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In the mathematical field of geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
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 the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. Kurt Reidemeister (1927) and, independently, James Waddell Alexander and Garland Baird Briggs (1926), demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves.
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 mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf.
In mathematics, Khovanov homology is an oriented link invariant that arises as the homology of a chain 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 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.
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 the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disk with only ribbon singularities. Intuitively, this kind of singularity can be formed by cutting a slit in the disk and passing another part of the disk through the slit. More precisely, this type of singularity is a closed arc consisting of intersection points of the disk with itself, such that the preimage of this arc consists of two arcs in the disc, one completely in the interior of the disk and the other having its two endpoints on the disk boundary.
In the mathematical theory of knots, the stick number is a knot invariant that intuitively gives the smallest number of straight "sticks" stuck end to end needed to form a knot. Specifically, given any knot K, the stick number of K, denoted by stick(K), is the smallest number of edges of a polygonal path equivalent to K.
In the mathematical theory of knots, a Berge knot or doubly primitive knot is any member of a particular family of knots in the 3-sphere. A Berge knot K is defined by the conditions:
- K lies on a genus two Heegaard surface S
- in each handlebody bound by S, K meets some meridian disc exactly once.
In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed. A non-invertible knot is any knot which does not have this property. The invertibility of a knot is a knot invariant. An invertible link is the link equivalent of an invertible knot.
In the mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot. It is a knot invariant.
In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself to untie it. If a knot has unknotting number , then there exists a diagram of the knot which can be changed to unknot by switching crossings. The unknotting number of a knot is always less than half of its crossing number.
Abigail A. Thompson is an American mathematician. She works as a professor of mathematics at the University of California, Davis, where she specializes in knot theory and low-dimensional topology.
In mathematics, the curve complex is a simplicial complex C(S) associated to a finite-type surface S, which encodes the combinatorics of simple closed curves on S. The curve complex turned out to be a fundamental tool in the study of the geometry of the Teichmüller space, of mapping class groups and of Kleinian groups. It was introduced by W.J.Harvey in 1978.
Jennifer Carol Schultens is an American mathematician specializing in low-dimensional topology and knot theory. She is a professor of mathematics at the University of California, Davis.
Introduction to 3-Manifolds is a mathematics book on low-dimensional topology. It was written by Jennifer Schultens and published by the American Mathematical Society in 2014 as volume 151 of their book series Graduate Studies in Mathematics.
References
- Clark, Bradd (1980), "The Heegaard Genus Of Manifolds Obtained By Surgery On Links And Knots", International Journal of Mathematics and Mathematical Sciences , 3 (3): 583–589, doi: 10.1155/S0161171280000440
- Boileau, Michel; Lustig, Martin; Moriah, Yoav (1994), "Links with super-additive tunnel number", Mathematical Proceedings of the Cambridge Philosophical Society , 115 (1): 85–95, Bibcode:1994MPCPS.115...85B, doi:10.1017/S0305004100071930, MR 1253284 .
- Kobayashi, Tsuyoshi; Rieck, Yo'av (2006), "On the growth rate of the tunnel number of knots", Journal für die Reine und Angewandte Mathematik , 2006 (592): 63–78, arXiv: math/0402025 , doi:10.1515/CRELLE.2006.023, MR 2222730 .
- Scharlemann, Martin (1984), "Tunnel number one knots satisfy the Poenaru conjecture", Topology and Its Applications , 18 (2–3): 235–258, doi:10.1016/0166-8641(84)90013-0, MR 0769294 .
- Scharlemann, Martin (2004), "There are no unexpected tunnel number one knots of genus one", Transactions of the American Mathematical Society , 356 (4): 1385–1442, arXiv: math/0106017 , doi:10.1090/S0002-9947-03-03182-9, MR 2034312 .
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- ↑ Boileau, Michel; Rost, Markus; Zieschang, Heiner (1 January 1988). "On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces". Mathematische Annalen. 279 (3): 553–581. doi:10.1007/BF01456287. ISSN 1432-1807.