Twist Triviality of Canonical Seifert Surfaces

TL;DR

This paper introduces the concept of twist triviality for Seifert surfaces, showing that canonical Seifert surfaces can be untwisted using ribbon twists, extending knot unknotting ideas to surfaces.

Contribution

It defines ribbon twists as an operation analogous to crossing changes and proves that canonical Seifert surfaces are twist trivial, broadening understanding of surface isotopy.

Findings

Canonical Seifert surfaces are twist trivial.

Ribbon twists can untwist Seifert surfaces.

Extension of unknotting concepts to surfaces.

Abstract

We generalize the idea of unknotting knots to Seifert surfaces. We define an operation called ribbon twist which serves as the equivalent of a crossing change for knots. A Seifert surface is considered untwisted, the equivalent to unknotted, if it is isotopic to a standardly embedded n-fold punctured torus. A Seifert surface is said to be twist trivial if it can be untwisted by ribbon twists. We show that canonical Seifert surfaces are twist trivial.