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The symplectic theory of polygon spaces, which was developed by Kapovich and Millson among others, can be used to define a fast and provably ergodic Markov chain on spaces of polygonal knots in 3-space. While such Markov chains have traditionally been of interest for numerical experimentation with simple ring polymer models, they can also be used to find an abundance of examples of small but complex knots. After describing the symplectic structure on polygon spaces and the Markov chain, I will report joint work with Ryan Blair, Thomas Eddy, and Nathaniel Morrison in which we have generated and analyzed more than 390 billion random polygonal knots with an eye to improving our knowledge of two geometric knot invariants: the stick number and the superbridge index. In particular, we have found 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, as well as determining the exact superbridge index of 53 knots, including 25 8- and 9-crossing knots.