Editing Knot polynomial

[[Image:Skein (HOMFLY).svg|thumb|Many knot polynomials are computed using [[skein relations]], which allow one to change the different crossings of a knot to get simpler knots.]]

In the [[mathematics|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 (mathematics)|knot]].

==History==
The first knot polynomial, the [[Alexander polynomial]], was introduced by [[James Waddell Alexander II]] in 1923. Other knot polynomials were not found until almost 60 years later.

In the 1960s, [[John Horton Conway|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 [[Partition function (statistical mechanics)|partition function]] (state-sum model), which involved the [[bracket polynomial]], an invariant of [[Framed knot|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]]. [[Victor Anatolyevich Vasilyev|Viktor Vasilyev]] 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 [[Heegaard Floer homology|knot Floer homology]] of [[Peter Ozsváth]] and [[Zoltán Szabó (mathematician)|Zoltan Szabó]] is the Alexander polynomial.

==Examples==
{| class="wikitable"
|-
! [[Alexander–Briggs notation]] !! [[Alexander polynomial]] <math>\Delta(t)</math> !!  [[Alexander polynomial#Alexander.E2.80.93Conway polynomial|Conway polynomial]] <math>\nabla(z) </math> !! [[Jones polynomial]]  <math>V(q)</math> !! [[HOMFLY polynomial]] <math>H(a,z)</math>
|-
|<math>0_1</math> ([[Unknot]])|| <math>1</math> || <math>1</math> || <math>1</math> || <math>1</math>
|-
|<math>3_1</math> ([[Trefoil knot|Trefoil Knot]])|| <math>t - 1 + t^{-1}</math> || <math>z^2 + 1</math> || <math>q^{-1} + q^{-3} - q^{-4}</math> || <math>-a^{4}+a^{2}z^{2}+2a^{2}</math>
|-
| <math>4_1</math> ([[Figure-eight knot (mathematics)|Figure-eight Knot]]) || <math>-t + 3 - t^{-1}</math> || <math>-z^2+1</math> || <math>q^2 - q + 1 - q^{-1} + q^{-2}</math> || <math>a^{2}+a^{-2}-z^{2}-1</math>
|-
| <math>5_1</math> ([[Cinquefoil knot|Cinquefoil Knot]]) || <math>t^2 - t + 1 - t^{-1} + t^{-2}</math> || <math>z^4 + 3z^2 + 1</math> || <math>q^{-2} + q^{-4} - q^{-5} + q^{-6} - q^{-7}</math> || <math>-a^{6}z^{2}-2a^{6}+a^{4}z^{4}+4a^{4}z^{2}+3a^{4}</math>
|-
| <math>3_1 \# 3_1</math> ([[Granny knot (mathematics)|Granny Knot]]) || <math>\left(t - 1 + t^{-1}\right)^2</math> || <math>\left(z^2 + 1\right)^2</math> || <math>\left(q^{-1} + q^{-3} - q^{-4}\right)^2</math> || <math>\left(-a^{4}+a^{2}z^{2}+2a^{2}\right)^2</math>
|-
| <math>3_1 \# 3^*_1</math> ([[Square knot (mathematics)|Square Knot]]) || <math>\left(t - 1 + t^{-1}\right)^2</math> || <math>\left(z^2 + 1\right)^2</math> || <math>\left(q^{-1} + q^{-3} - q^{-4}\right)\left(q + q^{3} - q^{4}\right)</math> || <math>\left(-a^{4}+a^{2}z^{2}+2a^{2}\right) \times</math> <br/> <math>\left(-a^{-4}+a^{-2}z^{-2}+2a^{-2}\right)</math>
|}

[[Alexander–Briggs notation]] organizes knots by their crossing number.

[[Alexander polynomial]]s and [[Alexander polynomial#Alexander.E2.80.93Conway polynomial|Conway polynomial]]s can ''not'' recognize the difference of left-trefoil knot and right-trefoil knot.
<gallery widths="60px" heights="60px" align="center">
Image:Trefoil knot left.svg|The left-trefoil knot.
Image:TrefoilKnot_01.svg|The right-trefoil knot.
</gallery>
So we have the same situation as the granny knot and square knot since the [[Knot theory#Adding knots|addition]] of knots in <math>\mathbb{R}^3</math> is the product of knots in [[Knot theory#Knot polynomials|knot polynomials]].

==See also==

===Specific knot polynomials===
*[[Alexander polynomial]]
*[[Bracket polynomial]]
*[[HOMFLY polynomial]]
*[[Jones polynomial]]
*[[Kauffman polynomial]]

===Related topics===
*[[Graph polynomial]], a similar class of polynomial invariants in graph theory
*[[Tutte polynomial]], a special type of graph polynomial related to the Jones polynomial
*[[Skein relation]] for a formal definition of the Alexander polynomial, with a worked-out example.

==Further reading==
*{{cite book |first=Colin |last=Adams |title=The Knot Book |publisher=American Mathematical Society |isbn=0-8050-7380-9 }}
*{{cite book |first=W. B. R. |last=Lickorish |authorlink=W. B. R. Lickorish |title=An Introduction to Knot Theory |series=[[Graduate Texts in Mathematics]] |volume=175 |publisher=Springer-Verlag |location=New York |year=1997 |isbn=0-387-98254-X }}

{{Knot theory|state=collapsed}}

[[Category:Knot invariants]]
[[Category:Polynomials]]