Ribbonness of a stable-ribbon surface-link, I. A stably trivial surface-link

TL;DR

This paper proves that handle-irreducible summands of stably trivial surface-links are trivial, leading to the conclusion that surface-knots with infinite cyclic fundamental groups are unknotted.

Contribution

It establishes the uniqueness of trivial summands in stably trivial surface-links and connects this to the triviality of certain surface-knots.

Findings

Handle-irreducible summands of stably trivial surface-links are trivial.

Surface-knots with infinite cyclic fundamental group are unknotted.

The result confirms a special case of a broader conjecture about surface-link triviality.

Abstract

There is a question asking whether a handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link. This question for the case of a trivial surface-link is affirmatively answered. That is, a handle-irreducible summand of every stably trivial surface-link is only a trivial 2-link. By combining this result with an old result of F. Hosowaka and the author that every surface-knot with infinite cyclic fundamental group is a stably trivial surface-knot, it is concluded that every surface-knot with infinite cyclic fundamental group is a trivial (i.e., an unknotted) surface-knot.