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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces In Mathematics, a knot is an Embedding of a Circle in 3-dimensional Euclidean space, R 3 considered up to continuous deformations The equivalence is often given by ambient isotopy but can be given by homeomorphism. In the mathematical subject of Topology, an ambient isotopy, also called an h-isotopy is a kind of continuous distortion of an "ambient space" a Manifold Topological equivalence redirects here see also Topological equivalence (dynamical systems. Some invariants are indeed numbers, but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory . In Mathematics, homology theory is the Axiomatic study of the intuitive geometric idea of homology of cycles on Topological spaces It can be broadly 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.
From the modern perspective, it is natural to define a knot invariant from a knot diagram. In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces Of course, it must be unchanged (that is to say, invariant) under the Reidemeister moves. In the mathematical area of Knot theory, a Reidemeister move refers to one of three local moves on a Link diagram. Tricoloring is a particularly simple example. In the mathematical field of Knot theory, Fox n -coloring is a method of specifying a representation of a Knot group (or a Link group Other examples are knot polynomials, such as the Jones polynomial, which are currently among the most useful invariants for distinguishing knots from one another, though at the time of writing it is not known whether any of these distinguishes all knots from each other or even just the unknot from all other knots. In the mathematical field of Knot theory, a knot polynomial is a Knot invariant in the form of a Polynomial whose coefficients encode some of In the mathematical field of Knot theory, the Jones polynomial is a Knot polynomial discovered by Vaughan Jones in 1983 The unknot arises in the mathematical theory of knots. Intuitively the unknot is a closed loop of rope without a Knot in it
Other invariants can be defined by taking the minimum "value" over all possible diagrams of a knot. This category includes the crossing number, which is the minimum number of crossings for any diagram of the knot, and the bridge number, which is the minimum number of bridges for any diagram of the knot. In Mathematics, crossing numbers arise in two related contexts in Knot theory and in Graph theory. In a mathematical field of Knot theory, the Bridge number is an invariant of a knot
Historically, many of the early knot invariants are not defined by first selecting a diagram but defined intrinsically, which can make computing some of these invariants a challenge. For example, knot genus is particularly tricky to compute, but can be effective (for instance, in distinguishing mutants). In Mathematics, a Seifert surface is a Surface whose boundary is a given knot or link. In the mathematical field of Knot theory, a mutation is an operation on a knot that can produce different knots
The complement of a knot itself (as a topological space) is known to be a "complete invariant" of the knot by the Gordon-Luecke theorem in the sense that it distinguishes the given knot from all other knots up to ambient isotopy and mirror image. In Mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a Solid torus Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, the Gordon-Luecke theorem on Knot complements states that every Homeomorphism between two complements of knots in the 3-sphere In the mathematical subject of Topology, an ambient isotopy, also called an h-isotopy is a kind of continuous distortion of an "ambient space" a Manifold Some invariants associated with the knot complement include the knot group which is just the fundamental group of the complement. In Mathematics, a knot is an Embedding of a Circle into 3-dimensional Euclidean space. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. The knot quandle is also a complete invariant in this sense but it is difficult to determine if two quandles are isomorphic.
Most knots are hyperbolic, which means the hyperbolic volume is an invariant for these knots. In Mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative In the mathematical field of Knot theory, the hyperbolic volume of a Hyperbolic link is simply the volume of the link's complement with respect to its complete hyperbolic Volume, and other hyperbolic invariants, have proven very effective, utilized in some of the extensive efforts at knot tabulation.
In recent years, there has been much interest in homological invariants of knots which categorify well-known invariants. In Mathematics, homology theory is the Axiomatic study of the intuitive geometric idea of homology of cycles on Topological spaces It can be broadly In Mathematics, categorification refers to the process of replacing set-theoretic Theorems by category-theoretic analogues Heegaard Floer homology is a homology theory whose Euler characteristic is the Alexander polynomial of the knot. Floer homology is a mathematical tool used in the study of Symplectic geometry and low-dimensional Topology. In Mathematics, homology theory is the Axiomatic study of the intuitive geometric idea of homology of cycles on Topological spaces It can be broadly In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant In Mathematics, the Alexander polynomial is a Knot invariant which assigns a Polynomial with integer coefficients to each knot type It has been proven effective in deducing new results about the classical invariants. Along a different line of study, there is a combinatorially defined cohomology theory of knots called Khovanov homology whose Euler characteristic is the Jones polynomial. In Mathematics, Khovanov homology is a Homology theory for knots and links. In the mathematical field of Knot theory, the Jones polynomial is a Knot polynomial discovered by Vaughan Jones in 1983 This has recently been shown to be useful in obtaining bounds on slice genus whose earlier proofs required gauge theory. In Mathematics, the slice genus of a smooth knot K in S3 (sometimes called its Murasugi genus or 4-ball Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Khovanov and Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants. Mikhail Khovanov is an associate professor of mathematics at Columbia University.
There is also growing interest from both knot theorists and scientists in understanding "physical" or geometric properties of knots and relating it to topological invariants and knot type. An old result in this direction is the Fary-Milnor theorem states that if the total curvature of a knot K in 
satisfies
where κ(p) is the curvature at p, then K is an unknot. In Mathematics, the Fary-Milnor theorem in Knot theory states that for any knot K in R 3 if the Total curvature This article only considers curves in Euclidean space Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian Therefore, for knotted curves,
An example of a "physical" invariant is ropelength, which is the amount of 1-inch diameter rope needed to realize a particular knot type. In Knot theory each realization of a link or knot has an associated ropelength.
Other invariants
In Mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in Three-dimensional space.Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic