# Corrigendum to “Taut Foliations, Left-Orderability, and Cyclic Branched Covers”

## Abstract

We correct an error in the statement and proof of Theorem 1.4 of our paper in Acta Mathematica Vietnamica (2014) 39(4), 599-635.

## Keywords

Taut foliation Left-orderability Cyclic branched cover## Mathematics Subject Classification (2010)

57M12 57M25 57R30 06F15 **Corrigendum to: Acta Math Vietnam (2014) 39(4):599-635**

**DOI 10.1007/s40306-014-0091-y**

Let *C* _{ p,q }(*K*) be the (*p*,*q*)-cable of a non-trivial knot *K* in *S* ^{3}, where *q* > 1 denotes the longitudinal winding, and let Σ_{ n } (*C* _{ p,q }(*K*)), *n* ≥ 2, be its *n*-fold cyclic branched cover. In Theorem 1.3 of [3], we showed that, unless *n* = *q* = 2, Σ_{ n }(*C* _{ p,q }(*K*)) has a co-orientable taut foliation and *π* _{1}(Σ_{ n }(*C* _{ p,q }(*K*))) is left-orderable (i.e., Σ_{ n }(*C* _{ p,q }(*K*)) is *excellent* in the terminology of [3]). Note that the first property implies that Σ_{ n }(*C* _{ p,q }(*K*)) is not an L-space [1, 6, 9]. In [3], we claimed that the conclusion of Theorem 1.3 fails in general when *n* = *q* = 2. Specifically, in Theorem 1.4, we asserted that it fails for Σ_{2}(*C* _{ p,2}(*K*)) where *K* is the right-handed trefoil and *p* ≥ 3. This is incorrect. In fact, we have the following.

### **Theorem 1**

*Let* *C* _{ p,2}(*K*)*be the* (*p*,2)*-cable* *of a non-trivial knot* K *in* *S* ^{3} *.* *Then,* Σ_{2}(*C* _{ p,2}(*K*))*is not an L-space. If* K *is a torus knot or iterated torus knot, then* Σ_{2}(*C* _{ p,2}(*K*))*is excellent.*

### *Proof*

In Section 4.2 of [3], we showed that Σ_{2}(*C* _{ p,2}(*K*))≅*X* _{0} ∪ *X* _{1}, where *X* _{0} and *X* _{1} are copies of the exterior X of K, glued along their boundaries so that a slope *a*/*b* on *∂* *X* _{0} is identified with the slope *p* − *a*/*b* on *∂* *X* _{1}. In particular, the meridians of *X* _{0} and *X* _{1} are identified.

*∂*

*X*, i.e., \(\mathcal {L}(X) = \{\alpha \mid X(\alpha ) \text { is an L-space}\}\), where

*X*(

*α*) is

*α*-Dehn filling on X. By [7, 8],

*g*is the genus of

*K*. Here, the intervals are to be interpreted as being in \(\mathbb {Q} \cup \{\infty \}\). Hence, the meridian

*μ*=

*∞*of

*K*is not in the interior \(\mathcal {L}^{o}(X)\). Therefore, by [5], Σ

_{2}(

*C*

_{ p,2}(

*K*)) is not an L-space.

If *K* is a torus knot or iterated torus knot, then Σ_{2}(*C* _{ p,2}(*K*)) is a graph manifold. By [2] and [4], for graph manifolds, the properties of not being an L-space, having a co-orientable taut foliation, and having left-orderable fundamental group, are equivalent. This proves the second part of the theorem. □

### *Remark 1*

The mistake in [3] occurs in the last three sentences of the proof of Theorem 1.4: clearly *∞* is a foliation-detected slope.

## Notes

### Acknowledgements

The first author would like to thank Steve Boyer and Jonathan Hanselman for helpful conversations. The first author was partially supported by NSF Grant DMS-1309021.

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