Milnor invariants and the HOMFLYPT polynomial (Q441105)

Given an oriented knot, the HOMFLYPT polynomial is a two-variable polynomial invariant of the knot. For a link \(L\), the Milnor invariants assign an integer to each finite, possibly repeating, sequence of link components of \(L\). However, this integer is only defined modulo the greatest common divisor of Milnor invariants of shorter sequences. Note that the Milnor invariants are classical link invariants, while the HOMFLYPT polynomial is a quantum invariant. This paper gives a formula relating these two types of invariant, as follows.NEWLINENEWLINELet \(L\) be a link whose Milnor invariants are zero for all sequences of length at most \(k\). A family of knots can be created from \(L\), and there is an explicit formula in terms of the HOMFLYPT polynomials of these knots for the Milnor invariants of \(L\) assigned to sequences with length between \(3\) and \(2k+1\).NEWLINENEWLINEThe proof of this formula makes use of the theory of claspers. A clasper for a link \(L\) is a surface embedded in \(S^3\) that decomposes in a particular way like a finite graph where each vertex has either valence \(1\) or valence \(3\). A clasper can be used to define a new link from \(L\) via surgery.NEWLINENEWLINEAnother idea that contributes to the proof is to express \(L\) as the closure of a string link. The Milnor invariants in this case are well-defined integers. A formula analogous to that for links is given for string links.