nLab knot

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.

Definition

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.

Example

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.

Relevant nLab Pages

Knots, Links, and other Variants

The theory of knots can be extended to include various similar things:

• links
• braids
• strings
• tangles
• singular knots

Invariants

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.

• Vassiliev skein relations
• Reidemeister moves

Images

The study of knots is very pictorial. There are various knot-related SVGs that can be included in to nLab pages.

• SVG images

Related concepts

• space of knots

• knot complement

• knot diagram

• isotopy, smooth isotopy

• Kirby calculus

• torus knot

• hyperbolic knot

• surface knot

• slice knot

• boxed knot, knot sum

• MKR dictionary in arithmetic topology

• chord diagram

• Wilson loop

• knots-quivers correspondence

• quantum topology

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

knots	braids
chord diagram, Jacobi diagram	horizontal chord diagram
1T&4T relation	2T&4T relation/ infinitesimal braid relations
weight system	horizontal weight system
Vassiliev knot invariant	Vassiliev braid invariant
weight systems are associated graded of Vassiliev invariants	horizontal weight systems are cohomology of loop space of configuration space

References

General

Historically, a motivation for Peter Tait to start thinking about classification of knots was the book

• Lord Kelvin, On Vortex Atoms

which presented the speculation in physics that atoms/elementary particles are fundamentally vortices in a spacetime-filling fluid-like substance.

Expositions of modern knot theory:

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

• Sam Nelson: The Combinatorial Revolution in Knot Theory, Notices of the AMS 58 11 (2011) [pdf, full issue: pdf]

• Aaron Lauda, Knot theory explained, (1:24 min lightning idea), USC Dornsife College of Letters, Arts and Sciences (2015) [video: YT]

• Abhijit Champanerkar, The geometry of knot complements [pdf, pdf]

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

Monographs:

• Richard H. Crowell, R. H. Fox, Introduction to knot theory, Graduate Texts in Mathematics 57, Springer (1963) [doi:10.1007/978-1-4612-9935-6]

• G. Burde, H. Zieschang, Knots, De Gruyter (1989).

• Michael Atiyah, The Geometry and Physics of Knots, Cambridge University Press (1990) [doi:10.1017/CBO9780511623868]

• Vladimir Turaev: Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics 18, de Gruyter & Co. (1994) [doi:10.1515/9783110435221, pdf]

• Colin C. Adams, The Knot Book – An elementary Introduction to the Mathematical Theory of Knots, W. H. Freedman and Co. (1994) [ISBN:978-0-8218-3678-1]

• Nick Gilbert, Tim Porter, Knots and surfaces, Oxford University Press (1994) [ISBN:9780198514909]

• Tomotada Ohtsuki: Quantum Invariants – A Study of Knots, 3-Manifolds, and Their Sets, World Scientific (2001) [doi:10.1142/4746]

• Dale Rolfsen, Knots and links, AMS Chelsea 346 (2003) [ams:chel-346-h, pdf]

• Alexei B. Sossinsky, Knots, Links and Their Invariants: An Elementary Course in Contemporary Knot Theory, The Student Mathematical Library 101, AMS (2023) [doi:10.1090/stml/101]

In higher dimensions:

• Andrew Ranicki: High-dimensional knot theory – Algebraic Surgery in Codimension 2, Monographs in Mathematics, Springer (1998, 2009) [doi:10.1007/978-3-662-12011-8, pdf]

  (via surgery and algebraic K-theory and L-theory)

Relation to physics

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):

• Edward Witten, Fivebranes and Knots, Quantum Topology, Volume 3, Issue 1, 2012, pp. 1-137 (arXiv:1101.3216, doi:10.4171/QT/26)

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