In this talk, we discuss two combinatorial objects used to study knots: knot diagrams and ideal triangulations. We explain how the octahedral decomposition connects these two objects and describe the process of collapsing it to an ideal triangulation of the knot complement. Finally, I will explain how this construction can be used to prove the 1-loop conjecture for hyperbolic two-bridge knots. This is joint work with Stavros Garoufalidis.