A tree traversal algorithm for decision problems in knot theory and 3-manifold topology

TL;DR

This paper introduces a novel tree traversal algorithm for normal surface enumeration in low-dimensional topology, offering practical improvements over traditional methods in terms of efficiency, scalability, and parallelization.

Contribution

The paper presents the first practical non-double description algorithm for normal surface enumeration, utilizing a tree traversal approach with feasibility and domination tests.

Findings

Lower time and space complexity compared to classical methods

Incremental output facilitates early results and partial enumeration

Enhanced suitability for parallel computation

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.