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I am reading "Quantum Invariants of Knots and 3-Manifolds" by Turaev. I have a dificulty to understand the proof of Lemma 2.6 on page 172.

The lemma says that a special ribbon graph drawn on page 167 presents a cylinder. I am sorry that I don't know how to show that ribbon graph here.

I especially don't understand the statement starting "One may check that the cylindrical structures are compatible on the boundary..."

Could you show me the detail and/or an intuitive(geometric) proof?

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up vote 5 down vote accepted

Turaev's book assumes familiarity with basic 3-dimensional geometric topology and especially Dehn surgery presentations of 3-manifolds. If you want to understand all the details in Tureav's book, then I strongly recommend first reading Rolfsen's "Knots and Links", or some similar text.

It's hard to explain without pictures, but briefly: Start with $S^2\times I$. Remove a regular neighborhood of the arcs (not the loops) of the tangle in Figure 2.4. Do Dehn surgery along the (framed) loops. The boundary of the resulting 3-manifold is the union of a "vertical" annulus for each straight arc of the tangle and "upper" and "lower" surface. The upper surface contains $S^2\times \{1\}$ (minus some disks) and an annulus for each curved arc of the tangle. Call this surface $Y$. Then the 3-manifold, after Dehn surgery, is homeomorphic to $Y\times I$. Turaev's Figure 2.5 shows a 3-punctured disk which, after Dehn surgery, becomes an instance of curve$\times I$ inside $Y\times I$.

If the above explanation makes no sense to you then you should read Rolfsen.

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Thank you for the answer. I took a look at Rolfsen and I think I understand the result of the surgery is the cylinder $\Sigma \times I$. But I don't see the parametrization of the upper boundary is the identity. By the construction, the top parametrization is the compositon of mir and $f^+: -R_{t^+} \rightarrow \Sigma^+ $ (by the book's notation on p.159 & 168). How does the surgery affect the parametrization? I guess we need one more mir to have the identity parametrization. – knot Feb 10 '12 at 5:48
I don't have a detailed answer for your question about parameterizations, but in general you want to identify annuli of the form (curve)$\times I$ in $\Sigma\times I$ (such as the one in Turaev's Figure 2.5) and then use these annuli to construct an identification between pants decompositions of the two boundary components of $\Sigma\times I$. – Kevin Walker Feb 10 '12 at 17:19
I reduced this question to the new question Could you take a look at it and if you have time to write answer, please do so. Thanks. – knot Feb 13 '12 at 8:11

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