Abstract
This paper analyzes the physical resources necessary and sufficient to tie a knot of given structure. We present the first sufficient bound on the number of fingers required to tie a given knot; the bound is linear in the number of crossings appearing in the knot diagram for a given knot. We also present a lower bound on the required number of fingers, under a particular model of knot tying. We study how many re-grasps are sufficient to tie an arbitrary knot, and present an algorithm that can yield a small sufficient number of re-grasps to tie the given knot. Physical experiments in which different knots are tied and untied by robots, alone and in collaboration with a human, serve as a proof of concept to show the simplicity and correctness of the approach.
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