In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
knot theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
, a knot invariant is a quantity (in a broad sense) defined for each
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
which is the same for
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
knots. The equivalence is often given by
ambient isotopy but can be given by
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
. Some invariants are indeed numbers (algebraic
), but invariants can range from the simple, such as a yes/no answer, to those as complex as a
homology theory (for example, "a ''knot invariant'' is a rule that assigns to any knot a quantity such that if and are equivalent then ."
). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification,
[Purcell, Jessica (2020). ''Hyperbolic Knot Theory'', p.7. American Mathematical Society. "A ''knot invariant'' is a function from the set of knots to some other set whose value depends only on the equivalence class of the knot."][Messer, Robert and Straffin, Philip D. (2018). ''Topology Now!'', p.50. American Mathematical Society. "A ''knot invariant'' is a mathematical property or quantity associated with a knot that does not change as we perform triangular moves on the knot.] both in "enumeration" and "duplication removal".
[Ricca, Renzo L.; ed. (2012). ''An Introduction to the Geometry and Topology of Fluid Flows'', p.67. Springer Netherlands. .]
From the modern perspective, it is natural to define a knot invariant from a
knot diagram. Of course, it must be unchanged (that is to say, invariant) under the
Reidemeister move
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany.
Life
He was a brother of Marie Neurath.
Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Göttinge ...
s ("triangular moves"
[). ]Tricolorability
In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two ...
(and ''n''-colorability) is a particularly simple and common example. Other examples are knot polynomial
In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.
History
The first knot polynomial, the Alexander polynomial, was introdu ...
s, such as the Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynom ...
, which are currently among the most useful invariants for distinguishing knots from one another, though currently it is not known whether there exists a knot polynomial which distinguishes all knots from each other. However, there are invariants which distinguish the unknot
In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...
from all other knots, such as Khovanov homology and knot Floer homology.
Other invariants can be defined by considering some integer-valued function of knot diagrams and taking its minimum value over all possible diagrams of a given knot. This category includes the crossing number, which is the minimum number of crossings for any diagram of the knot, and the bridge number
In the mathematical field of knot theory, the bridge number is an invariant of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot.
Definition
Given a knot or link, draw a diagram of the ...
, which is the minimum number of bridges for any diagram of the knot.
Historically, many of the early knot invariants are not defined by first selecting a diagram but defined intrinsically, which can make computing some of these invariants a challenge. For example, knot genus is particularly tricky to compute, but can be effective (for instance, in distinguishing mutants
In biology, and especially in genetics, a mutant is an organism or a new genetic character arising or resulting from an instance of mutation, which is generally an alteration of the DNA sequence of the genome or chromosome of an organism. It ...
).
The complement of a knot itself (as a topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
) is known to be a "complete invariant" of the knot by the Gordon–Luecke theorem in the sense that it distinguishes the given knot from all other knots up to ambient isotopy and mirror image. Some invariants associated with the knot complement include the knot group
In mathematics, a knot (mathematics), knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot ''K'' is defined as the fundamental group of the knot complement of ''K'' in R3,
:\pi_1(\mathbb^3 \setminus K).
Oth ...
which is just the fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of the complement. The knot quandle is also a complete invariant in this sense but it is difficult to determine if two quandles are isomorphic. The peripheral subgroup can also work as a complete invariant.
By Mostow–Prasad rigidity, the hyperbolic structure on the complement of a hyperbolic link
In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry. A hyperbolic knot is a hyperbolic link with one component.
As a ...
is unique, which means the hyperbolic volume
In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological inv ...
is an invariant for these knots and links. Volume, and other hyperbolic invariants, have proven very effective, utilized in some of the extensive efforts at knot tabulation
Ever since Sir William Thomson's vortex theory, mathematicians have tried to classify and tabulate all possible knots. As of May 2008, all prime knots up to 16 crossings have been tabulated. The major challenge of the process is that many appar ...
.
In recent years, there has been much interest in homological
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
* Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chrom ...
invariants of knots which categorify well-known invariants. Heegaard Floer homology
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 in ...
is a homology theory whose Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
is the Alexander polynomial
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 ve ...
of the knot. It has been proven effective in deducing new results about the classical invariants. Along a different line of study, there is a combinatorially defined cohomology theory of knots called Khovanov homology whose Euler characteristic is the Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynom ...
. This has recently been shown to be useful in obtaining bounds on slice genus
In mathematics, the slice genus of a smooth knot ''K'' in ''S''3 (sometimes called its Murasugi genus or 4-ball genus) is the least integer g such that ''K'' is the boundary of a connected, orientable 2-manifold ''S'' of genus ''g'' properly embedd ...
whose earlier proofs required gauge theory. Mikhail Khovanov
Mikhail Khovanov (russian: Михаил Гелиевич Хованов; born 1972) is a Russian-American professor of mathematics at Columbia University who works on representation theory, knot theory, and algebraic topology. He is known for int ...
and Lev Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants. Catharina Stroppel gave a representation theoretic interpretation of Khovanov homology by categorifying quantum group invariants.
There is also growing interest from both knot theorists and scientists in understanding "physical" or geometric properties of knots and relating it to topological invariants and knot type. An old result in this direction is the Fáry–Milnor theorem
In the mathematical theory of knots, the Fáry–Milnor theorem, named after István Fáry and John Milnor, states that three-dimensional smooth curves with small total curvature must be unknotted. The theorem was proved independently by Fáry i ...
states that if the total curvature
In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length:
:\int_a^b k(s)\,ds.
The total curvature of a closed curve i ...
of a knot in satisfies
:
where is the curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
at , then is an unknot. Therefore, for knotted curves,
:
An example of a "physical" invariant is ropelength
In physical knot theory, each realization of a link or knot has an associated ropelength. Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot. Knots and links that minimize ropelength are ...
, which is the length of unit-diameter rope needed to realize a particular knot type.
Other invariants
*
* (or Vassiliev or Vassiliev–Goussarov invariant)
*
Sources
Further reading
*
*
*
External links
*
*
{{Knot theory, state=collapsed