Thurston described the figure eight knot complement by identifying faces and edges of two tetrahedra here. Just prior to that he briefly introduced Heegaard decomposition. Is there a method to get Heegaard splittings from the tetrahedron identification?
If I'm not mistaken there is an algorithm to go from surgery diagrams of knots to Heegaard decompositions so I suppose you could go from tetrahedra to knots to Heegaard decompositions but my intuition is telling me it should be possible to go straight from tetrahedra identification to Heegaard decomposition.
Yes. Take the union of the knot with the $1$-skeleton of the triangulation. A regular neighborhood of this is one handle body, the closure of the complement is the other. The knot excised yields a Heegaard splitting of the knot complement.