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EDIT: In an attempt to not let the bounty go to waste, I will consider responses that give reasonable guesses of what the involved surfaces are, WITHOUT requiring parametrizations.

While using Mathematica to alter manifolds and numerically verify the Gauss-Bonnet Theorem, I generated the figure whose pictures I've attached in this post (multiple perspectives of the same shape are provided).

Does anyone know whether it is a known manifold with a name?

I generated this surface by using $h_{x}, h_{y}, h_{z}$ below, setting $t_{2}=t=1, M=0, \phi=\pi/2$ on the interval $0 \leq k_{x} \leq 2\pi$ and $0 \leq k_{x} \leq 2\pi$:

hamiltonian

After much reading, my biggest guess is that this is a pseudosphere glued to a duplin cyclide - however, I am having trouble knowing this for sure. I do not even know how I would go about proving this.

Note that I plotted some of the normal vectors on its surface and it appears that normal vectors in the pseudosphere-like region point INTO the surface, whereas other normal vectors point out. Perhaps this could be useful information.

Since this is a question with a bounty on it now, I should know what this 2D surface is with certainty.

For your convenience, I have a link to MATLAB and Mathematica files that have manipulable figures here: https://1drv.ms/f/s!Ak6chxAgMs9Pg_tYixvFy6MfO9531A

Here are some animations that show sections of the surface (I change parameters in a fixed frame so that you get certain perspectives inside the surface as it leaves the fixed frame).

enter image description here enter image description here

Here are some static images (essentially screenshots from the models in the link above): enter image description here enter image description here enter image description here enter image description here enter image description here enter image description here

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  • $\begingroup$ It would be helpful if you described how this surface is defined. The pictures are somewhat muddy (both due to the projections onto the two-dimensional space that is my monitor, and due to the choice of colors), and a clear, precise definition would go a long way... $\endgroup$
    – Xander Henderson ♦
    Aug 30, 2018 at 22:43
  • $\begingroup$ @XanderHenderson, thank you for the suggestion. I just edited my post to reflect those. A note about the colors, it was Mathematica's 'white' (with adjusted opacity) that gave me the clearest figures, given how the figure looked blurry because of it took into account each of the ~30000 coordinates it was generated from. $\endgroup$
    – TribalChief
    Aug 30, 2018 at 23:00
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    $\begingroup$ An image of an immersion of a surface in $\Bbb{R}^3$ that is not an imbedding doesn't give enough information to determine the surface. $\endgroup$
    – Rob Arthan
    Aug 30, 2018 at 23:04
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    $\begingroup$ @minimax, this is a Hamiltonian for Haldane's model (Fruchart et al.). I will try to generate a GIF later today! $\endgroup$
    – TribalChief
    Sep 23, 2018 at 0:55
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    $\begingroup$ Thanks! I got this paper: "An Introduction to Topological Insulators" Michel Fruchart, David Carpentier. This seems really nice stuff. $\endgroup$
    – minmax
    Sep 23, 2018 at 1:00

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