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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. 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 Image:Blue 6_2 Knot.png, 6₂ knot unknotting number 1 Imag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unknotting Trefoil , a mathematical problem
{{Mathematics disambiguation ...
Unknotting may refer to: *Unknotting number, the minimum number of times the knot must be passed through itself to untie it *Unknotting problem In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algorithms. A major unresolved challenge is to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three-twist Knot
In knot theory, the three-twist knot is the twist knot with three-half twists. It is listed as the 52 knot in the Alexander-Briggs notation, and is one of two knots with crossing number five, the other being the cinquefoil knot. Properties The three-twist knot is a prime knot, and it is invertible but not amphichiral. Its Alexander polynomial is :\Delta(t) = 2t-3+2t^, \, its Conway polynomial is :\nabla(z) = 2z^2+1, \, and its Jones polynomial is :V(q) = q^ - q^ + 2q^ - q^ + q^ - q^. \, Because the Alexander polynomial is not monic, the three-twist knot is not fibered. The three-twist knot is a hyperbolic knot, with its complement having a volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ... of approximately 2.82812. If the fibre of the knot in the initial ima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Linking Number
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In Euclidean space, the linking number is always an integer, but may be positive or negative depending on the orientation of the two curves (this is not true for curves in most 3-manifolds, where linking numbers can also be fractions or just not exist at all). The linking number was introduced by Gauss in the form of the linking integral. It is an important object of study in knot theory, algebraic topology, and differential geometry, and has numerous applications in mathematics and science, including quantum mechanics, electromagnetism, and the study of DNA supercoiling. Definition Any two closed curves in space, if allowed to pass through themselves but not each other, can be moved into exactly one of the following standard positions. Thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bridge Number
In the mathematical field of knot theory, the bridge number is an invariant of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot. Definition Given a knot or link, draw a diagram of the link using the convention that a gap in the line denotes an undercrossing. Call an arc in this diagram a bridge if it includes at least one overcrossing. Then the bridge number of a knot can be found as the minimum number of bridges required for any diagram of the knot.. Bridge number was first studied in the 1950s by Horst Schubert. The bridge number can equivalently be defined geometrically instead of topologically. In bridge representation, a knot lies entirely in the plane apart for a finite number of bridges whose projections onto the plane are straight lines. Equivalently the bridge number is the minimal number of local maxima of the projection of the knot onto a vector, where we minimize over all projections and over all conformat ... [...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] |
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] |
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 A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7₁ Knot
In knot theory, the 71 knot, also known as the septoil knot, the septafoil knot, or the (7, 2)-torus knot, is one of seven prime knots with crossing number (knot theory), crossing number seven. It is the simplest torus knot after the trefoil knot, trefoil and cinquefoil knot, cinquefoil. Properties The 71 knot is invertible knot, invertible but not amphichiral knot, amphichiral. Its Alexander polynomial is :\Delta(t) = t^3 - t^2 + t - 1 + t^ - t^ + t^, \, its Alexander–Conway polynomial, Conway polynomial is :\nabla(z) = z^6 + 5z^4 + 6z^2 + 1, \, and its Jones polynomial is :V(q) = q^ + q^ - q^ + q^ - q^ + q^ - q^. \, Example See also *Heptagram References {{DEFAULTSORT:7 1 knot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6₃ Knot
In knot theory, the 63 knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 62 knot. It is alternating, hyperbolic, and fully amphichiral. It can be written as the braid word :\sigma_1^\sigma_2^2\sigma_1^\sigma_2. \, Symmetry Like the figure-eight knot, the 63 knot is fully amphichiral. This means that the 63 knot is amphichiral, meaning that it is indistinguishable from its own mirror image. In addition, it is also invertible, meaning that orienting the curve in either direction yields the same oriented knot. Invariants The Alexander polynomial of the 63 knot is :\Delta(t) = t^2 - 3t + 5 - 3t^ + t^, \, Conway polynomial is :\nabla(z) = z^4 + z^2 + 1, \, Jones polynomial is :V(q) = -q^3 + 2q^2 - 2q + 3 - 2q^ + 2q^ - q^, \, and the Kauffman polynomial is :L(a,z) = az^5 + z^5a^ + 2a^2z^4 + 2z^4a^ + 4z^4 + a^3z^3 + az^3 + z^3a^ + z^3a^ - 3a^2z^2 - 3z^2a^ - 6z^2 - a^3z - 2az - 2za^ - za^-3 + a^2 + a^ +3. \, The 63 kno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6₂ Knot
In knot theory, the 62 knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 63 knot. This knot is sometimes referred to as the Miller Institute knot, because it appears in the logo of the Miller Institute for Basic Research in Science at the University of California, Berkeley. The 62 knot is invertible but not amphichiral. Its Alexander polynomial is :\Delta(t) = -t^2 + 3t -3 + 3t^ - t^, \, its Conway polynomial is :\nabla(z) = -z^4 - z^2 + 1, \, and its Jones polynomial is :V(q) = q - 1 + 2q^ - 2q^ + 2q^ - 2q^ + q^. \, The 62 knot is a hyperbolic knot, with its complement having a volume of approximately 4.40083. Surface File:Superfície - bordo Nó 6,2.jpg, Surface of knot 6.2 Example Ways to assemble of knot 6.2 File:6₂ knot.webm, Example 1 File:6₂ knot (2).webm, Example 2 If a bowline The bowline ( or ) is an ancient and simple knot used to form a fixed loop at the end of a rope. It has the virtue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stevedore Knot (mathematics)
In knot theory, the stevedore knot is one of three prime knots with crossing number six, the others being the 62 knot and the 63 knot. The stevedore knot is listed as the 61 knot in the Alexander–Briggs notation In the mathematical field of topology, knot theory is the study of 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 ..., and it can also be described as a twist knot with four twists, or as the (5,−1,−1) pretzel link, pretzel knot. The mathematical stevedore knot is named after the common stevedore knot, which is often used as a stopper knot, stopper at the end of a rope. The mathematical version of the knot can be obtained from the common version by joining together the two loose ends of the rope, forming a knotted loop (topology), loop. The stevedore knot is invertible knot, invertible but not amphichiral knot, amphichi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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