  We develop a new algebraic theory of positive braids and conjugacy classes in
the braid group $B_3$. We use our theory to establish a complete classification
of isotopy classes of degree three symplectic curves in $\mathbb{CP}^2$ with
only $A_n$-singularities for $n\geq 1$ (an $A_n$-singularity is locally
modelled by the equation $z^2 = w^n$) independent of Gromov's theory of
pseudoholomorphic curves. We show that if $C$ and $C'$ are degree three
symplectic curves in $\mathbb{CP}^2$ with the same numbers of
$A_n$-singularities for each $n\geq 1$, then $C$ is isotopic to $C'$.
Furthermore, our theory furnishes a single method of proof that independently
establishes and unifies several fundamental classification results on degree
three symplectic curves in $\mathbb{CP}^2$. In particular, we prove using our
theory: (1) there is a unique isotopy class of degree three smooth symplectic
curves in $\mathbb{CP}^2$ (a result due to Sikorav), (2) the number of nodes is
a complete invariant of the isotopy class of a degree three nodal symplectic
curve in $\mathbb{CP}^2$ (the case of irreducible nodal curves is due to
Shevchishin and the case of reducible nodal curves is due to Golla-Starkston),
and (3) there is a unique isotopy class of degree three cuspidal symplectic
curves in $\mathbb{CP}^2$ (a generalization of a result due to Ohta-Ono). The
present work represents the first step toward resolving the symplectic isotopy
conjecture using purely algebraic techniques in the theory of braid groups.
Finally, we independently establish a complete classification of genus one
Lefschetz fibrations over $\mathbb{S}^2$ (a result due to Moishezon-Livne).
