Three-page indices of torus links

TL;DR

This paper explores three-page presentations of links, providing an alternative proof that all links admit such representations, and calculates the exact three-page indices for certain torus links.

Contribution

It offers an alternative proof that every link can be represented in three pages and defines the three-page index, applying it to torus links to find exact values.

Findings

Every link admits a three-page presentation.

Defined the three-page index $eta_3(L)$ for links.

Determined exact three-page indices for several torus links.

Abstract

An arc presentation of a link is an embedding into the open book decomposition of $\mathbb{R}^3$ with a finite number of pages. An important rule of arc presentations is that different arcs must be placed on separate pages. In 1999, Dynnikov proposed a three-page presentation that bends this rule by restricting the total number of pages to three. Dynnikov showed that every link admits a three-page presentation. In this paper, we provide an alternative proof of this result. Also we define the three-page index $\alpha_3(L)$ of a link $L$ that the minimum number of arcs needed to represent $L$ in a three-page presentation. We examine three-page presentations for torus links, leading to the determination of the exact three-page indices for several torus links.