A robot that unknots knots
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arXiv
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| Format: | Preprint |
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2025
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| author | Hui, Connie On Yu Ibarra, Dionne Kauffman, Louis H. McQuire, Emma N. Montoya-Vega, Gabriel Mukherjee, Sujoy Reid, Corbin |
| author_facet | Hui, Connie On Yu Ibarra, Dionne Kauffman, Louis H. McQuire, Emma N. Montoya-Vega, Gabriel Mukherjee, Sujoy Reid, Corbin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01254 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A robot that unknots knots Hui, Connie On Yu Ibarra, Dionne Kauffman, Louis H. McQuire, Emma N. Montoya-Vega, Gabriel Mukherjee, Sujoy Reid, Corbin Geometric Topology Combinatorics 57K10 (primary), 57M15, 57Z10, 92-10 (secondary) 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. |
| title | A robot that unknots knots |
| topic | Geometric Topology Combinatorics 57K10 (primary), 57M15, 57Z10, 92-10 (secondary) |
| url | https://arxiv.org/abs/2504.01254 |