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Cyclic branched covers of Seifert links and properties related to the $ADE$ link conjecture

Steven Boyer, Cameron McA. Gordon, Ying Hu

Abstract

In this article we show that all cyclic branched covers of a Seifert link have left-orderable fundamental groups, and therefore admit co-oriented taut foliations and are not $L$-spaces, if and only if it is not an $ADE$ link up to orientation. This leads to a proof of the $ADE$ link conjecture for Seifert links. When $L$ is an $ADE$ link up to orientation, we determine which of its canonical $n$-fold cyclic branched covers $Σ_n(L)$ have non-left-orderable fundamental groups. In addition, we give a topological proof of Ishikawa's classification of strongly quasipositive Seifert links and we determine the Seifert links that are definite, resp. have genus zero, resp. have genus equal to its smooth $4$-ball genus, among others. In the last section, we provide a comprehensive survey of the current knowledge and results concerning the $ADE$ link conjecture.

Cyclic branched covers of Seifert links and properties related to the $ADE$ link conjecture

Abstract

In this article we show that all cyclic branched covers of a Seifert link have left-orderable fundamental groups, and therefore admit co-oriented taut foliations and are not -spaces, if and only if it is not an link up to orientation. This leads to a proof of the link conjecture for Seifert links. When is an link up to orientation, we determine which of its canonical -fold cyclic branched covers have non-left-orderable fundamental groups. In addition, we give a topological proof of Ishikawa's classification of strongly quasipositive Seifert links and we determine the Seifert links that are definite, resp. have genus zero, resp. have genus equal to its smooth -ball genus, among others. In the last section, we provide a comprehensive survey of the current knowledge and results concerning the link conjecture.
Paper Structure (28 sections, 74 theorems, 84 equations, 14 figures, 1 table)

This paper contains 28 sections, 74 theorems, 84 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. ADE links and their canonical cyclic branched covers
  3. Seifert links
  4. Left-orderable cyclic branched covers of Seifert links
  5. Strategy and results
  6. A sequence of orbifolds associated to a Seifert link
  7. Seifert links with chi(bar Bn(L))> 0
  8. Seifert links with chi(Bn(L)) = 0
  9. The fundamental group of Bn(L)
  10. Constructing representations of pi1(X(L))
  11. Seifert links with chi(B2(L))< 0
  12. Canonical cyclic branched covers with non-left-orderable fundamental groups
  13. The classification of strongly quasipositive Seifert links
  14. Strategy and results
  15. Alexander polynomials of Seifert links and Seifert links of genus zero
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $L$ be a prime Seifert link that is not an $ADE$ link as an unoriented link. Then, for any cyclic branched cover $\Sigma_\psi(L)$ of $L$, $\pi_1(\Sigma_\psi(L))$ has a non-trivial representation into $\widetilde{PSL_2}(\mathbb R)$. Hence $\Sigma_\psi(L)$ is $LO$, $CTF$, and $NLS$.

Figures (14)

  • Figure 1: The $ADE$ links
  • Figure 2: The link $L(p, q; 1, 1; -)$ with $p=2, q= 3$ and the diagram $D$ of $L(p, q; k, k; -)$ when $p=2, q= 3$ and $k=4$. In the figure, the box with the letter $q$ inside represents $q$ full twists.
  • Figure 3: Band $C_1$ to $C$ which results in the component $C'$ in $L'$. The diagram also shows how to isotope $C'$ in the case that $p<q$.
  • Figure 4:
  • Figure 5:
  • ...and 9 more figures

Theorems & Definitions (116)

  • Theorem 1.1
  • Conjecture 1.2: The $ADE$ Link Conjecture
  • Theorem 1.3: The $ADE$ Link Conjecture for Seifert links
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • proof
  • Corollary 1.9
  • ...and 106 more