  A multigraph drawn in the plane is non-homotopic if no two edges connecting
the same pair of vertices can be continuously deformed into each other without
passing through a vertex, and is $k$-crossing if every pair of edges
(self-)intersects at most $k$ times. We prove that the number of edges in an
$n$-vertex non-homotopic $k$-crossing multigraph is at most $6^{13 n (k + 1)}$,
which is a big improvement over previous upper bounds.
  We also study this problem in the setting of monotone drawings where every
edge is an x-monotone curve. We show that the number of edges, $m$, in such a
drawing is at most $2 \binom{2n}{k + 1}$ and the number of crossings is
$\Omega\bigl(\frac{m^{2 + 1/k}}{n^{1 + 1/k}}\bigr)$. For fixed $k$ these bounds
are both best possible up to a constant multiplicative factor.
