Narrow Results

We first introduce the null-homotopically peripheral quadratic function of a surface-link to obtain a lot of pseudo-ribbon, non-ribbon surface-links, generalizing a known property of the turned spun torus-knot of a non-trivial knot. Next, we study the torsion linking of a surface-link to show that the torsion linking of every pseudo-ribbon surface-link is the zero form, generalizing a known property of a ribbon surface-link. Further, we introduce and algebraically estimate the triple point cancelling number of a surface-link.

We focus an interest on the torsion linking of a surface-knot. It is a knot invariant independent of the surface-knot group and its peripheral subgroup. It is identified with the torsion linking of any associated closed 4-manifold with infinite cyclic first homology. In the case of such a 4-manifold with an exact leaf, the linking of the leaf is identified with an orthogonal sum of it and a hyperbolic linking.