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Milnor invariants and the HOMFLYPT Polynomial

Jean-Baptiste Meilhan and Akira Yasuhara

Geometry & Topology 16 (2012) 889–917
Abstract

We give formulas expressing Milnor invariants of an n–component link L in the 3–sphere in terms of the HOMFLYPT polynomial as follows. If the Milnor invariant μ̄L(J) vanishes for any sequence J with length at most k, then any Milnor μ̄–invariant μ̄L(I) with length between 3 and 2k + 1 can be represented as a combination of HOMFLYPT polynomial of knots obtained from the link by certain band sum operations. In particular, the “first nonvanishing” Milnor invariants can be always represented as such a linear combination.

Keywords
Milnor invariant, HOMFLYPT polynomial, clasper, string link, link-homotopy
Mathematical Subject Classification 2010
Primary: 57M25, 57M27
References
Publication
Received: 28 June 2011
Accepted: 2 February 2012
Published: 19 May 2012
Proposed: Peter Teichner
Seconded: Colin Rourke, Joan Birman
Authors
Jean-Baptiste Meilhan
Institut Fourier
Université Grenoble 1
100 rue des Maths
BP 74
38402 Saint-Martin d’Hères
France
http://www-fourier.ujf-grenoble.fr/~meilhan/
Akira Yasuhara
Department of Mathematics
Tokyo Gakugei University
4-1-1 Nukuikita-Machi
Koganei-shi
Tokyo 184-8501
Japan
http://www.u-gakugei.ac.jp/~yasuhara/