A special ribbon graph presents a cylinder. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T09:08:48Z http://mathoverflow.net/feeds/question/87567 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87567/a-special-ribbon-graph-presents-a-cylinder A special ribbon graph presents a cylinder. knot 2012-02-05T08:15:28Z 2012-02-05T14:39:41Z <p>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.</p> <p>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.</p> <p>I especially don't understand the statement starting "One may check that the cylindrical structures are compatible on the boundary..."</p> <p>Could you show me the detail and/or an intuitive(geometric) proof?</p> http://mathoverflow.net/questions/87567/a-special-ribbon-graph-presents-a-cylinder/87588#87588 Answer by Kevin Walker for A special ribbon graph presents a cylinder. Kevin Walker 2012-02-05T14:39:41Z 2012-02-05T14:39:41Z <p>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 <a href="http://books.google.com/books?id=s4eGEecSgHYC&amp;dq=rolfsen+knots+and+links&amp;source=gbs_navlinks_s" rel="nofollow">"Knots and Links"</a>, or some similar text.</p> <p>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 <code>$S^2\times \{1\}$</code> (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 $Y\times I$.</p> <p>If the above explanation makes no sense to you then you should read Rolfsen.</p>