[2008.09665] Efficient Geodesic Super Efficiency on Complex Curves

Download the PDF of the paper entitled “Efficient Geodesic Super Efficiency on Complex Curves” by Xifeng Jin and William W. Menasco.

Download PDF

overview: We show that efficient geodesics have a powerful property of ‘super-efficiency’. For any two vertices $v , w \in \mathcal{C}(S_g)$ in the complex curve of the closed oriented surface of genus $g \geq 2 $ and any efficient geodesic $v = v_1 , \cdots , v_{\text d}=w$ , but explicitly computable maximum $v_1$ vertex candidates. In this memo, we establish the bounds of this computable list, independent of ${\text d}$- distance and dependent only on genus (super efficient trait). The proof is based on a new intersection growth inequality between the number of curve intersections and the distance in a compound curve, and an exhaustive analysis of dot graphs associated with intersection sequences.