In 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 be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.
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 K a quantity φ(K) such that if K and K' are equivalent then φ(K) = φ(K')."). 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, both in "enumeration" and "duplication removal".
A knot invariant is a quantity defined on the set of all knots, which takes the same value for any two equivalent knots. For example, a knot group is a knot invariant.
Typically a knot invariant is a combinatorial quantity defined on knot diagrams. Thus if two knot diagrams differ with respect to some knot invariant, they must represent different knots. However, as is generally the case with topological invariants, if two knot diagrams share the same values with respect to a [single] knot invariant, then we still cannot conclude that the knots are the same.
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 the properties of a given knot.
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 with integer coefficients.
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.
Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and computational complexity theory.
In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial.
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called symplectic Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds.
In the mathematical field of knot theory, the bracket polynomial is a polynomial invariant of framed links. Although it is not an invariant of knots or links, a suitably "normalized" version yields the famous knot invariant called the Jones polynomial. The bracket polynomial plays an important role in unifying the Jones polynomial with other quantum invariants. In particular, Kauffman's interpretation of the Jones polynomial allows generalization to invariants of 3-manifolds.
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 determine if the problem admits a polynomial time algorithm; that is, whether the problem lies in the complexity class P.
In knot theory, a virtual knot is a generalization of knots in 3-dimensional Euclidean space, R3, to knots in thickened surfaces modulo an equivalence relation called stabilization/destabilization. Here is required to be closed and oriented. Virtual knots were first introduced by Kauffman (1999).
In the mathematical field of knot theory, the Arf invariant of a knot, named after Cahit Arf, is a knot invariant obtained from a quadratic form associated to a Seifert surface. If F is a Seifert surface of a knot, then the homology group H1(F, Z/2Z) has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an embedded circle representing an element of the homology group. The Arf invariant of this quadratic form is the Arf invariant of the knot.
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
Mikhail Khovanov is a Russian-American professor of mathematics at Columbia University who works on representation theory, knot theory, and algebraic topology. He is known for introducing Khovanov homology for links, which was one of the first examples of categorification.
Knots have been used for basic purposes such as recording information, fastening and tying objects together, for thousands of years. The early significant stimulus in knot theory would arrive later with Sir William Thomson and his vortex theory of the atom.
Tomasz Mrowka is an American mathematician specializing in differential geometry and gauge theory. He is the Singer Professor of Mathematics and former head of the Department of Mathematics at the Massachusetts Institute of Technology.
In the mathematical field of knot theory, a quantum knot invariant or quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement.
In theoretical physics, the six-dimensional (2,0)-superconformal field theory is a quantum field theory whose existence is predicted by arguments in string theory. It is still poorly understood because there is no known description of the theory in terms of an action functional. Despite the inherent difficulty in studying this theory, it is considered to be an interesting object for a variety of reasons, both physical and mathematical.
Hernando Burgos Soto is a Canadian writer and mathematician, professor of mathematics at George Brown College. He is the author of several math papers in which he introduced some mathematics concepts and extended to tangles some celebrated results of knot theory about the Khovanov homology and the Jones polynomial. During his career as a mathematician, his interests have included Mathematical Statistics, Knot Theory, Algebraic Topology and more recently Mathematical Finance. He is comfortable writing in English and Spanish. When writing in Spanish, he works in the area of prose fiction writing short stories. Some of his short stories were published at the website Cuentos y Cuentos.
