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 mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.
Sir Simon Kirwan Donaldson is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. He is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University in New York, and a Professor in Pure Mathematics at Imperial College London.
In the mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables m and l.
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.
In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram.
In mathematics, the Lickorish–Wallace theorem in the theory of 3-manifolds states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with ±1 surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted.
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: drilling then filling.
In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obtained by performing (non-trivial) Dehn surgery on the knot is not simply-connected. The conjecture states that all knots, except the unknot, have Property P.
The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the Tait conjectures have been solved, the most recent being the Flyping conjecture.
Colin Conrad Adams 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.
Louis Hirsch Kauffman is an American mathematician, topologist, and professor of mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago. He is known for the introduction and development of the bracket polynomial and the Kauffman polynomial.
Joan Sylvia Lyttle Birman is an American mathematician, specializing in low-dimensional topology. She has made contributions to the study of knots, 3-manifolds, mapping class groups of surfaces, geometric group theory, contact structures and dynamical systems. Birman is research professor emerita at Barnard College, Columbia University, where she has been since 1973.
Kenneth C. Millett is a professor of mathematics at the University of California, Santa Barbara. His research concerns low-dimensional topology, knot theory, and the applications of knot theory to DNA structure; his initial is the "M" in the name of the HOMFLY polynomial.
Marc Lackenby is a professor of mathematics at the University of Oxford whose research concerns knot theory, low-dimensional topology, and group theory.
Vladimir Georgievich Turaev is a Russian mathematician, specializing in topology.
