A crossing lemma for multigraphs

Let $G$ be a drawing of a graph with $n$ vertices and $e>4n$ edges, in which no two adjacent edges cross and any pair of independent edges cross at most once. According to the celebrated Crossing Lemma of Ajtai, Chv'atal, Newborn, Szemer'edi and Leighton, the number of crossings in $G$ is at least $c\frac{e^3}{n^2}$, for a suitable constant $c>0$. In a seminal paper, Sz'ekely generalized this result to multigraphs, establishing the lower bound $c\frac{e^3}{m{n}^2}$, where $m$ denotes the maximum multiplicity of an edge in $G$. We get rid of the dependence on $m$ by showing that, as in the original Crossing Lemma, the number of crossings is at least $c^{\mathrm{\prime }}\frac{e^3}{n^2}$ for some $c^{\mathrm{\prime }}>0$, provided that the “lens” enclosed by every pair of parallel edges in $G$ contains at least one vertex. This settles a conjecture of Kaufmann.