Assume that two piecewise smooth homotopic closed curves $\gamma_1$ and $\gamma_2$ with same endpoints, divide the plane into finite number of regions $R_0, R_1, \cdots R_n$, where $R_0$ is the unique infinite region. Let $F$ denote the fundamental group of the space $\mathbb{R}^2$ with every $R_i (1 \leq i \leq n)$ removed. Then F is freely generated by the elements $x_i (1 \leq i \leq n)$, where each $x_i$ represents a generator of $\pi_1(\mathbb{R}^2 - R_i) \cong \mathbb{Z}$. Let $g_1$ and $g_2$ be the elements in F representing $\gamma_1$ and $\gamma_2$.
I am trying to find the elements representing curves $\gamma_1$ and $\gamma_2$ for this figure
$\gamma_1$ and $\gamma_2$ are both shown in the figure and they share common basepoint $b_1$.
From the above text explanation, each element $x_i$ represents a curve which goes around a unique region $ R_i$. To represent the curves, let us denote the top part of $\gamma_2$ as $y$ and the bottom part $z$ as shown in the figure.
Now $x_1$ represents $\gamma_1$. $x_3$ represents the curve around region $R_3$, so, $x_3$ represents $ywy^{-1}$ with basepoint $b_1$.
$x_2$ represents $yw^{-1}zx_1^{-1}$ with base point $b_1$. From this I am unable to find the element representing $\gamma_2$. For $\gamma_1$, $x_1$ represents it.
After some operations, which I am assuming we can do. We get $x_2$ represents $x_3^{-1}yzx_1^{-1}$.
I can represent for curves like concentric circles which meet at a point. For that $x_1$ represents $\gamma_1$ which is the inner circle. $x_2$ represents $x_1^{-1}\gamma_2$. Now we can find the element representing $\gamma_2$. It is $x_1x_2$. In this case, $\gamma_2$ is the outer circle.
Any help would be much appreciated. The initial part of the text here is found in https://arxiv.org/abs/1412.0101 It is on page 10 of that paper.