  We model the typical behavior of knots and links using grid diagrams. Links
are ubiquitous in the sciences, and their "normal" or "typical" behavior is of
significant importance in understanding situations such as the topological
state of DNA or the statistical mechanics of ring polymers. We examine three
invariants: the expected size of a random knot; the expected number of
components of a random link; and the expected writhe of a random knot. We
investigate the first two numerically and produce generating functions which
codify the observed patterns: knot size is uniformly distributed and linearly
dependent upon grid size, and the number of components follows a distribution
whose mean and variance grow with log_2 of grid size; in particular, for any
fixed k, the k-component links grow vanishingly rare as grid size increases.
Finally, we observe that the odd moments of writhe vanish, and we perform an
exploratory data analysis to discover that variance grows with the square of
grid size and kurtosis is constant at approximately 3.5. We continue this
project in a future work, where we investigate genus and the effects of
crossing change on it.
