5 research outputs found
Enumerating fundamental normal surfaces: Algorithms, experiments and invariants
Computational knot theory and 3-manifold topology have seen significant
breakthroughs in recent years, despite the fact that many key algorithms have
complexity bounds that are exponential or greater. In this setting,
experimentation is essential for understanding the limits of practicality, as
well as for gauging the relative merits of competing algorithms.
In this paper we focus on normal surface theory, a key tool that appears
throughout low-dimensional topology. Stepping beyond the well-studied problem
of computing vertex normal surfaces (essentially extreme rays of a polyhedral
cone), we turn our attention to the more complex task of computing fundamental
normal surfaces (essentially an integral basis for such a cone). We develop,
implement and experimentally compare a primal and a dual algorithm, both of
which combine domain-specific techniques with classical Hilbert basis
algorithms. Our experiments indicate that we can solve extremely large problems
that were once though intractable. As a practical application of our
techniques, we fill gaps from the KnotInfo database by computing 398
previously-unknown crosscap numbers of knots.Comment: 17 pages, 5 figures; v2: Stronger experimental focus, restrict
attention to primal & dual algorithms only, larger and more detailed
experiments, more new crosscap number
On computing Hilbert bases via the Elliot−MacMahon algorithm
AbstractThe ways of using the Elliot–MacMahon algorithm to compute the Hilbert base of a system of linear Diophantine equations known so far are either not efficient or can fail to terminate. We present a version of an algorithm exploiting this range of ideas, which however is reasonably efficient as well as finite