On the Intersections of Non-homotopic Loops.

Let \(V = \{v_1, \dots , v_n\}\) be a set of n points in the plane and let \(x \in V\). An x-loop is a continuous closed curve not containing any point of V, except of passing exactly once through the point x. We say that two x-loops are non-homotopic if they cannot be transformed continuously into each other without passing through a point of V. For \(n=2\), we give an upper bound \(2^{O(k)}\) on the maximum size of a family of pairwise non-homotopic x-loops such that every loop has fewer than k self-intersections and any two loops have fewer than k intersections. This result is inspired by a very recent result of Pach, Tardos, and Toth who proved the upper bounds \(2^{16k^4}\) for the slightly different scenario when \(x\not \in V\).