The latter two collaborators were undergraduates at the University of Sydney when we worked on this project. We developed an alternative approach to the problem of finding closed surfaces in the exteriors of knots in the three-sphere. Whereas the Burton-Coward-Tillmann algorithm searches exclusively for closed surfaces by imposing an additional linear condition in the usual matching-equation linear programming problem, our algorithm instead uses the solutions from this usual problem to find closed surfaces. We have not yet determined which method is faster. There is reason to believe our method may be faster, since the intersection of a polytope with a plane can have many more vertices than the polytope itself. This project is still work in progress.