Natural and artificially created light fields in three-dimensional space contain lines of zero intensity, known as optical vortices. Here, we describe a scheme to create optical beams with isolated optical vortex loops in the forms of knots and links using algebraic topology. The required complex fields with fibred knots and links are constructed from abstract functions with braided zeros and the knot function is then embedded in a propagating light beam. We apply a numerical optimization algorithm to increase the contrast in light intensity, enabling us to observe several optical vortex knots. These knotted nodal lines, as singularities of the wave's phase, determine the topology of the wave field in space, and should have analogues in other three-dimensional wave systems such as superfluids and Bose-Einstein condensates.