Surface-link families with arbitrarily large triple point number

TL;DR

This paper demonstrates that for non-split surface-links with multiple trivial components, the minimal number of triple points in their broken sheet diagrams can be made arbitrarily large, generalizing previous results.

Contribution

It generalizes Oshiro's family to show surface-links with arbitrarily many trivial components can have arbitrarily large triple point numbers.

Findings

Existence of non-split surface-links with arbitrarily large triple point numbers

Construction of surface-links with specified triple point complexities

Extension of previous results to broader classes of surface-links

Abstract

Analogous to a classical knot diagram, a surface-link can be generically projected to 3-space and given crossing information to create a broken sheet diagram. The triple point number of a surface-link is the minimal number of triple points among all broken sheet diagrams that lift to that surface-link. This paper generalizes a family of Oshiro to show that there are non-split surface-links of arbitrarily many trivial components whose triple point number can be made arbitrarily large.