Abstract
Polyak showed that any Milnor’s [Formula: see text]-invariant of length 3 can be represented as a combination of the Conway polynomials of knots obtained by certain band sum of the link components. On the other hand, Habegger and Lin showed that Milnor invariants are also invariants for string links, called [Formula: see text]-invariants. We show that any Milnor’s [Formula: see text]-invariant of length [Formula: see text] can be represented as a combination of the HOMFLYPT polynomials of knots obtained from the string link by some operation, if all [Formula: see text]-invariants of length [Formula: see text] vanish. Moreover, [Formula: see text]-invariants of length [Formula: see text] are given by a combination of the Conway polynomials and linking numbers without any vanishing assumption.
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