Braids in trivial braid diagrams

• Published in 2003
• Added on 2026-04-10

We show that every trivial 3-strand braid diagram contains a disk, defined as a ribbon ending in opposed crossings. Under a convenient algebraic form, the result extends to every Artin--Tits group of dihedral type, but it fails to extend to braids with 4 strands and more. The proof uses a partition of the Cayley graph and a continuity argument.

Links

• https://arxiv.org/abs/math/0311326v1
• https://arxiv.org/pdf/math/0311326v1

Other information

BibTeX entry

@article{Braidsintrivialbraiddiagrams,
	key = {Braidsintrivialbraiddiagrams},
	type = {article},
	title = {Braids in trivial braid diagrams},
	author = {Patrick Dehornoy},
	abstract = {We show that every trivial 3-strand braid diagram contains a disk, defined as a ribbon ending in opposed crossings. Under a convenient algebraic form, the result extends to every Artin--Tits group of dihedral type, but it fails to extend to braids with 4 strands and more. The proof uses a partition of the Cayley graph and a continuity argument.},
	comment = {},
	date_added = {2026-04-10},
	date_published = {2003-04-10},
	urls = {https://arxiv.org/abs/math/0311326v1,https://arxiv.org/pdf/math/0311326v1},
	collections = {fun-maths-facts,the-groups-group,things-to-make-and-do},
	url = {https://arxiv.org/abs/math/0311326v1 https://arxiv.org/pdf/math/0311326v1},
	year = 2003,
	urldate = {2026-04-10},
	archivePrefix = {arXiv},
	eprint = {math/0311326},
	primaryClass = {math.GT}
}