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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.
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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 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.
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In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique.
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 Lagrangian 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.
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Thomas Schick is a German mathematician, specializing in algebraic topology and differential geometry.
In the mathematical field of quantum topology, the Reshetikhin–Turaev invariants (RT-invariants) are a family of quantum invariants of framed links. Such invariants of framed links also give rise to invariants of 3-manifolds via the Dehn surgery construction. These invariants were discovered by Nicolai Reshetikhin and Vladimir Turaev in 1991, and were meant to be a mathematical realization of Witten's proposed invariants of links and 3-manifolds using quantum field theory.
References
- ↑ "Efstratia Kalfagianni – The Mathematics Genealogy Project". www.mathgenealogy.org. Retrieved 2023-12-13.
- ↑ Office of the Provost MSU
- 1 2 cv
- ↑ Futer, D.; Kalfagianni, E.; Purcel, J. "Dehn filling, volume, and the Jones polynomial". Journal of Differential Geometry. 78 (3): 429–464. arXiv: math/0612138 . doi: 10.4310/jdg/1207834551 . ISSN 0022-040X.
- ↑ "Gromov norm and Turaev-Viro invariants of 3-manifolds | Société Mathématique de France". smf.emath.fr. Retrieved 2023-12-13.
- ↑ Belletti, Giulio; Detcherry, Renaud; Kalfagianni, Efstratia; Yang, Tian. "Growth of quantum $6j$-symbols and applications to the volume conjecture". Journal of Differential Geometry. 120 (2): 199–229. arXiv: 1807.03327 . doi:10.4310/jdg/1645207506. ISSN 0022-040X.
- ↑ Reviews of Guts of Surfaces and the Colored Jones Polynomial:MR 3024600; Zbl 1270.57002
- ↑ New York Journal of Mathematics
- ↑ Journal of Knot Theory and its Ramifications
- ↑ AMS Bookstore
- ↑ IAS Scholars
- ↑ 2019 Class of the Fellows of the AMS, American Mathematical Society
