By braiding non-Abelian anyons it is possible to realize fault-tolerant quantum algorithms through the computation of Jones polynomials. So far, this has been an experimentally formidable task. In this Letter, a photonic quantum system employing two-photon correlations and nondissipative imaginary-time evolution is utilized to simulate two inequivalent braiding operations of Majorana zero modes. The resulting amplitudes are shown to be mathematically equivalent to Jones polynomials. The high fidelity of our optical platform allows us to distinguish between a wide range of links, such as Hopf links, Solomon links, Trefoil knots, Figure Eight knots and Borromean rings, through determining their corresponding Jones polynomials. Our photonic quantum simulator represents a significant step towards executing fault-tolerant quantum algorithms based on topological quantum encoding and manipulation.