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On the 3-dimensional invariant for cyclic contact branched coverings.pdf
Topology and its Applications 216 (2017) 1–7 Contents lists available at ScienceDirect
Topology and its Applications
/locate/topol
On the 3-dimensional invariant for cyclic contact branched coverings
Tetsuya Ito 1
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama Toyonaka, Osaka 560-0043, Japan
article info
Article history: Received 27 July 2016 Accepted 8 November 2016 Available online 15 November 2016
MSC: primary 57M27 secondary 53D35, 57R17
Keywords: Contact branched covering 3-dimensional invariant
abstract
We give a formula of the 3-dimensional invariant for a cyclic contact branched covering of the standard contact S3.
? 2016 Elsevier B.V. All rights reserved.
1. Introduction
Let M → M be a branched covering of a 3-manifold M , branched along a link K ? M . When M has a contact structure ξ and K is a transverse link in the contact 3-manifold (M, ξ), M has a contact structure ξ which is a perturbation of the pull-back π?ξ. Such a contact structure is unique up to isotopy, and we call the contact 3-manifold (M , ξ) the contact branched covering of (M, ξ), branched along the transverse link K.
Let (M, ξ) be a p-fold cyclic contact branched covering of (S3, ξstd) (the standard contact S3), branched along a transverse link K. In [5, Theorem 1.4], it is shown that the Euler class e(ξ) is zero, and the 3-dimensional invariant d3(ξ) ∈ Q (see [3] for de?nition) only depends on a topological link type of K and its self-linking number. However, no explicit formula of d3(ξ) had been given and it is not an easy task to compute d3(ξ) when p is large or K is complicated.
E-mail address: tetito@math.sci.osaka-u.ac.jp. URL: http://www.math.sci.osaka-u.ac.jp/~tetito/. 1 The author was partially supported by JSPS KAKENHI, Grant Number 15K17540.
/10.1016/j.topol.2016.11.007 0166-8641/? 2016 Elsevier B.V. All rights reserved.
2 T. Ito / Topology and its Applications 216 (2017) 1–7
Fig. 1. Page S of the open book (S, ψ) inside S3.
In this n
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