# Clarification for arXiv:1606.08656

*Geometry and Topology*. Shortly thereafter, Yong-Geun Oh contacted us with some excellent questions where our exposition was less than clear. In case anyone else has the same questions, I thought I would write a blog post clarifying these points.

- Lemma 2.16(c) should say \(T=\mathbb{Z}/(m)\) rather than \(\mathbb{Z}/(\Delta/m)\).
- There is a discrepancy between what we write in Section 2.3:
- ``The generic orbit is homologous to \(p^2e/g\).'' (correct)

- ``the generic Reeb orbits are homologically trivial in \(\Sigma_{p,q}\)'' (not true in general)

I will now explain the two situations in which we needed this fact and how it generalises to arbitrary values of \(p\). In the case when \(p\) is odd (and \(g=1\)), this reduces to what we wrote.

- In Theorem 4.16, we have a holomorphic cylinder which we have
extracted from an SFT neck-stretching argument. One of its ends is
asymptotic to a Reeb orbit on a component of the boundary called
\(\Sigma_{p_2,q_2}\) (a contact lens space). We have already
established that the Reeb orbits fall into two categories:
``exceptional'' and ``generic'' (the Reeb orbits give a Seifert
fibration of the lens space).
**We want to show that this asymptote is to an exceptional orbit.**The homology class of the generic orbit is \(p^2/g\) times the class of the exceptional orbit. The relative homology class of the cylinder is (in notation from the paper) \(D\mathcal{E}\) and the connecting homomorphism to \(H_1(\Sigma_{p_2,q_2};\mathbf{Z})=\mathbf{Z}/(p_2^2)\) is to reduce \(D\) modulo \(p_2^2\). We know that \(D\leq p_2/3\) by Lemma 4.11(B). If the asymptote were generic then \(D\) would reduce modulo \(p_2^2\) to something divisible by \(p_2^2/g\) (where \(g\) is \(1\), \(2\) or \(4\), depending on \(p_2\) by Remark 2.12). But then we get \(p_2^2/g\leq p_2/3\), which implies \(p_2\leq g/3\leq 4/3\) so we don't need to worry about this case as \(p_2\) is at least \(2\).

- In the other situation, if the SFT limit is a plane (the case
\(|Z|=1\) in the paper) and the asymptote is generic then the
homology class of the plane is, again, divisible by \(p^2/g\) and
this contradicts the inequality \(D\leq 2p/3\) from Lemma 4.8.