[SnapPea-planning] snappea request (fwd)

[Saul Schleimer]

Dear Nathaniel -

  Nathan mentioned to me your plan to start working on SnapPea.  Let me
just say that this is a wonderful idea.  I've read the archived messages
with great interest, and subscribed to the list.
  Let me make two requests of the future SnapPea:

1. [Easy] Finer control of cusp settings: In the Cusp Neighborhoods
window, there is a slider for each cusp.  The maximum setting for each
slider is where that cusp first bumps itself.  The minimum setting is
where the cusp has volume 3 sqrt(3) / 16.  SnapPea then computes (and
shows in the cusp viewing window) the canonical triangulation with respect
to the given cusp settings.

When there is only one cusp this is a non-issue: there is only one
canonical triangulation.  When there is more than one cusp there are
finitely many canonical triangulations.  (If you play with the sliders
while looking at the cusp viewing window you can see the retriangulation
happening.)  The current minimum settings of the slider prevents us from
seeing all of the canonical triangulations.  I am particularly interested
in seeing the "stable" canonical triangulation - ie the one that you get
as the blue cusp volume goes to zero.

[Example: look at m295, from the red cusp, with blue on the minimal
setting: the canonical triangulation has vertices on the blue cusp
of height 0.631703 and of height 0.427413.  If we shrank the blue cusp the
canonical triangulation would only have vertices on the larger.  This can
be seen via a nice trick shown me by Futer and Gueritaud - take a double
cover and look from red with blue on minimal setting.  This gives the
correct canonical triangulation.]

Implementation: There is no minimum setting that will work for all
manifolds.  But allowing the cusp volume go to 3 sqrt(3) / 32 certainly
deals with all examples I've looked at.  Perhaps the correct thing to do
is leave the default setting where it is and provide a window where the
user can type in a smaller value (ie a negative one.)

2. [Harder] A window to view Dehn surgery space for the k^th cusp of the
manifold.  Perhaps click on a point and the vital statistics of the filled
manifold are shown.  In this window there should be little dots on the
integral points and the "good region" (where solution_type =
'all tetrahedra positively oriented' or 'contains negatively oriented
tetrahedra') should be in white while the bad region should be in black.
In particular this gives a visual way to see the set of exceptional
surgeries.  The shape of the bad region reflects very interesting
questions about the representation variety of the manifold.

Easy implementation: for every pixel reinitalize the manifold and compute.
(This is very slow, but looks very accurate and is dead easy to code.)

Irritating implementation: Work in polar coordinates - for every radial
ray reinitalize and then compute for every pixel along the ray, working
from the boundary of the viewing window towards zero.  As soon as you
enter the bad region, go on to the next ray.  (This is much faster
and hopefully still accurate.  Trickier to code.)

Very irritating implementation: crawl along the boundary of the bad
region.  Fill in the disk bounded by the curve you find.  This might be
lightning fast, but who knows how stable it will be: see the examples
below.

Here is a bit of ASCII art - the bad region for m004:

    ######################################
      ##################################
       ################################
       ##########            ##########
        ############ #####  ##########
        #########   ######   #########
       ##########            ##########
       ################################
      ##################################
    ######################################

and for m378

                                              #
                                            ###
                                          #####
                                        #######
                                      #########
                                    ###########
           ##                   ###  ##########
             #####################   ##########
             ####################   ###########
             ###################    ###########
              ##################    ###########
              ##################    ###########
              ##################    ########## #
              ##################   ##########  #
              ##################   # ######    #
              ##########  #######     ####    ##
              ##########    #       # ##      ###
              ##########     ####  ##       #####
               #########     # # ##      #########
                ###########  #  ###################
                 ############  #######  ###########
                  ########## # ###### #   ##########
                  #######     ## ######     #########
                   ####     #  ##     #     ##########
                   ##      ##           #   ##########
                   ##    ####   #  ######## ##########
                    #   ########   ###################
                    # ##########    ##################
                    ############    ##################
                    ############    ##################
                    ############    ##################
                    ###########     ##################
                    ###########   #####################
                    ###########  #######################
                    #############                       #
                    ##########
                    ########
                    ######
                    ####
                    ##

best,

saul