  The stick number of a knot is the minimum number of segments needed to build
a polygonal version of the knot. Despite its elementary definition and
relevance to physical knots, the stick number is poorly understood: for most
knots we only know bounds on the stick number. We adopt a Monte Carlo approach
to finding better bounds, producing very large ensembles of random polygons in
tight confinement to look for new examples of knots constructed from few
segments. We generated a total of 220 billion random polygons, yielding either
the exact stick number or an improved upper bound for more than 40% of the
knots with 10 or fewer crossings for which the stick number was not previously
known. We summarize the current state of the art in Appendix A, which gives the
best known bounds on stick number for all knots up to 10 crossings.
