Corrigendum to “Taut Foliations, Left-Orderability, and Cyclic Branched Covers”
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.
KeywordsTaut foliation Left-orderability Cyclic branched cover
Mathematics Subject Classification (2010)57M12 57M25 57R30 06F15
Corrigendum to: Acta Math Vietnam (2014) 39(4):599-635
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 , 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 ). Note that the first property implies that Σ n (C p,q (K)) is not an L-space [1, 6, 9]. In , 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.
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.
In Section 4.2 of , we showed that Σ2(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.
If K is a torus knot or iterated torus knot, then Σ2(C p,2(K)) is a graph manifold. By  and , 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. □
The mistake in  occurs in the last three sentences of the proof of Theorem 1.4: clearly ∞ is a foliation-detected slope.
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.