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Let β* be the unoriented Sato–Levine invariant, an invariant of two component links with even linking number. For a three component link L, we will develop a relation between the Milnor number , β*(L′) of certain two component sublinks L′ of L, and β* of the link formed by merging two of the components of L by band sum. Using this relation, we will show that every embedding of K₁₀ contains a two component link with β*(L) ≠ 0.

Milnor defined two kinds of link-homotopy invariants $\overline{\mu }$ and ${\mu }^{\ast }$. By definition it would seem that the ${\mu }^{\ast }$-invariant is weaker than the $\overline{\mu }$-invariant. In this paper, we show that this is indeed the case by giving an example of length greater than 4 where $\overline{\mu }=1$ and ${\mu }^{\ast }=0$. For non-repeated sequences of length not greater than 4, Milnor has shown that $\overline{\mu }={\mu }^{\ast }$.

Polyak showed that any Milnor’s $\overline{\mu }$-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 $\mu$-invariants. We show that any Milnor’s $\mu$-invariant of length $\le k+2$ 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 $\le k$ vanish. Moreover, $\mu$-invariants of length $3$ are given by a combination of the Conway polynomials and linking numbers without any vanishing assumption.

The space of Gauss diagram formulas that are knot invariants is introduced by Goussarov–Polyak–Viro in 2000; it is extended to nanophrases by Gibson–Ito in 2011. However, known invariants in concrete presentations of Gauss diagram formulas are very limited, even in the one-component case. This paper gives a recipe to obtain explicit forms of Gauss diagram formulas that are invariants of virtual links with base points or tangles. As an application, we introduce a new construction of Gauss diagram formulas of $3$-bouquets and how to give link invariants that do not change with base point moves, including a reconstruction of the Milnor’s triple linking number.