Extending closed plane curves to immersions of the disk with $n$ handles

Abstract:

Let $f:S \to E$ be a normal curve in the plane. The extensions of $f$ to immersions of the disk with $n$ handles $({T_n})$ can be determined as follows. A word for $f$ is constructed using the definitions of Blank and Marx and a combinatorial structure, called a ${T_n}$-assemblage, is defined for such words. There is an immersion extending $f$ to ${T_n}$ iff the tangent winding number of $f$ is $1 - 2n$ and $f$ has a ${T_n}$-assemblage. For each $n$, a canonical curve ${f_n}$ with a topologically unique extension to ${T_n}$ is described (${f_0}$ = Jordan curve). Any extendible curve with the minimum number $(2n + 2\;{\text {for}}\;n > 0)$ of self-intersections is equivalent to ${f_n}$.