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A Tree Traversal Algorithm for Decision Problems in Knot Theory and 3-Manifold Topology

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Abstract

In low-dimensional topology, many important decision algorithms are based on normal surface enumeration, which is a form of vertex enumeration over a high-dimensional and highly degenerate polytope. Because this enumeration is subject to extra combinatorial constraints, the only practical algorithms to date have been variants of the classical double description method. In this paper we present the first practical normal surface enumeration algorithm that breaks out of the double description paradigm. This new algorithm is based on a tree traversal with feasibility and domination tests, and it enjoys a number of advantages over the double description method: incremental output, significantly lower time and space complexity, and a natural suitability for parallelisation. Experimental comparisons of running times are included.

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Notes

  1. For some topological algorithms, vertex enumeration is not enough: instead we must enumerate a Hilbert basis for a polyhedral cone.

  2. Efficient algorithms do exist for certain classes of polytopes. For example, in the case of non-degenerate polytopes, the reverse search algorithm of Avis and Fukuda has virtually no space requirements beyond storing the input, and has a running time polynomial in the combined input and output size [3].

  3. Recall that for the domination test we replace every unknown symbol − with 0.

  4. The dual simplex method usually has an extra requirement that this initial basis be dual feasible. For our application there is no objective function to optimise, and so this extra requirement can be ignored.

  5. When we begin the tree traversal algorithm the matrix \(\widetilde{M}\) has precisely 3n columns, but recall from Sect. 3.4 that we might delete columns from \(\widetilde{M}\) as we move through the type tree.

  6. Of course we must be careful: for instance, row operations of the form x←λ x+μ y must be performed using 64-bit arithmetic, even though the inputs and outputs are guaranteed to fit into 32-bit integers.

  7. Both of these previously-known bounds were derived in standard coordinates (ℝ7n), not quadrilateral coordinates (ℝ3n). However, it is simple to convert between coordinate systems [9], and it can be shown that the upper bounds differ by a factor of at most 4n.

  8. Early indications of this appear in [10] (which works in the larger space ℝ7n), and a detailed study will appear in [11]. To illustrate how extreme these results are in ℝ3n: across all 139 103 032 closed 3-manifold triangulations of size n=9, the maximum output size is just 37, and the mean output size is a mere 9.7.

  9. These are standard linear regressions of the form logy=αn+β, where y is the quantity being measured on the vertical axis.

  10. Specifically, the vertical axis measures the double description running time divided by the tree traversal running time.

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Acknowledgements

The authors are grateful to RMIT University for the use of their high-performance computing facilities. The first author is supported by the Australian Research Council under the Discovery Projects funding scheme (project DP1094516).

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  1. School of Mathematics and Physics, The University of Queensland, Brisbane, QLD, 4072, Australia

    Benjamin A. Burton

  2. School of Mathematical and Geospatial Sciences, RMIT University, GPO Box 2476V, Melbourne, VIC, 3001, Australia

    Melih Ozlen

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Burton, B.A., Ozlen, M. A Tree Traversal Algorithm for Decision Problems in Knot Theory and 3-Manifold Topology. Algorithmica 65, 772–801 (2013). https://doi.org/10.1007/s00453-012-9645-3

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