A relation between Milnor's $\mu$-invariants and HOMFLYPT polynomials

TL;DR

This paper establishes a new relationship between Milnor's $ar{}$-invariants and HOMFLYPT polynomials, extending previous results and providing formulas for invariants of certain lengths based on knot polynomials.

Contribution

It generalizes the connection between Milnor invariants and knot polynomials, expressing higher-length invariants via HOMFLYPT polynomials under specific conditions.

Findings

Milnor invariants of length  can be expressed using HOMFLYPT polynomials.

Milnor invariants of length 3 are represented by Conway polynomials and linking numbers.

The results extend known relations for length 3 to higher lengths under vanishing conditions.

Abstract

Polyak showed that any Milnor's $\overline{\mu}$-invariant of length 3 can be represented as a combination of 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 $\mu$-invariants. We show that any Milnor's ${\mu}$-invariant of length $\leq k+2$ can be represented as a combination of the HOMFLYPT polynomials of knots obtained from the string link by some operation, if all ${\mu}$-invariants of length $\leq k$ vanish. Moreover, ${\mu}$-invariants of length $3$ are given by a combination of the Conway polynomials and linking numbers without any vanishing assumption.