The classification of local homeomorphisms from a closed line segment into $\mathbb{R}^2$ Does anyone know a reference to the classifications of local homeomorphisms from a closed line segment into $\mathbb{R^2}$ ? I suspect it is given by the minimal number of self intersections
of the image curve.
 A: Fact. Every local homeomorphism of $[0,1]$ into $\mathbb R^2$ is homotopic (through local homeomorphisms, without moving the endpoints) to a polygonal path. 
Once the fact is proved, one can reduce the number of intermediate vertices by homotoping the broken line $a_{i-1}a_ia_{i+1}$  to $a_{i-1}a_{i+1}$. When the intermediate vertices are gone, we have a line segment. Hence, the proposed classification has only one class.
Proof of the fact: By partitioning $[0,1]$, the problem reduces to the case of a homeomorphism  $f:[0,1]\to\mathbb R^2$. Let $\Phi$ be a conformal map of $\mathbb C\setminus [0,1]$ onto $\mathbb C\setminus f([0,1])$ with boundary values $\Phi(0)=f(0)$ and $\Phi(1)=f(1)$. Let $\gamma_\kappa$, $0<\kappa<1$, be the circular arc with endpoints $0$ and $1$ and center $1/2-i/\kappa$. The curves $\Phi\circ \gamma_\kappa$, after suitable reparametrization, form a homotopy between $f$ and a smooth  embedding of $ [0,1]$ into the plane. A smooth embedding can be approximated by a polygonal curve, to which it is related by a straight-line homotopy. $\Box$
(There may be an easier proof of the fact, but it escapes me now.)
