Knot theory

In 1927, working with this diagrammatic form of knots, J. W. Alexander and Garland Baird Briggs, and independently Kurt Reidemeister, demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown below. These operations, now called the Reidemeister moves, are:

Reidemeister moves

Type I	Type II
Type III

The proof that diagrams of equivalent knots are connected by Reidemeister moves relies on an analysis of what happens under the planar projection of the movement taking one knot to another. The movement can be arranged so that almost all of the time the projection will be a knot diagram, except at finitely many times when an "event" or "catastrophe" occurs, such as when more than two strands cross at a point or multiple strands become tangent at a point. A close inspection will show that complicated events can be eliminated, leaving only the simplest events: (1) a "kink" forming or being straightened out; (2) two strands becoming tangent at a point and passing through; and (3) three strands crossing at a point. These are precisely the Reidemeister moves (Sossinsky 2002, ch. 3) (Lickorish 1997, ch. 1).

A knot polynomial is a knot invariant that is a polynomial. Well-known examples include the Jones and Alexander polynomials. A variant of the Alexander polynomial, the Alexander–Conway polynomial, is a polynomial in the variable z with integer coefficients (Lickorish 1997).

The Alexander–Conway polynomial is actually defined in terms of links, which consist of one or more knots entangled with each other. The concepts explained above for knots, e.g. diagrams and Reidemeister moves, also hold for links.

Consider an oriented link diagram, i.e. one in which every component of the link has a preferred direction indicated by an arrow. For a given crossing of the diagram, let ${\displaystyle L_{+},L_{-},L_{0}}$ be the oriented link diagrams resulting from changing the diagram as indicated in the figure:

The original diagram might be either ${\displaystyle L_{+}}$ or ${\displaystyle L_{-}}$, depending on the chosen crossing's configuration. Then the Alexander–Conway polynomial, ${\displaystyle C(z)}$, is recursively defined according to the rules:

• ${\displaystyle C(O)=1}$ (where ${\displaystyle O}$ is any diagram of the unknot)
• ${\displaystyle C(L_{+})=C(L_{-})+zC(L_{0}).}$

The second rule is what is often referred to as a skein relation. To check that these rules give an invariant of an oriented link, one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way.

The following is an example of a typical computation using a skein relation. It computes the Alexander–Conway polynomial of the trefoil knot. The yellow patches indicate where the relation is applied.

C() = C() + z C()

gives the unknot and the Hopf link. Applying the relation to the Hopf link where indicated,

C() = C() + z C()

gives a link deformable to one with 0 crossings (it is actually the unlink of two components) and an unknot. The unlink takes a bit of sneakiness:

C() = C() + z C()

which implies that C(unlink of two components) = 0, since the first two polynomials are of the unknot and thus equal.

Putting all this together will show:

${\displaystyle C(\mathrm {trefoil} )=1+z(0+z)=1+z^{2}}$

Since the Alexander–Conway polynomial is a knot invariant, this shows that the trefoil is not equivalent to the unknot. So the trefoil really is "knotted".

• 
  The left-handed trefoil knot.
• 
  The right-handed trefoil knot.

Actually, there are two trefoil knots, called the right and left-handed trefoils, which are mirror images of each other (take a diagram of the trefoil given above and change each crossing to the other way to get the mirror image). These are not equivalent to each other, meaning that they are not amphichiral. This was shown by Max Dehn, before the invention of knot polynomials, using group theoretical methods (Dehn 1914). But the Alexander–Conway polynomial of each kind of trefoil will be the same, as can be seen by going through the computation above with the mirror image. The Jones polynomial can in fact distinguish between the left- and right-handed trefoil knots (Lickorish 1997).

William Thurston proved many knots are hyperbolic knots, meaning that the knot complement (i.e., the set of points of 3-space not on the knot) admits a geometric structure, in particular that of hyperbolic geometry. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant (Adams 2004).

The Borromean rings are a link with the property that removing one ring unlinks the others.

SnapPea's cusp view: the Borromean rings complement from the perspective of an inhabitant living near the red component.

Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the geodesics of the geometry. An example is provided by the picture of the complement of the Borromean rings. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of horoball neighborhoods of the link. By thickening the link in a standard way, the horoball neighborhoods of the link components are obtained. Even though the boundary of a neighborhood is a torus, when viewed from inside the link complement, it looks like a sphere. Each link component shows up as infinitely many spheres (of one color) as there are infinitely many light rays from the observer to the link component. The fundamental parallelogram (which is indicated in the picture), tiles both vertically and horizontally and shows how to extend the pattern of spheres infinitely.

