On the triple point number of surface-links in Yoshikawa's table

TL;DR

This paper analyzes the minimal triple point numbers of surface-links listed in Yoshikawa's table, providing calculations and bounds to advance understanding of knotted surfaces in four-dimensional space.

Contribution

It compiles known triple point numbers and establishes bounds for remaining surface-links, enhancing the classification of knotted surfaces.

Findings

Known triple point numbers are compiled.

Bounds are provided for remaining surface-links.

Enhanced understanding of surface-link complexity in R^4.

Abstract

Yoshikawa made a table of knotted surfaces in R^4 with ch-index 10 or less. This remarkable table is the first to enumerate knotted surfaces analogous to the classical prime knot table. A broken sheet diagram of a surface-link is a generic projection of the surface in R^3 with crossing information along its singular set. The minimal number of triple points among all broken sheet diagrams representing a given surface-knot is its triple point number. This paper compiles the known triple point numbers of the surface-links represented in Yoshikawa's table and calculates or provides bounds on the triple point number of the remaining surface-links.