On surface-knots with triple point number at most three

TL;DR

This paper investigates surface-knots with diagrams containing up to three triple points, demonstrating that their cocycle invariant is an integer and establishing a lower bound on triple point number for genus one surface-knots.

Contribution

It proves that surface-knots with at most three triple points have integer cocycle invariants and sets a minimum triple point number for genus one surface-knots.

Findings

Cocycle invariant is integer for diagrams with ≤3 triple points

Genus one surface-knots have triple point number at least four

Provides constraints on surface-knot invariants based on diagram complexity

Abstract

In this paper, we show that if a diagram of a surface-knot $F$ has at most three triple points, then the cocyle invariant of $F$ is an integer. In particular, for a surface-knot of genus one, the triple point number invariant is at least four.