1.
Crossing number (knot theory)
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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. By way of example, the unknot has crossing number zero, 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. The listing goes 31,41,51,52,61 and this order has not changed significantly since P. G. Tait published a tabulation of knots in 1877. There has been little progress on understanding the behavior of crossing number under rudimentary operations on knots. A big open question if the crossing number is additive when taking knot sums. It is also expected that a satellite of a knot K should have larger crossing number than K, additivity of crossing number under knot sum has been proven for special cases, for example if the summands are alternating knots, or if the summands are torus knots. Marc Lackenby has also given a proof that there is a constant N >1 such that 1 N ≤ c r, but his method, there are connections between the crossing number of a knot and the physical behavior of DNA knots. For prime DNA knots, crossing number is a predictor of the relative velocity of the DNA knot in agarose gel electrophoresis. Basically, the higher the number, the faster the relative velocity. For composite knots, this not appear to be the case. There are related concepts of average crossing number and asymptotic crossing number, both of these quantities bound the standard crossing number. Asymptotic crossing number is conjectured to be equal to crossing number, other numerical knot invariants include the bridge number, linking number, stick number, and unknotting number

2.
Hyperbolic volume
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In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the links complement with respect to its complete hyperbolic metric. The volume is necessarily a finite number, and is a topological invariant of the link. As a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture, the components of the link will become cusps of the 3-manifold, and the manifold itself will have finite volume. In particular, the volume of the complement is a knot invariant. In order to make it well-defined for all knots or links, there are only finitely many hyperbolic knots for any given volume. In practice, hyperbolic volume has proven effective in distinguishing knots. Jeffrey Weekss computer program SnapPea is the tool used to compute hyperbolic volume of a link. The Weeks manifold has the smallest possible volume of any closed manifold, Thurston and Jørgensen proved that the set of real numbers that are hyperbolic volumes of 3-manifolds is well-ordered, with order type ωω

3.
Unknotting number
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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 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 numbers for the first few knots, In general. Known cases include, The unknotting number of a nontrivial twist knot is equal to one. The unknotting number of a knot is equal to /2. The unknotting numbers of knots with nine or fewer crossings have all been determined. Crossing number Bridge number Linking number Stick number Unknotting problem Three_Dimensional_Invariants#Unknotting_Number, The Knot Atlas

4.
Conway notation (knot theory)
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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, 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, tangles 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 to into a position with the Reidemeister moves, it is called the 0 or ∞ tangle. If a tangle, a, is reflected on the NW-SE line, tangles have three binary operations, sum, product, and ramification, however all can be explained using tangle addition and negation. The tangle product, a b, is equivalent to −a+b. rational tangles are equivalent if and only if their fractions are equal. An accessible proof of this fact is given in, a number before an asterisk, *, denotes the polyhedron number, multiple asterisks indicate that multiple polyhedra of that number exist. Dowker notation Alexander–Briggs notation Conway, J. H, an Enumeration of Knots and Links, and Some of Their Algebraic Properties. In J. Leech, Computational Problems in Abstract Algebra, pdf available online Louis H. Kauffman, Sofia Lambropoulou, On the classification of rational tangles. Advances in Applied Mathematics,33, No

5.
Dowker notation
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In the mathematical field of knot theory, the Dowker notation, also called the Dowker–Thistlethwaite notation or code, for a knot is a sequence of even integers. The notation is named after Clifford Hugh Dowker and Morwen Thistlethwaite, to generate the Dowker notation, traverse the knot using an arbitrary starting point and direction. Label each of the n crossings with the numbers 1, 2n in order of traversal, with the following modification, if the label is an even number and the strand followed crosses over at the crossing, then change the sign on the label to be a negative. When finished, each crossing will be labelled a pair of integers, one even, the Dowker notation is the sequence of even integer labels associated with the labels 1,3. For example, a diagram may have crossings labelled with the pairs. The Dowker notation for this labelling is the sequence,6 −1228 −4 −10, Knots tabulations typically consider only prime knots and disregard chirality, so this ambiguity does not affect the tabulation. The ménage problem, posed by Tait, concerns counting the number of different number sequences possible in this notation, conway notation Alexander–Briggs notation Adams, Colin Conrad. The Knot Book, An Elementary Introduction to the Mathematical Theory of Knots

6.
Hyperbolic link
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In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i. e. has a hyperbolic geometry. A hyperbolic knot is a link with one component. As a consequence of the work of William Thurston, it is known that every knot is one of the following, hyperbolic. As a consequence, hyperbolic knots can be considered plentiful, a similar heuristic applies to hyperbolic links. As a consequence of Thurstons hyperbolic Dehn surgery theorem, performing Dehn surgeries on a hyperbolic link enables one to many more hyperbolic 3-manifolds. Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco. 4₁ knot 5₂ knot 6₁ knot 6₂ knot 6₃ knot 7₄ knot 10161 knot 12n242 knot SnapPea hyperbolic volume Colin Adams The Knot Book, American Mathematical Society, William Menasco Closed incompressible surfaces in alternating knot and link complements, Topology 23, 37–44. William Thurston The geometry and topology of three-manifolds, Princeton lecture notes

7.
Fibered knot
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For example, The unknot, trefoil knot, and figure-eight knot are fibered knots. The Hopf link is a fibered link, fibered knots and links arise naturally, but not exclusively, in complex algebraic geometry. For instance, each point of a complex plane curve can be described topologically as the cone on a fibered knot or link called the link of the singularity. The trefoil knot is the link of the cusp singularity z 2 + w 3, in these cases, the family of Seifert surfaces is an aspect of the Milnor fibration of the singularity. A knot is fibered if and only if it is the binding of some open book decomposition of S3, the Alexander polynomial of a fibered knot is monic, i. e. the coefficients of the highest and lowest powers of t are plus or minus 1. Examples of knots with nonmonic Alexander polynomials abound, for example the twist knots have Alexander polynomials qt − + qt−1, in particular the Stevedores knot is not fibered. Pretzel knot How to construct all fibered knots and links

8.
Pretzel link
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In the mathematical theory of knots, a pretzel link is a special kind of link. A pretzel link which is also a knot is a pretzel knot, in the standard projection of the 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, the pretzel link is a knot iff both n and all the p i are odd or exactly one of the p i is even. The pretzel link is split if at least two of the p i are zero, but the converse is false, the pretzel link is the mirror image of the pretzel link. The pretzel link is link-equivalent to the pretzel link, thus, too, the pretzel link is link-equivalent to the pretzel link. The pretzel link is link-equivalent to the pretzel link, however, if one orients the links in a canonical way, then these two links have opposite orientations. The pretzel knot is the trefoil, the knot is its mirror image. The pretzel knot is the stevedore knot, if p, q, r are distinct odd integers greater than 1, then the pretzel knot is a non-invertible knot. The pretzel link is a formed by three linked unknots. The pretzel knot is the sum of two trefoil knots. The pretzel link is the union of an unknot and another knot. A Montesinos link is a kind of link that generalizes pretzel links. A Montesinos link which is also a knot is a Montesinos knot, a Montesinos link is composed of several rational tangles. One notation for a Montesinos link is K, in this notation, e and all the α i and β i are integers. Many results have been stated about the manifolds that result from Dehn surgery on the knot in particular. The hyperbolic volume of the complement of the link is 4 times Catalans constant. This pretzel link complement is one of two two-cusped hyperbolic manifolds with the possible volume, the other being the complement of the Whitehead link. Trotter, Hale F. Non-invertible knots exist, Topology,2, 272–280