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arXiv · 2504.01254

A robot that unknots knots

Connie On Yu Hui·Dionne Ibarra·Louis H. Kauffman·Emma N. McQuire·Gabriel Montoya-Vega·Sujoy Mukherjee·Corbin Reid
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Abstract

Consider a robot that remembers only the starting position and walks along a knot once on a knot diagram, switching every undercrossing it meets until it returns to the starting position. We observe that the robot produces an ascending diagram, and we provide a new combinatorial proof that every ascending or descending knot diagram can be transformed into the zero-crossing unknot diagram. Using the machinery developed from the combinatorial proof, we show that the minimal number of Reidemeister moves required for such a transformation is bounded above by (7C+1)C if the diagram has C crossings. Moreover, we provide a new alternative proof that there exist sequences of Reidemeister moves that do not increase the number of crossings and transform ascending or descending knot diagrams into zero-crossing unknot diagrams.

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Connie On Yu Hui, Dionne Ibarra, Louis H. Kauffman, Emma N. McQuire, Gabriel Montoya-Vega, Sujoy Mukherjee, Corbin Reid. 2025-11-20. A robot that unknots knots. https://arxiv.org/abs/2504.01254

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