Acta Mathematica Vietnamica

, Volume 42, Issue 4, pp 775–776

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

• Cameron Gordon
• Tye Lidman
Article

## 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 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 Σ2(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, Σ2(C p,2(K))is not an L-space. If K is a torus knot or iterated torus knot, then Σ2(C p,2(K))is excellent.

### Proof

In Section 4.2 of [3], we showed that Σ2(C p,2(K))≅X 0X 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 pa/b on X 1. In particular, the meridians of X 0 and X 1 are identified.

Let $$\mathcal {L}(X)$$ be the set of L-space slopes on 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],
$$\mathcal{L}(X) = \{\infty \}, \quad [2g-1,\infty], \quad \textup{or} \quad [- \infty, 1-2g],$$
where 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], Σ2(C p,2(K)) is not an L-space.

If K is a torus knot or iterated torus knot, then Σ2(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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