The computational complexity of basic decision problems in 3-dimensional topology

Abstract

We study the computational complexity of basic decision problems of 3-dimensional topology, such as to determine whether a triangulated 3-manifold is irreducible, prime, ∂-irreducible, or homeomorphic to a given 3-manifold M. For example, we prove that the problem to recognize whether a triangulated 3-manifold is homeomorphic to a 3-sphere, or to a 2-sphere bundle over a circle, or to a real projective 3-space, or to a handlebody of genus g, is decidable in nondeterministic polynomial time (NP) of size of the triangulation. We also show that the problem to determine whether a triangulated orientable 3-manifold is irreducible (or prime) is in PSPACE and whether it is ∂-irreducible is in coNP. The proofs improve and extend arguments of prior author's article on the recognition problem for the 3-sphere.

On the complexity of torus knot recognition

“…After triangulating the knot complement, we present the annulus A as a normal surface with bounded weight, using Proposition 3.4; then we cut the knot complement along A and triangulate the remaining pieces using work of Lackenby [Lac16]. Finally, we certify that A is essential and that the remaining pieces are indeed solid tori using an algorithm of Ivanov [Iva08], and this verifies that we have a torus knot.…”

“…(3) Apply Proposition 4.2 to triangulate E K N (A) and verify that it has two connected components, say M 1 and M 2 . (4) Verify the certificates that the given triangulations of M 1 and M 2 produce solid tori, following [Iva08].…”

“…(3) Verify that A is an annulus and that the components of ∂A are homologically nontrivial in ∂E K . (4) Verify as in [Iva08] the certificate that C ∼ = S 1 × D 2 . (5) Verify the certification of the Thurston norm of [φ], as in [Lac16].…”

How to Draw a Tait-Colorable Graph

“…Thus, each 4-edge-colored quartic graph represents a certain 3-dimensional pseudomanifold. Even in dimension 3, several decision problems are known to be NP-complete [1,8]. Thus, it is thought that decision problems in 3-dimensional manifolds must translate via polynomial-time algorithms to decision problems for 4-edge-colored quartic graphs.…”

The intersection of subgroups in free groups and linear programming

“…almost) normal surfaces in 3dimensional manifolds and their (resp. almost) normal vectors in the Haken theory of normal surfaces and its generalizations, see [8], [9], [10], [12], [15]. In particular, the idea of a vertex solution works equally well both in the context of almost normal surfaces [12], see also [9], [15], and in the context of subgroups of free groups, providing in either situation both the connectedness of the underlying object associated with a vertex solution and an upper bound on the size of the underlying object.…”