  Random walks and polygons are used to model polymers. In this paper we
consider the extension of writhe, self-linking number and linking number to
open chains. We then study the average writhe, self-linking and linking number
of random walks and polygons over the space of configurations as a function of
their length. We show that the mean squared linking number, the mean squared
writhe and the mean squared self-linking number of oriented uniform random
walks or polygons of length $n$, in a convex confined space, are of the form
$O(n^2)$. Moreover, for a fixed simple closed curve in a convex confined space,
we prove that the mean absolute value of the linking number between this curve
and a uniform random walk or polygon of $n$ edges is of the form $O(\sqrt{n})$.
Our numerical studies confirm those results. They also indicate that the mean
absolute linking number between any two oriented uniform random walks or
polygons, of $n$ edges each, is of the form O(n). Equilateral random walks and
polygons are used to model polymers in $\theta$-conditions. We use numerical
simulations to investigate how the self-linking and linking number of
equilateral random walks scale with their length.
