Baskets and fibred links realizing $A_{n}$

TL;DR

This paper characterizes basket links with symmetrized Seifert form matching the $A_{n}$ Cartan matrix, showing they are isotopic to the torus link $T(2,n+1)$, and provides examples of similar links that are not isotopic.

Contribution

It proves the isotopy of certain basket links to torus links and constructs examples of non-isotopic links with the same Seifert form.

Findings

Basket links with symmetrized Seifert form congruent to $A_{n}$ are isotopic to $T(2,n+1).

Examples of links with the same form are not isotopic to the torus link.

The paper distinguishes between links based on their isotopy class despite similar algebraic invariants.

Abstract

We prove that basket links, whose symmetrized Seifert form is congruent to the Cartan matrix of the simply laced Dynkin diagram $A_{n}$, are isotopic to the torus link $T(2,n+1)$. In addition, we provide examples of links, constructed by plumbing $n$ positive Hopf bands the core curves of which intersect at most once, with symmetrized Seifert form congruent to the Cartan matrix $A_{n}$, that are not isotopic to $T(2,n+1)$.