The complete list of prime knots whose flat plumbing basket numbers are 6 or less

TL;DR

This paper classifies all prime knots with a flat plumbing basket number of six or less, enhancing previous classifications up to nine crossings and providing a complete understanding of these knots' geometric properties.

Contribution

The paper provides the first complete classification of prime knots with flat plumbing basket numbers at most six, using sequential presentations to improve existing knot classifications.

Findings

Classified all prime knots with flat plumbing basket number ≤ 6.

Extended previous work to include knots with up to 9 crossings.

Provided a comprehensive framework for analyzing flat plumbing basket surfaces.

Abstract

Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. These flat plumbing basket surface can be presented by a sequential presentation known as flat plumbing basket code first found by Furihata, Hirasawa and Kobayashi. The minimum number of flat plumbings to obtain a flat plumbing basket surfaces of a link is defined to be the flat plumbing basket number of the given link. In present article, we use these sequential presentations to find the complete classification theorem of prime knots whose flat plumbing basket number $6$ or less. As applications, this result improves the work of Hirose and Nakashima which finds the flat plumbing basket number of prime knots up to 9 crossings.