  We consider the problem of computing the topology and describing the geometry
of a parametric curve in $\mathbb{R}^n$. We present an algorithm, PTOPO, that
constructs an abstract graph that is isotopic to the curve in the embedding
space. Our method exploits the benefits of the parametric representation and
does not resort to implicitization.
  Most importantly, we perform all computations in the parameter space and not
in the implicit space. When the parametrization involves polynomials of degree
at most $d$ and maximum bitsize of coefficients $\tau$, then the worst case bit
complexity of PTOPO is $
\widetilde{\mathcal{O}}_B(nd^6+nd^5\tau+d^4(n^2+n\tau)+d^3(n^2\tau+
n^3)+n^3d^2\tau)$. This bound matches the current record bound
$\widetilde{\mathcal{O}}_B(d^6+d^5\tau)$ for the problem of computing the
topology of a plane algebraic curve given in implicit form. For plane and space
curves, if $N = \max\{d, \tau \}$, the complexity of PTOPO becomes
$\widetilde{\mathcal{O}}_B(N^6)$, which improves the state-of-the-art result,
due to Alc\'azar and D\'iaz-Toca [CAGD'10], by a factor of $N^{10}$. In the
same time complexity, we obtain a graph whose straight-line embedding is
isotopic to the curve. However, visualizing the curve on top of the abstract
graph construction, increases the bound to $\widetilde{\mathcal{O}}_B(N^7)$.
For curves of general dimension, we can also distinguish between ordinary and
non-ordinary real singularities and determine their multiplicities in the same
expected complexity of PTOPO by employing the algorithm of Blasco and
P\'erez-D\'iaz [CAGD'19]. We have implemented PTOPO in Maple for the case of
plane and space curves. Our experiments illustrate its practical nature.
