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

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.