# contact structure on double branched covers of $S^3$

We can assign a natural contact structure to double branched cover of a transverse knot $k$ in the $3$-sphere with its standard contact structure $(S^3,\xi_{st})$, as described in:http://arxiv.org/pdf/math/0412183.pdf. I want to know how to get a contact surgery diagram of this contact manifold. In the same paper they give a proof but it is quite unclear to me. Do you know of a reference where this is worked out more clearly?

• Maybe you could tell what exactly you find unclear about that paper? – Marco Golla Jan 20 '16 at 9:02
• yes, I should have explained it; they start with braiding the transverse knot and writing down the braid word in standard generators of the braid group. Then each $\sigma_i$ (and its inverse) in the braid word corresponds to a contact surgery on an unknot. So far the explanation is clear, but then I cannot figure out how these surgery unknots link. They refer to antoher paper, but I almost could not find a clear proof there either. – nikita Jan 20 '16 at 18:10
• These unknots lie on (distinct!) pages of the open book, and the order in which you perform the Dehn twists give you the order of the pages on which they lie. Does this help? – Marco Golla Jan 20 '16 at 23:15
• so basically you are getting from the open book campatible with the contact structure to its surgery diagram. I know this is done in another paper of the same author, maybe this gives a better way of understanding the linking. – nikita Jan 21 '16 at 3:08
• Yes, precisely, you go from the open book to the surgery diagram. – Marco Golla Jan 21 '16 at 9:22