Algebraic Markov equivalence for links in 3-manifolds
| dc.creator | Lambropoulou, Sofia | |
| dc.creator | Rourke, Colin P. | |
| dc.date | 2004-05-26 | |
| dc.date | 2006-02-16 | |
| dc.date.accessioned | 2026-07-07T06:36:48Z | |
| dc.date.available | 2026-07-07T06:36:48Z | |
| dc.description | Let $B_n$ denote the classical braid group on $n$ strands and let the {\em mixed braid group} $B_{m,n}$ be the subgroup of $B_{m+n}$ comprising braids for which the first $m$ strands form the identity braid. Let $B_{m,\infty}=\cup_nB_{m,n}$. We will describe explicit algebraic moves on $B_{m,\infty}$ such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented 3--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 {\em `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 3--manifolds there is an additional set of {\it `combed' band moves} which correspond to sliding moves over the surgery link. The main tool in the proofs is the one-move Markov Theorem using {\it $L$--moves} \cite{LR} (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 3--manifolds. | |
| dc.description | 27 pages, 23 figures, LaTex document. To appear in Compositio Mathematica | |
| dc.identifier | https://arxiv.org/abs/math/0405493 | |
| dc.identifier | http://arxiv.org/abs/math/0405493 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100208 | |
| dc.subject | Geometric Topology | |
| dc.subject | Algebraic Topology | |
| dc.subject | 57M25; 57M27; 57N10 | |
| dc.title | Algebraic Markov equivalence for links in 3-manifolds | |
| dc.type | text |