Computation Of Riemann Matrices Of Hyperbolic Quartic Curves

We classify the graphs which describe the monodromy structure of the Riemann surfaces of smooth quartic hyperbolic curves. The combinatorial classification provides a method to construct the homology basis for computing the Riemann period matrix of the Riemann surface of a hyperbolic quartic curve. We give a smooth quartic curve and produce a simpler Riemann matrix than that of the Vinnikov standard form.