Volume 9, issue 1 (2009)

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Classification of string links up to self delta-moves and concordance

Akira Yasuhara

Algebraic & Geometric Topology 9 (2009) 265–275
Abstract

For an $n$–component string link, the Milnor’s concordance invariant is defined for each sequence $I={i}_{1}{i}_{2}\cdots {i}_{m}\phantom{\rule{1em}{0ex}}\left({i}_{j}\in \left\{1,\dots ,n\right\}\right)$. Let $r\left(I\right)$ denote the maximum number of times that any index appears. We show that two string links are equivalent up to self $\Delta$–moves and concordance if and only if their Milnor invariants coincide for all sequences $I$ with $r\left(I\right)\le 2$.

Keywords
string link, $\Delta$–move, self $\Delta$–move, link-homotopy, concordance, self $\Delta$–equivalence, Milnor invariant
Mathematical Subject Classification 2000
Primary: 57M25, 57M27