This article is part of a series on topological quantum computing.

The explanation below highlights the relation between the Jones polynomial and TQFT. As first shown by Edward Witten, the Jones polynomial of a given knot $\gamma$ can be obtained by considering Chern–Simons theory on the three-sphere with gauge group $\mathrm{SU}(2)$, and computing the vacuum expectation value of a Wilson loop $W_{F}(\gamma)$, associated to $\gamma$, and the fundamental representation $F$ of $\mathrm{SU}(2)$.

Let’s recap first a few things we explained in previous articles.

A knot is a manifold in $\mathbb{R}^3$ diffeomorphic to $S^1$ while a link is diffeomeorphic to multiple copies of $S^1$. A knot or link is said to be ambient equivalent to another if you can transform one into another via the Reidemeister moves. Assigning an orientation to knot and links amount to assigning a way to walk/traverse a knot. It amount to saying how to perform a line-integral. At a crossing $C_{ij}$ of two knots $K_i, K_j$ you can have two options

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the sum is called the linking number of the knot

$$L(K_i,K_j) = \frac{1}{2}\sum_{c\in C_{ij}}\text{sign}(c)$$

and is an invariant of the knot. If you take a set of knots you can compute the sum of all the crossings. The sum can in fact be split in two parts; self-linking and cross linking

$$w(K_1,\ldots,K_n) = \sum_i w_i + \sum_{i\neq j} L(K_i,K_j)$$

and the self-linking number is called the writhe of the knot. This is not an invariant as can be seen below

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The Kauffman bracket $\langle K\rangle$ of a knot $K$ is the sum

$$\langle K\rangle = \sum_C \langle K|C\rangle\, d^{l-1}$$

where the sum is over the decomposition of the knot via

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and $l$ is the number of loops in the knot. The resulting polynomial is a function of $A, B, d$.

Finally, the Jones polynomial of the knot $K$ is

$$\mathcal{L}_K = (-A^3)^{-w(K)}\langle K \rangle$$ with

$$B = A^{-1},\, x = -A^2-A^{-2}$$

and $w(K)$ the writhe of the knot. It might look like a complex definition but if you work it out on a few examples the whole things is both fun and straightforward. Note that all of this is purely combinatorial and has nothing to do with QFT.

We saw that Chern-Simons is a TQFT with action

$$\Gamma(A) = \frac{k}{4\pi}\int \text{Tr}(A\wedge A + \frac{2}{3}A\wedge A \wedge A)$$

and with this you can do all the usual field theory stuff. Of particular importance are the Wilson loops and the expectation value of these loops is

$$\langle W_C\rangle = \int \mathcal{D}A\,W_C\, e^{\frac{i}{\hbar}S_{CS}}.$$

with

$$W_C = \text{Tr}\,\mathcal{P}\,\exp \oint_C A.$$

The Wilson loop can be seen as an inner-product between the spaces of loops and the space of connections. The space of connections is actually a moduli space $\mathcal{A}/\mathcal{G}$ with respect to the gauge invariance but that’s a whole different story.

The Chern-Simons actions is gauge invariant under the local transformation $\Omega$

$$A^\mu\mapsto \Omega^{-1}(x) \,A_\mu \Omega(x)- i\,\Omega^{-1}(x) \,\partial_\mu \Omega(x)$$

and the equation of motions are

$$F^a_{\mu\nu} = e^{-i \Gamma}\left(\frac{4\pi}{k i}\right)\epsilon_{\mu\nu\rho}\frac{\delta}{\delta A_\rho^a} e^{i \Gamma}.$$

Through this identity one has that the curvature inside an expectation value can be replaced by

$$F^a_{\mu\nu}\mapsto \frac{4\pi i}{k}\epsilon_{\mu\nu\rho}\frac{\delta}{\delta A_\rho^a},$$

specifically

$$\langle F^a_{\mu\nu} O_1 O_2\ldots\rangle = \langle \frac{4\pi i}{k}\epsilon_{\mu\nu\rho}\frac{\delta}{\delta A_\rho^a}O_1 O_2\ldots\rangle$$

which can be derived through a standard integration by parts.

Another curiosity of combining loops and gauge theory is that if you work out a tiny (rectangular) Wilson loop you get a curvature. Specifically, if you compute a Wilson loop $W_{\delta L}$ around the rectangle $\delta L$ below

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you get

$$W_{\delta L} = \exp i\sigma^{\mu\nu}F^a_{\mu\nu} R_a$$

with $R_a$ a representation of the gauge group under consideration. The $\sigma^{\mu\nu}$ corresponds to the infinitesimal area enclose by $\delta L$, i.e. $\sigma ^{\mu\nu} = dx^\mu\wedge dx^\nu.$