This pattern, the horoball pattern, is itself a useful invariant. Other hyperbolic invariants include the shape of the fundamental parallelogram, length of shortest geodesic, and volume. Modern knot and link tabulation efforts have utilized these invariants effectively. Fast computers and clever methods of obtaining these invariants make calculating these invariants, in practice, a simple task (Adams, Hildebrand & Weeks 1991).

Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a two-dimensional sphere (${\displaystyle \mathbb {S} ^{2}}$) embedded in 4-dimensional Euclidean space (${\displaystyle \mathbb {R} ^{4}}$). Such an embedding is knotted if there is no homeomorphism of ${\displaystyle \mathbb {R} ^{4}}$ onto itself taking the embedded 2-sphere to the standard "round" embedding of the 2-sphere. Suspended knots and spun knots are two typical families of such 2-sphere knots.

The mathematical technique called "general position" implies that for a given n-sphere in m-dimensional Euclidean space, if m is large enough (depending on n), the sphere should be unknotted. In general, piecewise-linear n-spheres form knots only in (n + 2)-dimensional space (Zeeman 1963), although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted ${\displaystyle (4k-1)}$-spheres in 6k-dimensional space; e.g., there is a smoothly knotted 3-sphere in ${\displaystyle \mathbb {R} ^{6}}$ (Haefliger 1962) (Levine 1965). Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth k-sphere embedded in ${\displaystyle \mathbb {R} ^{n}}$ with ${\displaystyle 2n-3k-3>0}$ is unknotted. The notion of a knot has further generalisations in mathematics, see: Knot (mathematics), isotopy classification of embeddings.

Every knot in the n-sphere ${\displaystyle \mathbb {S} ^{n}}$ is the link of a real-algebraic set with isolated singularity in ${\displaystyle \mathbb {R} ^{n+1}}$ (Akbulut & King 1981).

An n-knot is a single ${\displaystyle \mathbb {S} ^{n}}$ embedded in ${\displaystyle \mathbb {R} ^{m}}$. An n-link consists of k-copies of ${\displaystyle \mathbb {S} ^{n}}$ embedded in ${\displaystyle \mathbb {R} ^{m}}$, where k is a natural number. Both the ${\displaystyle m=n+2}$ and the ${\displaystyle m>n+2}$ cases are well studied, and so is the ${\displaystyle n>1}$ case.[5][6]

This is the most traditional notation, due to the 1927 paper of James W. Alexander and Garland B. Briggs and later extended by Dale Rolfsen in his knot table (see image above and List of prime knots). The notation simply organizes knots by their crossing number. One writes the crossing number with a subscript to denote its order amongst all knots with that crossing number. This order is arbitrary and so has no special significance (though in each number of crossings the twist knot comes after the torus knot). Links are written by the crossing number with a superscript to denote the number of components and a subscript to denote its order within the links with the same number of components and crossings. Thus the trefoil knot is notated 3₁ and the Hopf link is 2²
₁. Alexander–Briggs names in the range 10₁₆₂ to 10₁₆₆ are ambiguous, due to the discovery of the Perko pair in Charles Newton Little's original and subsequent knot tables, and differences in approach to correcting this error in knot tables and other publications created after this point.[7]

A knot diagram with crossings labelled for a Dowker sequence

The Dowker–Thistlethwaite notation, also called the Dowker notation or code, for a knot is a finite sequence of even integers. The numbers are generated by following the knot and marking the crossings with consecutive integers. Since each crossing is visited twice, this creates a pairing of even integers with odd integers. An appropriate sign is given to indicate over and undercrossing. For example, in this figure the knot diagram has crossings labelled with the pairs (1,6) (3,−12) (5,2) (7,8) (9,−4) and (11,−10). The Dowker–Thistlethwaite notation for this labelling is the sequence: 6, −12, 2, 8, −4, −10. A knot diagram has more than one possible Dowker notation, and there is a well-understood ambiguity when reconstructing a knot from a Dowker–Thistlethwaite notation.

The Conway notation for knots and links, named after John Horton Conway, is based on the theory of tangles (Conway 1970). The advantage of this notation is that it reflects some properties of the knot or link.

The notation describes how to construct a particular link diagram of the link. Start with a basic polyhedron, a 4-valent connected planar graph with no digon regions. Such a polyhedron is denoted first by the number of vertices then a number of asterisks which determine the polyhedron's position on a list of basic polyhedra. For example, 10** denotes the second 10-vertex polyhedron on Conway's list.

Each vertex then has an algebraic tangle substituted into it (each vertex is oriented so there is no arbitrary choice in substitution). Each such tangle has a notation consisting of numbers and + or − signs.

An example is 1*2 −3 2. The 1* denotes the only 1-vertex basic polyhedron. The 2 −3 2 is a sequence describing the continued fraction associated to a rational tangle. One inserts this tangle at the vertex of the basic polyhedron 1*.

