Is the slit of a rational slit mapping in 𝑆 straight?

Abstract

The question in the title is answered by showing that if f f is a rational function in C ^ {\mathbf {\hat C}} and maps some disk injectively onto the complement of a set E E of empty interior, then degree ⁡ ( f ) = 2 \operatorname {degree}(f) = 2 , and E E is either a circular arc or a line segment in C ^ = C ∪ { ∞ } {\mathbf {\hat C}} = {\mathbf {C}} \cup \{ \infty \} .