In this package we provide methods to compute with automorphisms of the free group of rank 2. The details of the algorithms are available in [{Fer}26]. We use the following notation:
\sigma=\begin{cases} x \mapsto y,\\ y \mapsto x, \end{cases}\quad \phi_1=\begin{cases} x \mapsto xy^{-1},\\ y \mapsto y, \end{cases}\quad \phi_2=\begin{cases} x \mapsto x,\\ y \mapsto yx, \end{cases}\quad \phi_3=\begin{cases} x \mapsto y^{-1}x,\\ y \mapsto y. \end{cases}
Then \mathrm{Aut}(F_2) has the following presentation.
\begin{array}{r@{}l} \mathrm{Aut}(F_2) = \langle \sigma,\phi_1,\phi_2,\phi_3 \mid & \phi_1\phi_3=\phi_3\phi_1, \quad \phi_2\phi_3\phi_2 = \phi_3\phi_2\phi_3, \quad \phi_2\phi_1\phi_2 = \phi_1\phi_2\phi_1, \\ & (\phi_1\phi_2\phi_3)^4=1, \quad \sigma^2=1, \quad \phi_1^{\sigma} = \phi_2^{-1}, \\ & \phi_3\sigma = \sigma\phi_3^{-1}\phi_2^{-1}\phi_1^{-1}\phi_2\phi_3, \quad \phi_3^{-1}\sigma= \sigma\phi_3^{-1}\phi_2^{-1}\phi_1\phi_2\phi_3\rangle. \end{array}
Denote the commutator subgroup of a group G by G^{\prime}. For \psi\in \mathrm{Aut}(F_2), we define the map \psi^\ast by \psi^\ast(x F_2^{\prime}) = \psi(x) F_2^{\prime}. If \overline{\cdot} denotes the isomorphism between \mathrm{Aut}(F_2/F_2^\prime) and \mathrm{GL}_2(\mathbb{Z}), then the map
\begin{array}{r@{}l} \Psi \colon & \mathrm{Aut}(F_2) \to \mathrm{GL}_2(\mathbb{Z}) \\ & \psi \mapsto \overline{(\psi^\ast)} \end{array}
is a group homomorphism and the kernel of this map is \mathrm{Inn}(F_2). We denote by \mathrm{SA}_2 the subgroup of \mathrm{Aut}(F_2) consisting of those automorphisms that are mapped into \mathrm{SL}_2(\mathbb{Z}) under \Psi. Let B_4 denote the braid group on 4 strands. Then \mathrm{SA}_2 \cong B_4 / Z(B_4). Using this, we have that each automorphism can be written uniquely as a word of the form
\alpha = \sigma^{\varepsilon_1}\Delta^{\varepsilon_2}\alpha_1\alpha_2\cdots\alpha_k
where \varepsilon_1,\varepsilon_2 \in \{0,1\}, \Delta = \phi_1(\phi_2\phi_1)(\phi_3\phi_2\phi_1), and \alpha_1,\dots,\alpha_k\in \mathrm{SA}_2 whose images under the isomorphism between \mathrm{SA}_2 and B_4 / Z(B_4) are simple braids. We call this expression the left canonical form of \alpha.
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