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Recognition of Seifert fibered spaces with boundary is in NP

Adele Jackson

Abstract

We show that the decision problem of recognising whether a triangulated 3-manifold admits a Seifert fibered structure with non-empty boundary is in NP. We also show that the problem of producing Seifert data for a triangulation of such a manifold is in the complexity class FNP. We do this by proving that in any triangulation of a Seifert fibered space with boundary there is both a fundamental horizontal surface of small degree and a complete collection of normal vertical annuli whose total weight is bounded by an exponential in the square of the triangulation size.

Recognition of Seifert fibered spaces with boundary is in NP

Abstract

We show that the decision problem of recognising whether a triangulated 3-manifold admits a Seifert fibered structure with non-empty boundary is in NP. We also show that the problem of producing Seifert data for a triangulation of such a manifold is in the complexity class FNP. We do this by proving that in any triangulation of a Seifert fibered space with boundary there is both a fundamental horizontal surface of small degree and a complete collection of normal vertical annuli whose total weight is bounded by an exponential in the square of the triangulation size.
Paper Structure (20 sections, 58 theorems, 6 equations, 10 figures)

This paper contains 20 sections, 58 theorems, 6 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background and conventions
  3. Computational complexity
  4. Triangulations and normal surface theory
  5. Seifert fibered spaces
  6. Conventions and notation
  7. Existence of a minimal degree fundamental horizontal surface
  8. Split handle structures
  9. Normal surfaces
  10. Fundamental normal surfaces
  11. A complete bounded collection of normal vertical annuli
  12. Fundamental Möbius bands
  13. Fundamental vertical annuli
  14. Recognising circle bundles over surfaces with boundary is in NP
  15. Recognising Seifert fibered spaces with boundary is in NP
  16. ...and 5 more sections

Key Result

theorem 1.1

The problem Seifert fibered space with boundary recognition is in NP.

Figures (10)

  • Figure 1: An example of the intersection of a normal surface with a tetrahedron.
  • Figure 2: The pieces resulting from cutting along an elementary triangle in a tetrahedron (in the interior of some larger triangulation), and the picture in the dual split handle structure. To depict the handles, we draw their boundary graphs (in $S^2$). The forbidden region is shaded in red. There are no parallelity pieces.
  • Figure 3: The induced split handle structure from cutting along an elementary disc in a split 0-handle $H$ that does not correspond to one in a tetrahedron. The forbidden region is shaded in red, and the sutures are the thick red lines.
  • Figure 4: The horizontal and vertical boundaries of the product of a disc with two punctures, $\Sigma$, with the interval, $I$. The fibration of the visible part of the vertical boundary $\partial\Sigma\times I$ is striped. One component of the horizontal boundary is visible.
  • Figure 5: Two non-examples of elementary disc boundaries in a split 0-handle. The disc boundaries are drawn in teal. The one on the left crosses the same lake in two arcs, while the one on the right intersects a bridge and an adjacent lake. Both therefore fail condition \ref{['defn:normalsurface:itm:0handles']} of Definition \ref{['defn:normalsurface']}.
  • ...and 5 more figures

Theorems & Definitions (118)

  • theorem 1.1
  • theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4: Proposition 3.3.24 Matveev
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7: Lemma 6.1 HassLagariasPippenger
  • ...and 108 more