nLab knot



Knot theory


topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory


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 1S^1 into 3\mathbb{R}^3, or equivalently into the 33-sphere S 3S^3 (one can also consider knots in other 3-manifolds).

Sometimes, higher dimensional knots are also considered. nn-dimensional knot (or simply an nn-knot) is a smooth embedding of nn-dimensional closed manifold (usually an nn-sphere) into the (n+2)(n+2)-dimensional sphere 𝕊 n\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 3S^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 KK is amphicheiral? if there exists an orientation-reversing homeomorphism of R 3R^3 mapping KK to itself.

Relevant nLab Pages

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).

Ancillary pages

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 diagramsweight systems
linear chord diagrams,
round chord diagrams
Jacobi diagrams,
Sullivan chord diagrams
Lie algebra weight systems,
stringy weight system,
Rozansky-Witten weight systems

chord diagram,
Jacobi diagram
horizontal chord diagram
1T&4T relation2T&4T relation/
infinitesimal braid relations
weight systemhorizontal weight system
Vassiliev knot invariantVassiliev braid invariant
weight systems are associated graded of Vassiliev invariantshorizontal weight systems are cohomology of loop space of configuration space




  • J. Hoste, M. Thistlethwaite, J. Weeks: The First 1,701,936 Knots, The Mathematical Intelligencer 20 (1998) 33–48 [doi:10.1007/BF03025227]

  • 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]

  • Roland van der Veen: A smooth introduction to knots, lecture notes (2016) [pdf, 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 to physics

Relation of knot theory to physics/quantum field theory:

In string theory (NS5-branes/M5-branes):

Higher dimensional knots

  • 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.

category: knot theory

Last revised on July 17, 2024 at 16:19:26. See the history of this page for a list of all contributions to it.