8d. Jones polynomial

[math] \newcommand{\mathds}{\mathbb}[/math]

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Now back to the knots and links, we have all the needed ingredients. Indeed, we can now put everything together, and we obtain, following Jones:

Theorem

We can define the Jones polynomial of an oriented knot or link as being the image of the corresponding braid producing it via the map

[[math]] tr:B_k\to TL_N(k)\to\mathbb C [[/math]]

with the following change of variables:

[[math]] N=q^{1/2}+q^{-1/2} [[/math]]

We obtain a Laurent polynomial in [math]q^{1/2}[/math], which is an invariant, up to planar isotopy.

Show Proof

There is a long story here, the idea being as follows:

(1) To start with, the result follows indeed by combining the above ingredients, the idea being that the various algebraic properties of [math]tr:TL_N(k)\to\mathbb C[/math] are exactly what is needed for the above composition, up to a normalization, to be invariant under the Reidemeister moves of type I, II, III, and so to produce indeed a knot invariant.

(2) More specifically, the result follows from Theorem 8.12, combined with what we have in Theorem 8.21, which is now fully proved, with the positivity part coming from Theorem 8.30, and with the change of variables [math]N=q^{1/2}+q^{-1/2}[/math] in the statement coming from the equation [math]t^2-Nt+1=0[/math] that we found in the proof of Theorem 8.21.

(3) As an illustration for how this works, consider first the unknot:

[[math]] \xymatrix@R=50pt@C=50pt{ \ar@{-}[r]&\ar@{-}[d]\\ \ar[u]&\ar@{-}[l]} [[/math]]

For this knot, or rather unknot, the corresponding Jones polynomial is:

[[math]] V=1 [[/math]]

(4) Next, let us look at the link formed by two unlinked unknots:

[[math]] \xymatrix@R=50pt@C=50pt{ \ar@{-}[r]&\ar@{-}[d]&\ar@{-}[r]&\ar@{-}[d]\\ \ar[u]&\ar@{-}[l]&\ar[u]&\ar@{-}[l] } [[/math]]

For this link, or rather unlink, the corresponding Jones polynomial is:

[[math]] V=-q^{-1/2}-q^{1/2} [[/math]]

(5) Next, let us look at the link formed by two linked unknots, namely:

[[math]] \xymatrix@R=14pt@C=40pt{ \ar@{-}[rr]&&\ar@{-}[dd]\\ &\ar@{-}[r]&|\ar@{-}[r]&\\ \ar@{-}[r]\ar[uu]&|\ar@{-}[r]&&\\ &\ar@{-}[rr]\ar[uu]&&\ar@{-}[uu] } [[/math]]

For this link, the corresponding Jones polynomial is given by:

[[math]] V=q^{1/2}+q^{5/2} [[/math]]

(6) Finally, let us look at the trefoil knot, which is as follows:

[[math]] \xymatrix@R=18pt@C=40pt{ \ar@{-}[rr]&&\ar@{-}[dd]\\ &\ar@{-}[r]&|\ar@{-}[r]&\\ \ar@{-}[r]\ar[uu]&|\ar@{-}[rr]&-&\ar@{-}[u]\\ &\ar@{-}[r]\ar@{-}[uu]&\ar@{-}[u] } [[/math]]

For this knot, the corresponding Jones polynomial is as follows:

[[math]] V=q+q^3-q^4 [[/math]]

Observe that, as previously for the unknot, this is a Laurent polynomial in [math]q[/math]. This is part of a more general phenomenon, the point being that for knots, or more generally for links having an odd number of components, we get a Laurent polynomial in [math]q[/math].

(7) In practice now, far more things can be said, about this. For instance the change of variables [math]N=q^{1/2}+q^{-1/2}[/math] in the statement, that we already used in chapter 3, in the random walk context for the small norm graphs, is something well-known in planar algebras, and with all this being related to operator algebras and subfactor theory. More on this in chapter 16 below, when discussing subfactors and planar algebras.

(8) From a purely topological perspective, however, nothing beats the skein relation interpretation of the Jones polynomial [math]V_L(q)[/math], which is as follows, with [math]L_+,L_-,L_0[/math] being knots, or rather links, differing at exactly 1 crossing, in the 3 possible ways:

[[math]] q^{-1}V_{L_+}-qV_{L_-}=(q^{1/2}+q^{-1/2})V_{L_0} [[/math]]

To be more precise, here are the conventions for [math]L_+,L_-,L_0[/math], that you need to know, in order to play with the above formula, and compute Jones polynomials at wish:

[[math]] \xymatrix@R=20pt@C=15pt{ &\ar[ddrr]&&&&&\ar@{-}[dr]&&&&&\ar@/_/[rr]&&\\ L_+:&&\ar[ur]&&&L_+:&&\ar[dr]&&&L_0:\\ &\ar@{-}[ur]&&&&&\ar@{-}[uurr]&&&&&\ar@/^/[rr]&&} [[/math]]

As for the proof of the above formula, this comes from our definition of the Jones polynomial, because thinking well, “unclosing” links as to get braids, and then closing Temperley-Lieb diagrams as to get scalars, as required by the construction of [math]V_L(q)[/math], seemingly is some sort of identity operation, but the whole point comes from the fact that the Artin braids [math]g_1,\ldots,g_{k-1}[/math] and the Jones projections [math]e_1,\ldots,e_{k-1}[/math] differ precisely by a crossing being replaced by a non-crossing. Exercise for you, to figure out all this.

(9) In short, up to you to learn all this, in detail, and its generalizations too, with link polynomials defined more generally via relations of the following type:

[[math]] xP_{L_+}+yP_{L_-}+zP_{L_0}=0 [[/math]]

Equivalently, we can define these more general invariants by using various versions of the Temperley-Lieb algebra. As usual, check here the papers of Jones , , .

(10) With the comment here that, among all these invariants, Jones polynomial included, the first came, historially, the Alexander polynomial. However, from a modern point of view, the Alexander polynomial is something more complicated than the Jones polynomial, which remains the central invariant of knots and links.

(11) As another comment, with all this pure mathematics digested, physics strikes back, via a very interesting relation with statistical mechanics, happening in 2D as well, the idea being that “interactions happen at crossings”, and it is these interactions that produce the knot invariant, as a kind of partition function. See Jones , .

(12) Quite remarkably, the above invariants can be directly understood in 3D as well, in a purely geometric way, with elegance, and no need for 2D projection. But this is a more complicated story, involving ideas from quantum field theory. See Witten .

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General references

Banica, Teo (2024). "Graphs and their symmetries". arXiv:2406.03664 [math.CO].