This answer is more or less a summary of the Aharonov-Jones-Landau paper you linked to, but with everything not directly related to defining the algorithm removed. Hopefully this is useful.
The Aharonov-Jones-Landau algorithm approximates the Jones polynomial of the plat closure of a braid $\sigma$ at a $k$th root of unity by realizing it as (some rescaling of) a matrix element of a certain unitary matrix $U_\sigma$, the image of $\sigma$ under a certain unitary representation of the braid group $B_{2n}$. Given an implementation of $U_\sigma$ as a quantum circuit, approximating its matrix elements is straightforward using the Hadamard test. The nontrivial part is approximating $U_\sigma$ as a quantum circuit.
If $\sigma$ is a braid on $2n$ strands with $m$ crossings, we can write $\sigma = \sigma_{a_1}^{\epsilon_1} \sigma_{a_2}^{\epsilon_2} \cdots \sigma_{a_m}^{\epsilon_m}$, where $a_1, a_2, \ldots, a_m \in \{1, 2, \ldots, 2n - 1\}$, $\epsilon_1, \epsilon_2, \ldots, \epsilon_m \in \{\pm 1\}$, and $\sigma_i$ is the generator of $B_{2n}$ that corresponds to crossing the $i$th strand over the $(i + 1)$st. It suffices to describe $U_{\sigma_i}$, since $U_\sigma = U_{\sigma_{a_1}}^{\epsilon_1} \cdots U_{\sigma_{a_m}}^{\epsilon_m}$.
To define $U_{\sigma_i}$, we first give a certain subset of the standard basis of $\mathbb{C}^{2^{2n}}$ on which $U_{\sigma_i}$ acts nontrivially. For $\psi = \lvert b_1 b_2 \cdots b_{2n} \rangle$, let $\ell_{i'}(\psi) = 1 + \sum_{j = 1}^{i'} (-1)^{1-b_j}$. Let's call $\psi$ admissible if $1 \leq \ell_{i'}(\psi) \leq k - 1$ for all $i' \in \{1, 2, \ldots, 2n\}$. (This corresponds to $\psi$ describing a path of length $2n$ on the graph $G_k$ defined in the AJL paper.) Let $$\lambda_r = \begin{cases}\sin(\pi r / k) & \textrm{if $1 \leq r \leq k - 1$},\\ 0 & \textrm{otherwise.}\end{cases}$$ Let $A = ie^{-\pi i/2k}$ (this is mistyped in the AJL paper; also note that here and only here, $i = \sqrt{-1}$ is not the index $i$). Write $\psi = \lvert \psi_i b_i b_{i+1} \cdots\rangle$, where $\psi_i$ is the first $i - 1$ bits of $\psi$, and let $z_i = \ell_{i-1}(\psi_i)$. Then
$$
\begin{align}
U_{\sigma_i}(\lvert\psi_i 00 \cdots\rangle) & = A^{-1}\lvert\psi_i 00 \cdots\rangle\\
U_{\sigma_i}(\lvert\psi_i 01 \cdots \rangle) & = \left( A\frac{\lambda_{z_i-1}}{\lambda_{z_i}} + A^{-1}\right)\lvert\psi_i 01 \cdots\rangle + A\frac{\sqrt{\lambda_{z_i+1}\lambda_{z_i-1}}}{\lambda_{z_i}}\lvert\psi_i 10 \cdots\rangle\\
U_{\sigma_i}(\lvert\psi_i 10 \cdots \rangle) & = A\frac{\sqrt{\lambda_{z_i+1}\lambda_{z_i-1}}}{\lambda_{z_i}}\lvert\psi_i 01 \cdots\rangle + \left(A\frac{\lambda_{z_i+1}}{\lambda_{z_i}} + A^{-1}\right)\lvert\psi_i 10 \cdots\rangle\\
U_{\sigma_i}(\lvert\psi_i 11 \cdots\rangle) & = A^{-1}\lvert\psi_i 11 \cdots\rangle
\end{align}
$$
We define $U_{\sigma_i}(\psi) = \psi$ for non-admissible basis elements $\psi$.
We would now like to describe $U_{\sigma_i}$ as a quantum circuit with polynomially many (in $n$ and $k$) gates. Notice that while $U_{\sigma_i}$ only changes two qubits, it also depends on the first $i - 1$ qubits through the dependence on $z_i$ (and indeed, it depends on all qubits for the admissibility requirement). However, we can run a counter to calculate and store $z_i$ (and also determine admissibility of the input) in logarithmically many (in $k$) ancilla qubits, and therefore we can apply the Solovay-Kitaev algorithm to get a good approximation to $U_{\sigma_i}$ using only polynomially many gates. (The paper appeals to Solovay-Kitaev twice: once for incrementing the counter at each step, and once for applying $U_{\sigma_i}$; I'm not sure if there is a more direct way to describe either of these as quantum circuits with standard gates. The paper also doesn't mention the need to check for admissibility here; I'm not sure if this is important, but certainly we at least need $1 \leq z_i \leq k - 1$.)
So to recap:
- Start with a braid $\sigma \in B_{2n}$ with $m$ crossings.
- Write $\sigma = \sigma_{a_1}^{\epsilon_1} \sigma_{a_2}^{\epsilon_2} \cdots \sigma_{a_m}^{\epsilon_m}$.
- For each $i \in \{1, 2, \ldots, m\}$, apply the Solovay-Kitaev algorithm to get an approximation of the unitary matrix $U_{\sigma_{a_i}}$ (or its inverse if $\epsilon_i = -1$).
- Compose all of the approximations from step 3 to get a quantum circuit with polynomially many gates that approximates $U_{\sigma}$.
- Apply the real and imaginary Hadamard tests polynomially many times with the circuit from step 4 and the state $\lvert 1010 \cdots 10\rangle$.
- Average the results of step 5 and multiply by some scaling factor to get an approximation to the real and imaginary parts of the Jones polynomial of the plat closure of $\sigma$ evaluated at $e^{2\pi i/k}$.
