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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.
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.
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In three-dimensional topology, a branch of mathematics, the cyclic surgery theorem states that, for a compact, connected, orientable, irreducible three-manifold M whose boundary is a torus T, if M is not a Seifert-fibered space and r,s are slopes on T such that their Dehn fillings have cyclic fundamental group, then the distance between r and s is at most 1. Consequently, there are at most three Dehn fillings of M with cyclic fundamental group. The theorem appeared in a 1987 paper written by Marc Culler, Cameron Gordon, John Luecke and Peter Shalen.
In mathematical knot theory, a Conway sphere, named after John Horton Conway, is a 2-sphere intersecting a given knot or link in a 3-manifold transversely in four points. In a knot diagram, a Conway sphere can be represented by a simple closed curve crossing four points of the knot, the cross-section of the sphere; such a curve does not always exist for an arbitrary knot diagram of a knot with a Conway sphere, but it is always possible to choose a diagram for the knot in which the sphere can be depicted in this way. A Conway sphere is essential if it is incompressible in the knot complement. Sometimes, this condition is included in the definition of Conway spheres.
Jonathan Micah Rosenberg is an American mathematician, working in algebraic topology, operator algebras, K-theory and representation theory, with applications to string theory in physics.
Lisa Marie Piccirillo is an American mathematician who works in the fields of geometry and low-dimensional topology. In 2020, Piccirillo published a mathematical proof in the journal Annals of Mathematics determining that the Conway knot is not a slice knot, answering an unsolved problem in knot theory first proposed over fifty years prior by English mathematician John Horton Conway.
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References
- ↑ M. Culler, C. Gordon, J. Luecke, P. Shalen (1987). Dehn surgery on knots. The Annals of Mathematics (Annals of Mathematics) 125 (2): 237-300.
- ↑ Cameron Gordon and John Luecke, Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989), no. 2, 371–415.
- ↑ The University of Texas at Austin; Faculty profile Archived 2011-09-27 at the Wayback Machine
- ↑ NSF Presidential and Honorary Awards
- ↑ "Sloan Research Fellowships". University of Texas at Austin. Retrieved 2024-03-03.
- ↑ List of Fellows of the American Mathematical Society, retrieved 2013-02-02.
