Note on the canonical genus of a knot

TL;DR

This paper demonstrates that every canonical Seifert surface can be represented by a knot diagram with pairwise disjoint open Seifert disks, providing a new perspective on the structure of Seifert surfaces.

Contribution

It establishes a characterization of canonical Seifert surfaces via knot diagrams with disjoint open disks, offering a novel geometric insight.

Findings

Every canonical Seifert surface corresponds to a knot diagram with disjoint open disks.

The result simplifies understanding the structure of canonical Seifert surfaces.

Provides a new geometric approach to studying knot genus.

Abstract

We show that every canonical Seifert surface is (up to isotopy) given by a knot diagram in which the (open) Seifert disks are pairwise disjoint.