We describe a new method of producing equations for the canonical component of representation variety of a knot group into PSL_2(C). Unlike known methods, this one does not involve any polyhedral decomposition or triangulation of the knot complement, and uses only a knot diagram satisfying a few mild restrictions. This gives a simple algorithm that can often be performed by hand, and in many cases, for an infinite family of knots at once. The algorithm yields an explicit description for the hyperbolic structures (complete or incomplete) that correspond to geometric representations of a hyperbolic knot. As an illustration, we will give the formulas for the equations for the variety of an infinite family of closed alternating braids. As another illustration, we will show how this approach helps to compute factors of the A-polynomial.