Flat plumbing basket, self-linking number and Thurston-Bennequin number

TL;DR

This paper explores the relationship between flat plumbing baskets, Legendrian links, and classical invariants like self-linking and Thurston-Bennequin numbers, providing new insights and calculating flat plumbing basket numbers for torus links.

Contribution

It introduces a method to construct Legendrian links from flat plumbing baskets and establishes relations among key link invariants, also determining flat plumbing basket numbers for torus links.

Findings

Established a relation among self-linking, Thurston-Bennequin, and flat plumbing basket numbers.

Constructed Legendrian links from flat plumbing baskets.

Determined flat plumbing basket numbers for torus links.

Abstract

A flat plumbing basket is a surface consisting a disk and finitely many bands which are contained in distinct pages of the trivial open book decomposition of $\mathbf{S}^{3}$. In this paper, we construct a Legendrian link from a flat plumbing basket, and we describe a relation among the self-linking number, the Thurston-Bennequin number and the flat plumbing basket number of the link. As a corollary, we determine the flat plumbing basket numbers of torus links.