Minimality of the constructed triangulations for double twist knots

Prove that for all integers p,q ≥ 2, the explicit ideal triangulations T_{K(p,q)} of the complements of the double twist knots K(p,q) constructed via layered solid tori in Section 3 realise the triangulation complexity of these manifolds; that is, show that each T_{K(p,q)} uses the minimal number of tetrahedra among all ideal triangulations of S^3 − K(p,q).

Background

The paper constructs two explicit triangulations for each double twist knot complement K(p,q): one that agrees with the canonical (Sakuma–Weeks) triangulation, and another that the authors prove is geometric for all p,q ≥ 2. The latter is built by triangulated Dehn fillings using layered solid tori and is designed to use fewer tetrahedra than the canonical triangulation.

SnapPy census data provides supporting evidence for small (p,q), and for larger parameters the triangulation arises by adjusting a layered solid torus in a Dehn-filled manifold. Despite this evidence, the minimality has not been proven in general, leading to the authors’ explicit conjecture stated below.