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Representing Milnor’s ${\mu}$-invariant by HOMFLY polynomials

Hold Date
2014-10-31 16:00〜2014-10-31 17:00
Seminar Room 1, Faculty of Mathematics building, Ito Campus
Object person
Yuka KOTORII (Tokyo University)

For an ordered oriented link in the 3-sphere, J. Milnor defined a family of invariants, known as Milnor's $\overline{\mu}$-invariants. Those invariants are determined by a sequence of integers. Polyak showed that any Milnor's $\overline{\mu}$-invariants of length 3 sequence can be represented as a combination of the Conway polynomials of knots. On the other hand, Habegger and Lin showed that Milnor invariants are also invariants for string links, called $\mu$-invariants. In this talk, we show that any Milnor's ${\mu}$-invariant can be represented as a combination of the HOMFLYPT polynomials of knots under some assumption of string links, by using a finite type property of Milnor's ${\mu}$-invariant.In particular, for any ${\mu}$-invariants of length $3$ sequence are given by a combination of the HOMFLYPT polynomials and linking numbers without the assumption of string links. This result is a string link version of Polyak's result.