A more complicated example is 8*3.1.2 0.1.1.1.1.1 Here again 8* refers to a basic polyhedron with 8 vertices. The periods separate the notation for each tangle.

Any link admits such a description, and it is clear this is a very compact notation even for very large crossing number. There are some further shorthands usually used. The last example is usually written 8*3:2 0, where the ones are omitted and kept the number of dots excepting the dots at the end. For an algebraic knot such as in the first example, 1* is often omitted.

Conway's pioneering paper on the subject lists up to 10-vertex basic polyhedra of which he uses to tabulate links, which have become standard for those links. For a further listing of higher vertex polyhedra, there are nonstandard choices available.

Gauss code, similar to the Dowker–Thistlethwaite notation, represents a knot with a sequence of integers. However, rather than every crossing being represented by two different numbers, crossings are labeled with only one number. When the crossing is an overcrossing, a positive number is listed. At an undercrossing, a negative number. For example, the trefoil knot in Gauss code can be given as: 1,−2,3,−1,2,−3

Gauss code is limited in its ability to identify knots. This problem is partially addressed with by the extended Gauss code.

• Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
• Adams, Colin; Crawford, Thomas; DeMeo, Benjamin; Landry, Michael; Lin, Alex Tong; Montee, MurphyKate; Park, Seojung; Venkatesh, Saraswathi; Yhee, Farrah (2015), "Knot projections with a single multi-crossing", Journal of Knot Theory and Its Ramifications, 24 (3): 1550011, 30, arXiv:1208.5742, doi:10.1142/S021821651550011X, MR 3342136, S2CID 119320887
• Adams, Colin; Hildebrand, Martin; Weeks, Jeffrey (1991), "Hyperbolic invariants of knots and links", Transactions of the American Mathematical Society, 326 (1): 1–56, doi:10.1090/s0002-9947-1991-0994161-2, JSTOR 2001854
• Akbulut, Selman; King, Henry C. (1981), "All knots are algebraic", Comm. Math. Helv., 56 (3): 339–351, doi:10.1007/BF02566217, S2CID 120218312
• Bar-Natan, Dror (1995), "On the Vassiliev knot invariants", Topology, 34 (2): 423–472, doi:10.1016/0040-9383(95)93237-2
• Burton, Benjamin A. (2020). "The Next 350 Million Knots". 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz Int. Proc. Inform. Vol. 164. Schloss Dagstuhl–Leibniz-Zentrum für Informatik. pp. 25:1–25:17. doi:10.4230/LIPIcs.SoCG.2020.25.
• Collins, Graham (April 2006), "Computing with Quantum Knots", Scientific American, 294 (4): 56–63, Bibcode:2006SciAm.294d..56C, doi:10.1038/scientificamerican0406-56, PMID 16596880
• Dehn, Max (1914), "Die beiden Kleeblattschlingen", Mathematische Annalen, 75 (3): 402–413, doi:10.1007/BF01563732, S2CID 120452571
• Conway, John H. (1970), "An enumeration of knots and links, and some of their algebraic properties", Computational Problems in Abstract Algebra, Pergamon, pp. 329–358, doi:10.1016/B978-0-08-012975-4.50034-5, ISBN 978-0-08-012975-4
• Doll, Helmut; Hoste, Jim (1991), "A tabulation of oriented links. With microfiche supplement", Math. Comp., 57 (196): 747–761, Bibcode:1991MaCom..57..747D, doi:10.1090/S0025-5718-1991-1094946-4
• Flapan, Erica (2000), When topology meets chemistry: A topological look at molecular chirality, Outlook, Cambridge University Press, ISBN 978-0-521-66254-3
• Haefliger, André (1962), "Knotted (4k − 1)-spheres in 6k-space", Annals of Mathematics, Second Series, 75 (3): 452–466, doi:10.2307/1970208, JSTOR 1970208
• Hass, Joel (1998), "Algorithms for recognizing knots and 3-manifolds", Chaos, Solitons and Fractals, 9 (4–5): 569–581, arXiv:math/9712269, Bibcode:1998CSF.....9..569H, doi:10.1016/S0960-0779(97)00109-4, S2CID 7381505
• Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeffrey (1998), "The First 1,701,935 Knots", Math. Intelligencer, 20 (4): 33–48, doi:10.1007/BF03025227, S2CID 18027155
• Hoste, Jim (2005), "The enumeration and classification of knots and links", Handbook of Knot Theory (PDF), Amsterdam: Elsevier
• Levine, Jerome (1965), "A classification of differentiable knots", Annals of Mathematics, Second Series, 1982 (1): 15–50, doi:10.2307/1970561, JSTOR 1970561
• Kontsevich, Maxim (1993), "Vassiliev's knot invariants", I. M. Gelfand Seminar, Adv. Soviet Math., 2, Providence, RI: American Mathematical Society, 16: 137–150, doi:10.1090/advsov/016.2/04, ISBN 9780821841174
• Lickorish, W. B. Raymond (1997), An Introduction to Knot Theory, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, doi:10.1007/978-1-4612-0691-0, ISBN 978-0-387-98254-0, S2CID 122824389
• Perko, Kenneth (1974), "On the classification of knots", Proceedings of the American Mathematical Society, 45 (2): 262–6, doi:10.2307/2040074, JSTOR 2040074
• Rolfsen, Dale (1976), Knots and Links, Mathematics Lecture Series, vol. 7, Berkeley, California: Publish or Perish, ISBN 978-0-914098-16-4, MR 0515288
• Schubert, Horst (1949), "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten", Heidelberger Akad. Wiss. Math.-Nat. Kl. (3): 57–104
• Silver, Dan (2006), "Knot theory's odd origins" (PDF), American Scientist, vol. 94, no. 2, pp. 158–165, doi:10.1511/2006.2.158
• Simon, Jonathan (1986), "Topological chirality of certain molecules", Topology, 25 (2): 229–235, doi:10.1016/0040-9383(86)90041-8
• Sossinsky, Alexei (2002), Knots, mathematics with a twist, Harvard University Press, ISBN 978-0-674-00944-8
• Turaev, Vladimir G. (1994), "Quantum invariants of knots and 3-manifolds", De Gruyter Studies in Mathematics, Berlin: Walter de Gruyter & Co., 18, arXiv:hep-th/9409028, Bibcode:1994hep.th....9028T, ISBN 978-3-11-013704-0
• Weisstein, Eric W. (2013). "Reduced Knot Diagram". MathWorld. Wolfram. Retrieved 8 May 2013.
• Weisstein, Eric W. (2013a). "Reducible Crossing". MathWorld. Wolfram. Retrieved 8 May 2013.
• Witten, Edward (1989), "Quantum field theory and the Jones polynomial", Comm. Math. Phys., 121 (3): 351–399, Bibcode:1989CMaPh.121..351W, doi:10.1007/BF01217730, S2CID 14951363
• Zeeman, Erik C. (1963), "Unknotting combinatorial balls", Annals of Mathematics, Second Series, 78 (3): 501–526, doi:10.2307/1970538, JSTOR 1970538

