Milnor invariants and the HOMFLYPT Polynomial

Meilhan, Jean-Baptiste¹ ; Yasuhara, Akira²

¹ Institut Fourier, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d’Hères, France
² Department of Mathematics, Tokyo Gakugei University, 4-1-1 Nukuikita-Machi, Koganei-shi, Tokyo 184-8501, Japan

Geometry & topology, Tome 16 (2012) no. 2, pp. 889-917.

Résumé

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 ${\bar{\mu }}_L\left(J\right)$ vanishes for any sequence $J$ with length at most $k$, then any Milnor $\bar{\mu }$–invariant ${\bar{\mu }}_L\left(I\right)$ 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.

Détail

DOI : 10.2140/gt.2012.16.889

Classification : 57M25, 57M27
Keywords: Milnor invariant, HOMFLYPT polynomial, clasper, string link, link-homotopy

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