Hey Simon, > > On 2015-11-05, Travis Scrimshaw <tsc...@ucdavis.edu <javascript:>> wrote: > > I would then be advocating for using UniqueRepresentation if that was > > the only issue. > > It really depends whether in comparing manifolds you would prefer to do > *some* > heuristics to detect homeomorphic manifolds, or prefer to consider > manifolds unequal if they were created with different input data. >
IMO, such checks are too expensive to be done at creation. For example, you want do so some tests over all n-spheres. > > > However the problem I see is this (even for using > > CachedRepresentation). Suppose you created a 2-sphere called 'S', and > then > > you created a manifold called 'M' and worked with that for a while, took > a > > lunch break, came back and did some more work. Then you wanted to create > a > > genus 3 surface called 'S' in the same session. All of a sudden, to your > > surprise, you have charts that tell you 'S' is a sphere (because you > forgot > > you had one previously defined). This also means we have to have > > `_clear_cache_()` calls all throughout the doctests. > > Pardon? The variable name of a manifold will certainly not be part of the > input > data. > > What I rather mean is this: > The three-dimensional sphere can be constructed, for example, > - as the boundary of the four-dimensional unit ball, > - by gluing two 3-balls together, > - by a Heegaard diagram (infinitely many possibilities) > > 1. Do you want that all possible ways to construct a 3-sphere yield the > same > object? Not feasible. > > 2. Do you want that distinct constructions yield distinct (but potentially > homeomorphic) objects, whereas two equal Heegaard diagrams will result > in > the same object? Feasible. > Then, do you want that comparison of manifolds does some heuristic of > solving the homeomorphism problem? > a) If you do, then use CachedRepresentation > b) If you don't, use UniqueRepresentation > > 3. Assume that you construct a three-manifold out of a Heegaard diagram, > and then do the same construction again. Would you accept that the > result of the two constructions are two distinct objects? Then don't > use CachedRepresentation. Do you, in addition, accept that the two > distinct objects evaluate unequal, even though they obviously are > equal? Then you can use EqualityById. > > Granted, there are nice and well-defined inputs for 2d and 3d manifolds, which would make good sense to do Cached/UniqueRepresentation around these structures. For a general (topological) manifold where there are not necessarily such nice inputs, what would you give as input data? Side note, I think it would be good to have a class for Heegaard diagrams and the associated 3 manifolds if there was a feasible way of working with these on a computer. Best, Travis -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
