics at Williams College, where he has been since 1985. He writes "Mathematically Bent", a column of math for the ''Mathematical Intelligencer''. His nephew is popular American singer Still Woozy. Academic career Adams received a B.Sc. from MIT in 1978 and a Ph.D. in mathematics from the University of Wisconsin–Madison in 1983. His dissertation was entitled "Hyperbolic Structures on Link Complements" and supervised by James Cannon. In 2012 he became a fellow of the American Mathematical Society. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Knot Sum Of Trefoils Stick Number
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ''ABCD'' would be denoted . Characterizations A convex quadrilateral is a square if and only if it is any one of the following: * A rectangle with two adjacent equal sides * A rhombus with a right vertex angle * A rhombus with all angles equal * A parallelogram with one right vertex angle and two adjacent equal sides * A quadrilateral with four equal sides and four right angles * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilateral with successiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces. More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of a connected sum on knots, called the knot sum or composition of knots. Connected sum at a point A connected sum of two ''m''-dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres. If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation. Although the construction uses the choice of the balls, the result is unique up to homeomorphism. One can also make this operation work in the smooth category, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crossing Number (knot Theory)
In the mathematical area of knot theory, the crossing number of a 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 zero, the trefoil knot three and the 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 significantly since P. G. Tait published a tabulation of knots in 1877. Additivity There ... [...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. 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 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 isotopic to a trefoil knot is also considered to be a trefoil. In addition, the mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
_torus_knot.svg "Click for more on -> Torus Knot")
