Authors: Taylor Applebaum, Sam Blackwell, Alex Davies, Thomas Edlich, András Juhász, Marc Lackenby, Nenad Tomašev, Daniel Zheng
Abstract: We have developed a reinforcement learning agent that often finds a minimal
sequence of unknotting crossing changes for a knot diagram with up to 200
crossings, hence giving an upper bound on the unknotting number. We have used
this to determine the unknotting number of 57k knots. We took diagrams of
connected sums of such knots with oppositely signed signatures, where the
summands were overlaid. The agent has found examples where several of the
crossing changes in an unknotting collection of crossings result in hyperbolic
knots. Based on this, we have shown that, given knots $K$ and $K’$ that satisfy
some mild assumptions, there is a diagram of their connected sum and $u(K) +
u(K’)$ unknotting crossings such that changing any one of them results in a
prime knot. As a by-product, we have obtained a dataset of 2.6 million distinct
hard unknot diagrams; most of them under 35 crossings. Assuming the additivity
of the unknotting number, we have determined the unknotting number of 43 at
most 12-crossing knots for which the unknotting number is unknown.
Source: http://arxiv.org/abs/2409.09032v1
