A robot that unknots knots

• Published in 2025
• Added on 2025-05-03

Consider a robot that walks along a knot once on a knot diagram and switches every undercrossing it meets, stopping when it comes back to the starting position. We show that such a robot always unknots the knot. In fact, we prove that the robot produces an ascending diagram, and we provide a purely combinatorial proof that every ascending or descending knot diagram with C crossings can be transformed into the zero-crossing unknot diagram using at most (7C+1)C Reidemeister moves. Moreover, we show that an ascending or descending knot diagram can always be transformed into a zero-crossing unknot diagram using Reidemeister moves that do not increase the number of crossings.

Links

• http://arxiv.org/abs/2504.01254v1
• http://arxiv.org/pdf/2504.01254v1

Other information

BibTeX entry

@article{Arobotthatunknotsknots,
	key = {Arobotthatunknotsknots},
	type = {article},
	title = {A robot that unknots knots},
	author = {Connie On Yu Hui and Dionne Ibarra and Louis H. Kauffman and Emma N. McQuire and Gabriel Montoya-Vega and Sujoy Mukherjee and Corbin Reid},
	abstract = {Consider a robot that walks along a knot once on a knot diagram and switches
every undercrossing it meets, stopping when it comes back to the starting
position. We show that such a robot always unknots the knot. In fact, we prove
that the robot produces an ascending diagram, and we provide a purely
combinatorial proof that every ascending or descending knot diagram with C
crossings can be transformed into the zero-crossing unknot diagram using at
most (7C+1)C Reidemeister moves. Moreover, we show that an ascending or
descending knot diagram can always be transformed into a zero-crossing unknot
diagram using Reidemeister moves that do not increase the number of crossings.},
	comment = {},
	date_added = {2025-05-03},
	date_published = {2025-05-07},
	urls = {http://arxiv.org/abs/2504.01254v1,http://arxiv.org/pdf/2504.01254v1},
	collections = {easily-explained,fun-maths-facts,protocols-and-strategies},
	url = {http://arxiv.org/abs/2504.01254v1 http://arxiv.org/pdf/2504.01254v1},
	year = 2025,
	urldate = {2025-05-03},
	archivePrefix = {arXiv},
	eprint = {2504.01254},
	primaryClass = {math.GT}
}