New classes of minimal knot diagrams

TL;DR

This paper introduces new classes of minimal link diagrams, including certain alternating and homogeneous diagrams, and a larger class with minimal Seifert circles, using the Morton-Franks-Williams inequality.

Contribution

It defines novel classes of minimal link diagrams and establishes their minimality through the Morton-Franks-Williams inequality, expanding understanding of link diagram minimality.

Findings

Identified new classes of minimal link diagrams including some with previously unproven minimality.

Proved minimality of certain homogeneous diagrams and standard torus link diagrams.

Established a larger class of diagrams with minimal Seifert circles for given links.

Abstract

We describe a new class of minimal link diagrams. This class includes certain alternating diagrams, the standard diagrams of all torus links, and numerous homogeneous diagrams whose minimality has not been proven before. Besides, we describe a new larger class of link diagrams with the least number of Seifert circles among all diagrams of a given link. Our approach refers to the Morton-Franks-Williams inequality.