Prime theta-curves on minimal genus Seifert surfaces

TL;DR

This paper proves that when a prime knot is combined with an essential arc on a minimal genus Seifert surface, the resulting structure is a prime theta-curve, revealing a new relationship in knot theory.

Contribution

It establishes a novel connection between prime knots, essential arcs, and prime theta-curves on minimal genus Seifert surfaces.

Findings

Prime knots with an essential arc form prime theta-curves.

The result applies specifically to minimal genus Seifert surfaces.

Provides new insights into the structure of knot complements.

Abstract

We prove that each prime knot union an essential arc on a minimal genus Seifert surface is a prime theta-curve.