  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 $
\tilde{\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
$\tilde{\mathcal{O}}_B(d^6+d^5\tau)$ 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
$\tilde{\mathcal{O}}_B(N^6)$, 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 $\tilde{\mathcal{O}}_B(N^7)$. 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.
