Jones Polynomial

Definition 0.1.

An N-component link is the image of a C embeddingMathworldPlanetmathPlanetmathPlanetmath f:S1××S1Ntimes3. If N=1, we call the link a knot.

Using links and knots as embeddings is not very convenient, as visualising the bends and curves in 3 is very hard. Therefore, we use the notion of a knot diagramMathworldPlanetmath.

Definition 0.2.

Given a knot K, a knot projection π:32 is a linear surjective map that satisfies:

  1. 1.


  2. 2.


  3. 3.

    There is a finite number of points in K for which card{π-1(x)}=2.

A knot diagram is the image of a projection of a knot.

Note that there is no universalPlanetmathPlanetmath definition of a knot, but this one is the one we use here. Specifically, this definition is used to rule out singular knots, whose projection has an infiniteMathworldPlanetmath number of crossings.

In this way, we have the set D of all possible knot diagrams. Here, again one would like to study properties of truly distinct knot diagrams, so naturally we study D/, where now represents an equivalence relationMathworldPlanetmath on D, consisting of 2-dimensional ambient isotopies and the Reidemeister moves.

Necessary is:

Definition 0.3.

The writhe (or Tait number) w(L) is the sum of all crossing numbers of a given projection, w(L)=isign(ci).

Definition 0.4.

Given a link projection D, let y be a crossing, and y^,y^ be that crossing opened vertically and horizontally. Then there exists a polynomial PL(ξ,η,ψ) that satisfies:

  1. 1.


  2. 2.


  3. 3.


where ξ,η,ψ. This PL is called the bracket, also denoted by [..].

An important remark here is that this definition of the polynomial is through a recursion on the number of crossings of a link diagram. This means that we can construct the polynomial by performing as many recursions as there are crossings and so this polynomial is well defined, because there are only a finite number of crossings in our knots.

Corollary 0.1.

From this also follows that PL1L2=ψPL1PL2

Example 0.1.

For the Hopf link, we see that the bracket polynomial [Hopf]=ηξ+ξ2ψ+ηξ+η2ψ.

Conjecture 0.1.

P^L=αPL,α, is a knot invariant for a suitable chosen α.

This is not true (reason will be added, just check the Reidemeister moves), but this is:

Proposition 0.2.

P^L(ξ):=(-ξ-3)w(L)PL(ξ,ξ-1,-ξ2-ξ-2) is a knot invariant.


Since w(L) and PL are invariant under Ω2, Ω3, P^L(ξ) naturally is invariant under those moves too. Under Ω1, a positive crossing attains an extra term -ξ3 which is compensated by the extra -ξ-3 of the prefactor, since Ω1 changes the writhe by +1. For a negative crossing, the result is analogous. Therefore, P^L(ξ) is invariant under Ω1 also. Finally, because all projections of a link can be obtained through a finite number of Reidemeister moves, P^L(ξ) is a knot invariant. ∎

Now the Jones polynomialMathworldPlanetmath, first conceived by Vaughan Jones in 1984, is exactly JL(t):=P^L(t-14).

Example 0.2.

For the Hopf link, w(L)=2 and PHopf=ξ2(-ξ2-ξ-2)+1+1+ξ-2(-ξ2-ξ-2)=-ξ4-ξ-4. Therefore, JHopf(t)=(-(t-14)-3)2((-(t-14)4)-(t-14)-4)=-(t12+t52)

Having defined the Jones polynomial, this allows us once again to classify knots. Notice that the Jones polynomial allows the powers of t to be negative. Each knot has a Jones polynomial associated to it and it has been shown that the Jones polynomial distinguishes between more knots than simple knot invariants, such as tricolorability. In this sense, the Jones polynomial is a better invariant to determine nonequivalency between two knots. Although it is known that there are pairs of knots that have the same Jones polynomial, the question whether or not there exists a nontrivial knot which has Jones polynomial 1 (same as the unknot), but is not equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the unknot, has still not been answered. Above all, it should be clear that the Jones polynomial still does not suffice to distinguish all possible knots.

Title Jones Polynomial
Canonical name JonesPolynomial
Date of creation 2013-03-22 18:51:58
Last modified on 2013-03-22 18:51:58
Owner Stephaninos (23208)
Last modified by Stephaninos (23208)
Numerical id 17
Author Stephaninos (23208)
Entry type Topic
Classification msc 57M25
Related topic KnotTheory