Laces: A generalisation of braids
2001; Osaka University; Volume: 38; Issue: 2 Linguagem: Inglês
10.18910/10868
ISSN0030-6126
AutoresRoger Fenn, Gyo Taek Jin, Richárd Rimányi,
Tópico(s)Advanced Operator Algebra Research
ResumoRIM ANYI´(Received May 10, 1999)1. IntroductionThe subject of this paper is a previously little studied object which we call a lace .A lace of components is represented by a disjoint union of arcs in the planewhich join xed points to other xed points. We take as the initial points of thearcs the points (1 1) (2 1) ::: ( 1) and the nal points of the arcs are (1 0) (2 0),::: ( 0) in some order.There are several notions of equivalence of laces. Apart from the obvious notionof isotopy in the plane there is a notion of 3-isotopy in which the interiors of the arcsare allowed to move in the upper half space. There is also a notion of cobordism andto each of the previous equivalences can be added a similar equivalence where the arcsare allowed to lie in the extended Riemannian plane or sphere. Clearly isotopy impliescobordism and, because the 3-isotopy has one extra dimensional freedom, it is weakerthan the cobordism.Lemma 8. Cobordant laces are 3-isotopic.Laces are a very natural generalisation of braids. Given an -lace we can con-struct an -braid as follows: Consider the -plane in which the lace lies as being thehorizontal plane = 0 in space with the -axis vertical
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