The triple point number of surface-knots of genus one is at least four

TL;DR

This paper proves that surface-knots of genus one cannot have a triple point number invariant of three, establishing a lower bound of four for this class of knots.

Contribution

It demonstrates that the triple point number of genus one surface-knots is at least four, filling a gap in the understanding of their minimal triple point configurations.

Findings

No genus one surface-knot has triple point number three.

The minimal triple point number for genus one surface-knots is at least four.

Provides a new lower bound for the triple point number in surface-knot theory.

Abstract

In this paper, we show that there is no surface-knot of genus one with triple point number invariant equal to three.