Running-time dependence on Euler characteristic for the Schaefer–Sedgwick–Štefankovič isotopy algorithm

Determine the dependence on the Euler characteristic χ(S) of the running time of the algorithm of Schaefer, Sedgwick, and Štefankovič (Theorem 2(a) in their work) for deciding whether two simple closed curves on a compact surface S with non-empty boundary are isotopic. Provide an explicit bound or characterization of the algorithm’s complexity in terms of χ(S).

Background

The paper introduces fast algorithms for curves on surfaces, including computing geometric intersection numbers and deciding isotopy, with running times polynomial in the triangulation size and in the logarithm of curve weights. It contrasts these with prior approaches that often require surfaces with boundary and rely on ideal triangulations.

In reviewing related work, the authors note that Schaefer, Sedgwick, and Štefankovič provided an algorithm (Theorem 2(a) in their paper) to decide isotopy of simple closed curves on surfaces with boundary. However, while this algorithm addresses the problem, the dependence of its running time on the Euler characteristic of the surface is not specified, leaving a gap in understanding its complexity relative to surface topology.