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Algebraic Markov equivalence for links in three-manifolds
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Lambropoulou, S. and Rourke, C. P. (2006) Algebraic Markov equivalence for links in three-manifolds. COMPOSITIO MATHEMATICA, 142 (4). pp. 1039-1062. doi:10.1112/S0010437X06002144 ISSN 0010-437X.
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Official URL: http://dx.doi.org/10.1112/S0010437X06002144
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.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Journal or Publication Title: | COMPOSITIO MATHEMATICA | ||||
Publisher: | LONDON MATH SOC | ||||
ISSN: | 0010-437X | ||||
Official Date: | July 2006 | ||||
Dates: |
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Volume: | 142 | ||||
Number: | 4 | ||||
Number of Pages: | 24 | ||||
Page Range: | pp. 1039-1062 | ||||
DOI: | 10.1112/S0010437X06002144 | ||||
Publication Status: | Published |
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