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Register a new userThe efficient certification of knottedness and Thurston norm
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Authors
Marc Lackenby
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We show that the problem of determining whether a knot in the 3-sphere is non-trivial lies in NP. This is a consequence of the following more general result. The problem of determining whether the Thurston norm of a second homology class in a compact orientable 3-manifold is equal to a given integer is in NP. As a corollary, the problem of determining the genus of a knot in the 3-sphere is in NP. We also show that the problem of determining whether a compact orientable 3-manifold has incompressible boundary is in NP.
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We present an overview of the study of the Thurston norm, introduced by W. P. Thurston in the seminal paper A norm for the homology of 3-manifolds (written in 1976 and published in 1986). We first review fundamental properties of the Thurston norm of a 3-manifold, including a construction of codimension-1 taut foliations from norm-minimizing embedded surfaces, established by D. Gabai. In the main part we describe relationships between the Thurston norm and other topological invariants of a 3-manifold: the Alexander polynomial and its various generalizations, Reidemeister torsion, the Seiberg-Witten invariant, Heegaard Floer homology, the complexity of triangulations and the profinite completion of the fundamental group. Some conjectures and questions on related topics are also collected. The final version of this paper will appear as a chapter in the book In the tradition of Thurston, II, edited by K. Ohshika and A. Papadopoulos (Springer, 2022).
Given a triangulation of a closed, oriented, irreducible, atoroidal 3-manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by non-negative integer weights, 14 for each 3-simplex, that describe how many copies of each oriented normal disc type there are. The Euler characteristic and homology class are both linear functions of the weights. There is a convex polytope in the space of weights, defined by linear equations given by the combinatorics of the triangulation, whose image under the homology map is the unit ball, B, of the Thurston norm. Applications of this approach include (1) an algorithm to compute B and hence the Thurston norm of any homology class, (2) an explicit exponential bound on the number of vertices of B in terms of the number of simplices in the triangulation, (3) an algorithm to determine the fibred faces of B and hence an algorithm to decide whether a 3-manifold fibres over the circle.
The classifying space for the framed Haefliger structures of codimension $q$ and class $C^r$ is $(2q-1)$-connected, for $1le rleinfty$. The corollaries deal with the existence of foliations, with the homology and the perfectness of the diffeomorphism groups, with the existence of foliated products, and of foliated bundles.
We establish a form of the h-principle for the existence of foliations quasi-complementary to a given one; the same methods also provide a proof of the classical Mather-Thurston theorem.
Thurston norms are invariants of 3-manifolds defined on their second homology vector spaces, and understanding the shape of their dual unit ball is a (widely) open problem. W. Thurston showed that every symmetric polygon in Z^2, whose vertices satisfy a parity property, is the dual unit ball of a Thurston norm on a 3-manifold. However, it is not known if the parity property on the vertices of polytopes is a sufficient condition in higher dimension or if their are polytopes, with mod 2 congruent vertices, that cannot be realized as dual unit balls of Thurston norms. In this article, we provide a family of polytopes in Z^2g that can be realized as dual unit balls of Thurston norms on 3-manifolds. These polytopes come from intersection norms on oriented closed surfaces and this article widens the bridge between these two norms.
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