User manifold-destiny - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T06:26:06Z http://mathoverflow.net/feeds/user/11621 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49628/when-are-nontopological-bistellar-flips-manifold-preserving When are nontopological bistellar flips manifold-preserving? manifold-destiny 2010-12-16T10:17:44Z 2010-12-22T23:31:44Z <p>A <em>topological bistellar flip</em> is the term used by <a href="http://www.springerlink.com/content/yjf3gpbej4yr71ml/" rel="nofollow">Dougherty, Faber, and Murphy</a> to describe a bistellar flip that does not cause any face of a complex to be duplicated. Suppose we consider a generalized notion of bistellar flips that are non-topological, ie, they can duplicate existing faces of complexes. For instance, on a 3-simplex, a (2,2) non-topological flip would map one edge of the tetrahedron into the edge it does not intersect.</p> <p>Is this notion already defined in the literature? Which classes of nontopological bistellar flips are known to preserve the type of a manifold? It seems likely to me that in low dimensions a good deal may already be known.</p> http://mathoverflow.net/questions/50100/whats-the-best-way-to-test-if-a-sphere-is-a-polytope-algorithms-for-the-simpli What's the best way to test if a sphere is a polytope? (algorithms for the Simplicial Steinitz Problem) manifold-destiny 2010-12-21T22:33:13Z 2010-12-21T22:33:13Z <p>The problem of recognizing whether a simplicial face lattice is polytopal is sometimes called the <a href="http://www.math.tu-bs.de/~pfetsch/apropo/simplicial_steinitz_problem.html" rel="nofollow">Steinitz problem</a>.</p> <p>Sturmfels and Bokowski advanced a <a href="http://www.springer.com/mathematics/geometry/book/978-3-540-50478-8" rel="nofollow">set of methods</a> in the late 80s to test whether the face lattice of a simplicial sphere was also realizable as a polytope. </p> <p>The method uses oriented matroids. The problem is NP-hard, so their algorithm requires exponential time in the worst case, but they reported that the algorithm often converged quickly.</p> <p>In the intervening two decades, I'm sure that newer approaches have been developed. Is there a better method known today? More interestingly, are there any software implementations available that solve this problem -- even using the older approach?</p> http://mathoverflow.net/questions/50100/whats-the-best-way-to-test-if-a-sphere-is-a-polytope-algorithms-for-the-simpli Comment by manifold-destiny manifold-destiny 2010-12-22T02:05:32Z 2010-12-22T02:05:32Z Thanks! However, I believe that computing the geometric realization of a simplicial sphere (which is what it looks like Polymake does) is not sufficient to guarantee polytopality. There are known instances where a complex is realizable as a sphere, but where the dual is nonpolytopal. http://mathoverflow.net/questions/49628/when-are-nontopological-bistellar-flips-manifold-preserving Comment by manifold-destiny manifold-destiny 2010-12-16T19:36:13Z 2010-12-16T19:36:13Z Perhaps I am describing the wrong operation. Consider a 3-simplex to which we apply a (2,2) bistellar flip. The operation drags several edges together. This would seem to collapse a 2-sphere into a 1-sphere.