As part of my work with Ben Burton (see arXiv:2403.11659), I have developed an implementation of an algorithm to determine the Heegaard genus of a triangulation of a 3-manifold.
This can currently be accessed on Github, but will (likely) be implemented into Regina. This also includes a .csv file of the computed genus for the first 11,031 triangulations in the closed hyperbolic census of Hodgson and Weeks.
The code also includes an implementation of the gadget discussed in the aforementioned paper.
In its current state, given the isomorphism signature of a triangulation of a closed hyperbolic 3-manifold, it determines the Heegaard genus (or, a tight upper bound).
I have made a small tool to create a rotate-able model of a tetrahedron with normal discs given by normal coordinates. It allows the model to be saved as an .svg. I currently have it hosted on CodePen, but be warned that it can be temperamental. It is also embedded here.