Geometry & Topology

Bounds for the genus of a normal surface

William Jaco, Jesse Johnson, Jonathan Spreer, and Stephan Tillmann

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Abstract

This paper gives sharp linear bounds on the genus of a normal surface in a triangulated compact orientable 3–manifold in terms of the quadrilaterals in its cell decomposition — different bounds arise from varying hypotheses on the surface or triangulation. Two applications of these bounds are given. First, the minimal triangulations of the product of a closed surface and the closed interval are determined. Second, an alternative approach to the realisation problem using normal surface theory is shown to be less powerful than its dual method using subcomplexes of polytopes.

Article information

Source
Geom. Topol., Volume 20, Number 3 (2016), 1625-1671.

Dates
Received: 24 November 2014
Accepted: 17 July 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510858999

Digital Object Identifier
doi:10.2140/gt.2016.20.1625

Mathematical Reviews number (MathSciNet)
MR3523065

Zentralblatt MATH identifier
1350.57024

Subjects
Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57Q15: Triangulating manifolds 57M20: Two-dimensional complexes 57N35: Embeddings and immersions 53A05: Surfaces in Euclidean space

Keywords
3–manifold normal surface minimal triangulation efficient triangulation realisation problem

Citation

Jaco, William; Johnson, Jesse; Spreer, Jonathan; Tillmann, Stephan. Bounds for the genus of a normal surface. Geom. Topol. 20 (2016), no. 3, 1625--1671. doi:10.2140/gt.2016.20.1625. https://projecteuclid.org/euclid.gt/1510858999


