Knot polynomial
Encyclopedia
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
.
, was introduced by J. W. Alexander
in 1923, but other knot polynomials were not found until almost 60 years later.
In the 1960s, John Conway
came up with a skein relation
for a version of the Alexander polynomial, usually referred to as the Alexander–Conway polynomial. The significance of this skein relation was not realized until the early 1980s, when Vaughan Jones
discovered the Jones polynomial. This led to the discovery of more knot polynomials, such as the so-called HOMFLY polynomial
.
Soon after Jones' discovery, Louis Kauffman
noticed the Jones polynomial could be computed by means of a state-sum model, which involved the bracket polynomial
, an invariant of framed knots. This opened up avenues of research linking knot theory and statistical mechanics
.
In the late 1980s, two related breakthroughs were made. Edward Witten
demonstrated that the Jones polynomial, and similar Jones-type invariants, had an interpretation in Chern–Simons theory. Viktor Vassiliev and Mikhail Goussarov
started the theory of finite type invariant
s of knots. The coefficients of the previously named polynomials are known to be of finite type (after perhaps a suitable "change of variables").
In recent years, the Alexander polynomial has been shown to be related to Floer homology
. The graded Euler characteristic of the knot Floer homology of Ozsváth and Szabó is the Alexander polynomial.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, a knot polynomial is a knot invariant
Knot invariant
In the mathematical field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the...
in the form of a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
whose coefficients encode some of the properties of a given knot
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...
.
History
The first knot polynomial, the Alexander polynomialAlexander 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...
, was introduced by J. W. Alexander
J. W. Alexander
J. W. Alexander may refer to:* James Waddel Alexander , American Presbyterian minister and author* John White Alexander , American portrait painter and illustrator...
in 1923, but other knot polynomials were not found until almost 60 years later.
In the 1960s, John Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
came up with a skein relation
Skein relation
A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One tool used to answer such questions is a knot polynomial which is an invariant of the knot. If two diagrams have different polynomials, they represent different knots. The reverse may not...
for a version of the Alexander polynomial, usually referred to as the Alexander–Conway polynomial. The significance of this skein relation was not realized until the early 1980s, when Vaughan Jones
Vaughan Jones
Sir Vaughan Frederick Randal Jones, KNZM, FRS, FRSNZ is a New Zealand mathematician, known for his work on von Neumann algebras, knot polynomials and conformal field theory. He was awarded a Fields Medal in 1990, and famously wore a New Zealand rugby jersey when he accepted the prize...
discovered the Jones polynomial. This led to the discovery of more knot polynomials, such as the so-called HOMFLY polynomial
HOMFLY polynomial
In the mathematical field of knot theory, the HOMFLY polynomial, sometimes called the HOMFLY-PT polynomial or the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables m and l....
.
Soon after Jones' discovery, Louis Kauffman
Louis Kauffman
Louis H. Kauffman is an American mathematician, topologist, and professor of Mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago...
noticed the Jones polynomial could be computed by means of a state-sum model, which involved the bracket polynomial
Bracket polynomial
In the mathematical field of knot theory, the bracket polynomial is a polynomial invariant of framed links. Although it is not an invariant of knots or links , a suitably "normalized" version yields the famous knot invariant called the Jones polynomial...
, an invariant of framed knots. This opened up avenues of research linking knot theory and statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
.
In the late 1980s, two related breakthroughs were made. Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
demonstrated that the Jones polynomial, and similar Jones-type invariants, had an interpretation in Chern–Simons theory. Viktor Vassiliev and Mikhail Goussarov
Mikhail Goussarov
Mikhail Goussarov was a Soviet mathematician who worked in low-dimensional topology. He and Victor Vassiliev independently discovered finite type invariants of knots and links. He accidentally drowned at the age of 41 in Tel Aviv, Israel.- See also :* maintained by Dror Bar-Natan....
started the theory of finite type invariant
Finite type invariant
In the mathematical theory of knots, a finite type invariant is a knot invariant that can be extended to an invariant of certain singular knots that vanishes on singular knots with m + 1 singularities and does not vanish on some singular knot with 'm' singularities...
s of knots. The coefficients of the previously named polynomials are known to be of finite type (after perhaps a suitable "change of variables").
In recent years, the Alexander polynomial has been shown to be related to Floer homology
Floer homology
Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
. The graded Euler characteristic of the knot Floer homology of Ozsváth and Szabó is the Alexander polynomial.
Specific knot polynomials
- Alexander polynomialAlexander polynomialIn 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...
- Bracket polynomialBracket polynomialIn the mathematical field of knot theory, the bracket polynomial is a polynomial invariant of framed links. Although it is not an invariant of knots or links , a suitably "normalized" version yields the famous knot invariant called the Jones polynomial...
- HOMFLY polynomialHOMFLY polynomialIn the mathematical field of knot theory, the HOMFLY polynomial, sometimes called the HOMFLY-PT polynomial or the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables m and l....
- Jones polynomial
- Kauffman polynomial
Related topics
- skein relationship for a formal definition of the Alexander polynomial, with a worked-out example.