Examples/classes:
Types
Related concepts:
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
The theory of knots is very visual. It can provide a link between the concrete and abstract. Some of the arguments are quite elementary, others very deep, and there are numerous connections with other parts of mathematics.
A knot is a smooth (or piecewise-linear) embedding of the circle $S^1$ into $\mathbb{R}^3$, or equivalently into the $3$-sphere $S^3$ (one can also consider knots in other 3-manifolds).
Sometimes, higher dimensional knots are also considered. $n$-dimensional knot (or simply an $n$-knot) is a smooth embedding of $n$-dimensional closed manifold (usually an $n$-sphere) into the $(n+2)$-dimensional sphere $\mathbb{S}^n$.
Typically, knots are considered up to ambient isotopy (or smooth isotopy).
Knots that are ambient isotopic are often said to have the same knot type or to be in the same isotopy class.
The trefoil knot is the simplest non-trivial knot. In its simplest representation, it has three crossings. It is a torus knot, that is it can be embedded on the surface of a solid torus, itself embedded in $S^3$. Here is a picture.
Classifying knots up to isotopy is usually done using knot invariants. Some of these are simple to define (these tend to be geometric and also tend to be hard to calculate) others are harder to define and to show they are invariants but are easier to calculate. A few are reasonably easy to define and to calculate… Yippee!
It is often useful to consider the domain circle of a knot as being oriented. This is then represented by putting a directional arrow on diagrams of the knot. Such oriented knots are usually considered up to ambient isotopy in which the isotopy is orientation preserving. This leads to the idea of invertible knots?. It is also possible to take the mirror reflection? of knots and thus to introduce the concept of knot chirality?, a knot invariant; Knots that remain equivalent to their mirror images possess a certain symmetry called achiral knots? or equivalently, amphicheiral?. An alternative definition of this notion is the following: A knot $K$ is amphicheiral? if there exists an orientation-reversing homeomorphism of $R^3$ mapping $K$ to itself.
The theory of knots can be extended to include various similar things:
A major line in the study of knots is to look for knot invariants (see also link invariants).
There are various pages related to knot theory that are linked from the main articles.
The study of knots is very pictorial. There are various knot-related SVGs that can be included in to nLab pages.
chord diagrams | weight systems |
---|---|
linear chord diagrams, round chord diagrams Jacobi diagrams, Sullivan chord diagrams | Lie algebra weight systems, stringy weight system, Rozansky-Witten weight systems |
Expositions:
Hoste, Thistlethwaite and Weeks, The First 1,701,936 Knots, Scientific American, 20, No. 4, 1998. (link to pdf)
Aaron Lauda, Knot theory explained (1:24 min lightning idea), USC Dornsife College of Letters, Arts and Sciences (video)
Abhijit Champanerkar, The geometry of knot complements (pdf, pdf)
General:
R. H. Crowell, R. H. Fox, Introduction to knot theory, Springer, Graduate Texts 57, 1963.
G. Burde, H. Zieschang, Knots, De Gruyter (1989).
Michael Atiyah, The Geometry and Physics of Knots, Cambridge University Press 1990 (doi:10.1017/CBO9780511623868)
Nick Gilbert, Tim Porter, Knots and surfaces, Oxford University Press (1994) [ISBN:9780198514909]
Dale Rolfsen, Knots and links, AMS Chelsea 346 (2003) [ams:chel-346-h, pdf]
Historically, a motivation for Peter Tait to start thinking about classification of knots was the book
which presented the speculation in physics that atoms/elementary particles are fundamentally vortices in a spacetime-filling fluid-like substance.
Relation of knot theory to physics/quantum field theory:
Louis Kauffman, Knots and physics, Series on Knots and Everything, Volume 1, World Scientific, 1991 (doi:10.1142/1116)
Louis Kauffman (ed.) The Interface of Knots and Physics, Proceedings of Symposia in Applied Mathematics
Volume 51 (1996) (pdf, doi:10.1090/psapm/051)
In string theory (NS5-branes/M5-branes):
D. Roseman, Reidemeister-type moves for surfaces in four dimensional space, Banach Center Publication, 42 (1998), 347-380 pdf doi
J. S. Carter, M. Saito, Knotted surfaces and their diagrams, Mathematical Surveys and Monographs 55, Amer. Math. Soc., Providence, RI, 1998
V. A. Rohlin, The embedding of non-orientable three-manifolds into five-dimensional Euclidean space, Dokl. Akad. Nauk SSSR 160 (1965) 549–551 (in Russian; English transl.: Soviet Math. Dokl. 6 (1965), 153–156)
I.G. Korepanov, G.I. Sharygin, D.V. Talalaev, Cohomology of the tetrahedral complex and quasi-invariants of 2-knots, arxiv/1510.03015
J. E. Fischer, Jr. 2-Categories and 2-knots, Duke Math. J. 75 (1994), 493–596.
Last revised on October 26, 2023 at 11:22:46. See the history of this page for a list of all contributions to it.