Milnor invariants and the HOMFLYPT Polynomial

Meilhan, Jean-Baptiste; Yasuhara, Akira

doi:10.2140/gt.2012.16.889

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Milnor invariants and the HOMFLYPT Polynomial

Jean-Baptiste Meilhan and Akira Yasuhara

Geometry & Topology 16 (2012) 889–917

DOI: 10.2140/gt.2012.16.889

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 ${\bar{μ}}_{L} (J)$ vanishes for any sequence $J$ with length at most $k$ , then any Milnor $\bar{μ}$ –invariant ${\bar{μ}}_{L} (I)$ with length between $3$ and $2 k + 1$ 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/

Volume 16, issue 2 (2012)