There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is (Rolfsen 1976). Other good texts from the references are (Adams 2004) and (Lickorish 1997). Adams is informal and accessible for the most part to high schoolers. Lickorish is a rigorous introduction for graduate students, covering a nice mix of classical and modern topics. (Cromwell 2004) is suitable for undergraduates who know point-set topology; knowledge of algebraic topology is not required.

• Burde, Gerhard; Zieschang, Heiner (1985), Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter, ISBN 978-3-11-008675-1
• Crowell, Richard H.; Fox, Ralph (1977). Introduction to Knot Theory. ISBN 978-0-387-90272-2.
• Kauffman, Louis H. (1987), On Knots, ISBN 978-0-691-08435-0
• Kauffman, Louis H. (2013), Knots and Physics (4th ed.), World Scientific, ISBN 978-981-4383-00-4
• Cromwell, Peter R. (2004), Knots and Links, Cambridge University Press, ISBN 978-0-521-54831-1

• Menasco, William W.; Thistlethwaite, Morwen, eds. (2005), Handbook of Knot Theory, Elsevier, ISBN 978-0-444-51452-3
  • Menasco and Thistlethwaite's handbook surveys a mix of topics relevant to current research trends in a manner accessible to advanced undergraduates but of interest to professional researchers.
• Livio, Mario (2009), "Ch. 8: Unreasonable Effectiveness?", Is God a Mathematician?, Simon & Schuster, pp. 203–218, ISBN 978-0-7432-9405-8

• Thomson, Sir William (1867), "On Vortex Atoms", Proceedings of the Royal Society of Edinburgh, VI: 94–105
• Silliman, Robert H. (December 1963), "William Thomson: Smoke Rings and Nineteenth-Century Atomism", Isis, 54 (4): 461–474, doi:10.1086/349764, JSTOR 228151, S2CID 144988108
• Movie of a modern recreation of Tait's smoke ring experiment
• History of knot theory (on the home page of Andrew Ranicki)

• KnotInfo: Table of Knot Invariants and Knot Theory Resources
• The Knot Atlas — detailed info on individual knots in knot tables
• KnotPlot — software to investigate geometric properties of knots
• Knotscape — software to create images of knots
• Knoutilus — online database and image generator of knots
• KnotData.html — Wolfram Mathematica function for investigating knots
• Regina — software for low-dimensional topology with native support for knots and links. Tables of prime knots with up to 19 crossings