Algebraic Markov equivalence for links in three-manifolds

Lambropoulou, S; Rourke, CP

dc.contributor.author	Lambropoulou, S	en
dc.contributor.author	Rourke, CP	en
dc.date.accessioned	2014-03-01T01:23:33Z
dc.date.available	2014-03-01T01:23:33Z
dc.date.issued	2006	en
dc.identifier.issn	0010-437X	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/17015
dc.subject	Braid equivalence	en
dc.subject	Combed band move	en
dc.subject	Knot complements	en
dc.subject	Markov equivalence	en
dc.subject	Mixed braids	en
dc.subject	Mixed links	en
dc.subject	Three-manifolds	en
dc.subject	Twisted conjugation	en
dc.subject.classification	Mathematics	en
dc.subject.other	KNOT-THEORY	en
dc.title	Algebraic Markov equivalence for links in three-manifolds	en
heal.type	journalArticle	en
heal.identifier.primary	10.1112/S0010437X06002144	en
heal.identifier.secondary	http://dx.doi.org/10.1112/S0010437X06002144	en
heal.language	English	en
heal.publicationDate	2006	en
heal.abstract	Let B-n denote the classical braid group on n strands and let the mixed braid group B-m,B-n be the subgroup of Bm+n comprising braids for which the first m strands form the identity braid. Let B-m,B-infinity = boolean OR(n) B-m,B-n. We describe explicit algebraic moves on B-m,B-infinity such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented three-manifold. The moves depend on a fixed link representing the manifold in S-3. More precisely, for link complements the moves are the two familiar moves of the classical Markov equivalence together with 'twisted' conjugation by certain loops a(i). This means premultiplication by a(i)(-1) and postmultiplication by a 'combed' version of a(i). For closed three-manifolds there is an additional set of 'combed' band moves that correspond to sliding moves over the surgery link. The main tool in the proofs is the one-move Markov theorem using L-moves (adding in-box crossings). The resulting algebraic classification is a direct extension of the classical Markov theorem that classifies links in S-3 up to isotopy, and potentially leads to powerful new link invariants, which have been explored in special cases by the first author. It also provides a controlled range of isotopy moves, useful for studying skein modules of three-manifolds.	en
heal.publisher	LONDON MATH SOC	en
heal.journalName	Compositio Mathematica	en
dc.identifier.doi	10.1112/S0010437X06002144	en
dc.identifier.isi	ISI:000239834100010	en
dc.identifier.volume	142	en
dc.identifier.issue	4	en
dc.identifier.spage	1039	en
dc.identifier.epage	1062	en