Now, let’s apply this to a line segment where we inject a little loop like below

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and let $\delta U_{12}$ be the Wilson integral over the line with the tiny bubble inserted:

$$\langle U_{12}\rangle = \langle U_{1x}\,\left(i\, \sigma^{\mu\nu}F_{\mu\nu}^a R_a\right)\, U_{x2}\rangle$$

using the relation above. At the same time, if we inject the equation of motion of the Chern-Simons field we get

$$\langle\delta U_{12}\rangle = i\,\sigma^{\mu\nu} R_a \frac{4\pi i}{k}\,\epsilon_{\mu\nu\rho}\langle U_{1x}\,\frac{\delta}{\delta A^a_{\rho}(x)}U_{x2}\rangle.$$

The functional derivative inside the expectation value of the Wilson integral amounts to an extra group generator:
$$\frac{\delta}{\delta A^a_{\rho}(x)} U = i\,R_a \delta(x-y)\,U_y\, dy^\rho$$

which once inserted above gives

$$\langle \delta U_{12}\rangle = \frac{4\pi}{i k} \left(\epsilon_{\mu\nu\rho}\sigma^{\mu\nu}\,dy^\rho\delta(x-y)\right)\langle U_{1x}\,c_2(R)\,U_{x2} \rangle.$$

The element $dy^\rho$ originates from take a line integral and so this element is tangential to the line. The area element is point in or out of the surface where the line is in. Hence, the expression between brackets is the volume defined by the surface vector and the field $A_\mu$. Depending on the direction the bubble is traversed you get $\pm \delta v$ and hence

$$\langle \frac{\delta U_{12}}{\delta v}\rangle = \frac{4\pi}{i k} \langle U_{1x}\,c_2(R)\,U_{x2} \rangle$$

and $c_2$ is called the second Casimir operator of the group

$$c_2(R) = \sum_a R_a R_a.$$

This is quite general and let’s focus now on a few concrete cases. We’ll use the following notation for the specific loops

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At this point we assume that the loops (like e.g. $\hat{W}_+$) are small in order to apply the derivation above. Also, one should note that the condition $k\gg 1$ corresponds to

$$\left|\frac{2\pi}{k i}c_2\right|\ll 1$$

in the sense that $k\sim 1/\hbar$ in the Chern-Simons path-integral. So, if we look at the difference

$$\langle \hat{W}+\rangle – \langle \hat{W}-\rangle = \frac{4\pi}{i k}\langle \hat{W}_0\rangle $$

and if one sets

$$\alpha := \frac{1}{a-\frac{2\pi}{k i}c_2}$$

then

$$\langle \hat{W}_+ \rangle = \alpha \langle \hat{W}_0 \rangle \\ \langle \hat{W}_{-\rangle} = \alpha^{-1} \langle \hat{W}_{0} \rangle$$

up to orders of $O(\frac{1}{k^2})$. With the same definition of $\alpha$ you get

$$\langle \hat{W}_{+} \rangle + \langle \hat{W}_{-\rangle} = 2 \langle \hat{W}_{0} \rangle. $$

Note that is is independent of the gauge group. If one focuses on $SU(2)$ and the difference $\langle W_+\rangle – \langle W_-\rangle $ you need the actual Casimir value and for $SU(2)$ this is the so-called Fierz identity

$$c_2(SU(2)) = \sum_a R^a_{ij}\, R^a_{kl} = \frac{1}{2}\delta_{il}\delta_{jk} – \frac{1}{2N}\delta_{ij}\delta_{kl}.$$

Pictorially this amounts to

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which can be seen if you write down the expectation value with the matrix indices. Inserting this in the difference $\langle W_+\rangle – \langle W_-\rangle $:

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and with $\langle W_+\rangle + \langle W_-\rangle = 2\langle W_0\rangle:$

$$(1+\frac{\pi}{iNk})\langle W_+\rangle – (1-\frac{\pi}{iNk})\langle W_-\rangle = \frac{2\pi}{ik}\langle W_0\rangle.$$

With the definition

$$\beta = \frac{1}{1 – \frac{\pi}{iNk}}, \; x = \frac{2\pi}{ik}$$

one gets finally

$$\beta\,\langle W_+\rangle – \beta^{-1}\,\langle W_-\rangle = z\,\langle W_0\rangle$$

which be recognized as the HOMFLY polynomial. The HOMFLY poynomial $P_L(t,z)$ is a Laurent polynomial defined through the relation

$$t P_+ – t^{-1}P_- = z P_0$$

and reduces to the Jones if one computes $P_L(t, z = \sqrt{t}-1/\sqrt{t})$ while the Alexander-Conway polynomial can be computed via $P_L(t=1,z).$ The HOMFLY polynomial is in this sense a generalization of the Jones polynomial and it shows that

computing expectation values of Wilson loops over knots in $SU(2)$ Chern-Simons quantum field theory is the same as computing the Jones knot polynomials.

This is the bridge between TQFT and knot theory. It also embodies how one can describe a topological quantum computer although we’ll see that quantum double models simplify things a whole lot.