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Acta Mathematica Vietnamica

, Volume 42, Issue 4, pp 775–776 | Cite as

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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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Texas at AustinAustinUSA
  2. 2.Department of MathematicsNorth Carolina State UniversityRaleighUSA

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