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References

  • A Altshuler, Polyhedral realization in $R\sp{3}$ of triangulations of the torus and $2$–manifolds in cyclic $4$–polytopes, Discrete Math. 1 (1971/1972) 211–238
  • A Altshuler, Combinatorial $3$–manifolds with few vertices, J. Combinatorial Theory Ser. A 16 (1974) 165–173
  • D Archdeacon, C P Bonnington, J A Ellis-Monaghan, How to exhibit toroidal maps in space, Discrete Comput. Geom. 38 (2007) 573–594
  • J Bokowski, On heuristic methods for finding realizations of surfaces, from: “Discrete differential geometry”, (A I Bobenko, P Schr öder, J M Sullivan, G M Ziegler, editors), Oberwolfach Semin. 38, Birkhäuser, Basel (2008) 255–260
  • J Bokowski, U Brehm, A new polyhedron of genus $3$ with $10$ vertices, from: “Intuitive geometry”, (K B ör öczky, G Fejes Tóth, editors), Colloq. Math. Soc. János Bolyai 48, North-Holland, Amsterdam (1987) 105–116
  • J Bokowski, A Eggert, Toutes les réalisations du tore de Möbius avec sept sommets, Structural Topology (1991) 59–78
  • M Bucher, R Frigerio, C Pagliantini, The simplicial volume of $3$–manifolds with boundary, J. Topol. 8 (2015) 457–475
  • B A Burton, Face pairing graphs and $3$–manifold enumeration, J. Knot Theory Ramifications 13 (2004) 1057–1101
  • B A Burton, Enumeration of non-orientable $3$–manifolds using face-pairing graphs and union-find, Discrete Comput. Geom. 38 (2007) 527–571
  • B A Burton, R Budney, W Petterson, Regina: Software for $3$–manifold topology and normal surface theory Available at \setbox0\makeatletter\@url http://regina.sourceforge.net/ {\unhbox0
  • B A Burton, A Coward, S Tillmann, Computing closed essential surfaces in knot complements, from: “Computational geometry”, ACM, New York (2013) 405–413
  • B A Burton, M Ozlen, A tree traversal algorithm for decision problems in knot theory and $3$–manifold topology, Algorithmica 65 (2013) 772–801
  • B A Burton, J a Paixão, J Spreer, Computational Topology and Normal Surfaces: Theoretical and Experimental Complexity Bounds, from: “Proceedings of the Meeting on Algorithm Engineering & Expermiments”, SIAM, Philadelphia (2013) 78–87
  • D Cooper, S Tillmann, The Thurston norm via normal surfaces, Pacific J. Math. 239 (2009) 1–15
  • Á Császár, A polyhedron without diagonals, Acta Univ. Szeged. Sect. Sci. Math. 13 (1949) 140–142
  • R A Duke, Geometric embedding of complexes, Amer. Math. Monthly 77 (1970) 597–603
  • F Effenberger, J Spreer, simpcomp (GAP package) version 2.1.1 Available at \setbox0\makeatletter\@url https://github.com/simpcomp-team/simpcomp {\unhbox0
  • R Frigerio, B Martelli, C Petronio, Complexity and Heegaard genus of an infinite class of compact $3$–manifolds, Pacific J. Math. 210 (2003) 283–297
  • B Grünbaum, Convex polytopes, 2nd edition, Graduate Texts in Mathematics 221, Springer, New York (2003)
  • W Haken, Theorie der Normalflächen, Acta Math. 105 (1961) 245–375
  • W Haken, Über das Homöomorphieproblem der $3$–Mannigfaltigkeiten, I, Math. Z. 80 (1962) 89–120
  • J Hass, J C Lagarias, N Pippenger, The computational complexity of knot and link problems, J. ACM 46 (1999) 185–211
  • G Hemion, On the classification of homeomorphisms of $2$–manifolds and the classification of $3$–manifolds, Acta Math. 142 (1979) 123–155
  • S Hougardy, F H Lutz, M Zelke, Surface realization with the intersection segment functional, Experiment. Math. 19 (2010) 79–92
  • W Jaco, U Oertel, An algorithm to decide if a $3$–manifold is a Haken manifold, Topology 23 (1984) 195–209
  • W Jaco, H Rubinstein, S Tillmann, Minimal triangulations for an infinite family of lens spaces, J. Topol. 2 (2009) 157–180
  • W Jaco, J H Rubinstein, $0$–efficient triangulations of $3$–manifolds, J. Differential Geom. 65 (2003) 61–168
  • W Jaco, J H Rubinstein, Inflations of ideal triangulations, Adv. Math. 267 (2014) 176–224
  • W Jaco, J H Rubinstein, S Tillmann, Coverings and minimal triangulations of $3$–manifolds, Algebr. Geom. Topol. 11 (2011) 1257–1265
  • T Kalelkar, Euler characteristic and quadrilaterals of normal surfaces, Proc. Indian Acad. Sci. Math. Sci. 118 (2008) 227–233
  • H Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten, Jahresber. Dtsch. Math.-Ver. 38 (1929) 248–260
  • F Luo, S Tillmann, A new combinatorial class of $3$–manifold triangulations, preprint (2015) To appear in Asian J. Math.
  • F H Lutz, The Manifold Page, electronic resource Available at \setbox0\makeatletter\@url http://www.math.tu-berlin.de/diskregeom/stellar {\unhbox0
  • F H Lutz, Enumeration and random realization of triangulated surfaces, from: “Discrete differential geometry”, (A I Bobenko, P Schr öder, J M Sullivan, G M Ziegler, editors), Oberwolfach Semin. 38, Birkhäuser, Basel (2008) 235–253
  • S V Matveev, Complexity theory of three-dimensional manifolds, Acta Appl. Math. 19 (1990) 101–130
  • S V Matveev, Algorithmic topology and classification of $3$–manifolds, Algorithms and Computation in Mathematics 9, Springer, Berlin (2003)
  • P McMullen, C Schulz, J M Wills, Polyhedral $2$–manifolds in $E\sp{3}$ with unusually large genus, Israel J. Math. 46 (1983) 127–144
  • G Ringel, Map color theorem, Grundl. Math. Wissen. 209, Springer, New York (1974)
  • J H Rubinstein, An algorithm to recognize the $3$–sphere, from: “Proceedings of the International Congress of Mathematicians, Volume 1”, (S D Chatterji, editor), Birkhäuser, Basel (1995) 601–611
  • J H Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for $3$–dimensional manifolds, from: “Geometric topology”, (W H Kazez, editor), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 1–20
  • L Schewe, Nonrealizable minimal vertex triangulations of surfaces: showing nonrealizability using oriented matroids and satisfiability solvers, Discrete Comput. Geom. 43 (2010) 289–302
  • J Spreer, Normal surfaces as combinatorial slicings, Discrete Math. 311 (2011) 1295–1309
  • E Steinitz, olyederrelationen}}, Arch. der Math. u. Phys. 11 (1906) 86–88
  • A Thompson, Thin position and the recognition problem for $S\sp 3$, Math. Res. Lett. 1 (1994) 613–630
  • S Tillmann, Normal surfaces in topologically finite $3$–manifolds, Enseign. Math. 54 (2008) 329–380
  • J L Tollefson, Normal surface $Q$–theory, Pacific J. Math. 183 (1998) 359–374
  • A Y Vesnin, E A Fominykh, Exact values of the complexity of Paoluzzi–Zimmermann manifolds, Dokl. Akad. Nauk 439 (2011) 727–729 In Russian; translated in Dokl. Math. 84 (2011) 542–544
  • G M Ziegler, Polyhedral surfaces of high genus, from: “Discrete differential geometry”, (A I Bobenko, P Schr öder, J M Sullivan, G M Ziegler, editors), Oberwolfach Semin. 38, Birkhäuser, Basel (2008) 191–213