Milnor's triple linking numbers and derivatives of genus three knots

Abstract

A derivative of an algebraically slice knot $K$ is an oriented link disjointly embedded in a Seifert surface of $K$ such that its homology class forms a basis for a metabolizer $H$ of $K$. We show that for a genus three algebraically slice knot $K$, the set $\{ \bar{\mu}_{\{\gamma_1,\gamma_2,\gamma_3\}}(123) - \bar{\mu}_{\{\gamma'_1,\gamma'_2,\gamma'_3\}}(123)| \{\gamma_1,\gamma_2,\gamma_3\}$ and $\{\gamma'_1,\gamma'_2,\gamma'_3\}$ are derivatives of $K$ associated with a metabolizer $H\}$ contains $n\cdot \mathbb{Z}$ where $n$ is determined by a Seifert form of $K$ and a metabolizer $H$. As a corollary, we show that it is possible to realize any integer as a Milnor's triple linking number of a derivative of the unknot on a fixed Seifert surface with a fixed metabolizer. In addition, we show that a knot, which is a connected sum of three genus one algebraically slice knots, has at least one derivative which has non-zero Milnor's triple linking number.