1–efficient triangulations and the index of a cusped hyperbolic 3–manifold

Garoufalidis, Stavros¹ ; Hodgson, Craig D² ; Rubinstein, J Hyam² ; Segerman, Henry³

¹ School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA
² Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Parkville VIC 3010, Australia
³ Department of Mathematics, Oklahoma State University, Stillwater, VIC 74078, USA, Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Parkville VIC 3010, Australia

Geometry & topology, Tome 19 (2015) no. 5, pp. 2619-2689.

Résumé

In this paper we will promote the 3D index of an ideal triangulation $\mathcal{T}$ of an oriented cusped $3$–manifold $M$ (a collection of $q$–series with integer coefficients, introduced by Dimofte, Gaiotto and Gukov) to a topological invariant of oriented cusped hyperbolic $3$–manifolds. To achieve our goal we show that (a) $\mathcal{T}$ admits an index structure if and only if $\mathcal{T}$ is $1$–efficient and (b) if $M$ is hyperbolic, it has a canonical set of $1$–efficient ideal triangulations related by $2$–$3$ and $0$–$2$ moves which preserve the 3D index. We illustrate our results with several examples.

Détail

DOI : 10.2140/gt.2015.19.2619

Classification : 57N10, 57M50, 57M25
Keywords: ideal triangulations, hyperbolic $3$–manifolds, gluing equations 3D index, invariants, $1$–efficient triangulations

Bibliographie

[1] H Akiyoshi, Finiteness of polyhedral decompositions of cusped hyperbolic manifolds obtained by the Epstein–Penner’s method, Proc. Amer. Math. Soc. 129 (2001) 2431 | DOI

[2] J E Andersen, R Kashaev, A TQFT from Quantum Teichmüller theory, Comm. Math. Phys. 330 (2014) 887 | DOI

[3] S Baseilhac, R Benedetti, Classical and quantum dilogarithmic invariants of flat PSL(2, C)–bundles over 3–manifolds, Geom. Topol. 9 (2005) 493 | DOI

[4] S Baseilhac, R Benedetti, Quantum hyperbolic geometry, Algebr. Geom. Topol. 7 (2007) 845 | DOI

[5] C Beem, T Dimofte, S Pasquetti, Holomorphic blocks in three dimensions, J. High Energy Phys. 2014 (2014) | DOI

[6] R Benedetti, C Petronio, Branched standard spines of 3–manifolds, 1653, Springer (1997) | DOI

[7] D Cooper, M Culler, H Gillet, D D Long, P B Shalen, Plane curves associated to character varieties of 3–manifolds, Invent. Math. 118 (1994) 47 | DOI

[8] M Culler, N M Dunfield, J R Weeks, SnapPy, a computer program for studying the topology of 3–manifolds

[9] J A De Loera, J Rambau, F Santos, Triangulations: Structures for algorithms and applications, 25, Springer (2010) | DOI

[10] T Dimofte, D Gaiotto, S Gukov, 3–manifolds and 3d indices, Adv. Theor. Math. Phys. 17 (2013) 975 | DOI

[11] T Dimofte, D Gaiotto, S Gukov, Gauge theories labelled by three-manifolds, Comm. Math. Phys. 325 (2014) 367 | DOI

[12] T Dimofte, S Garoufalidis, The quantum content of the gluing equations, Geom. Topol. 17 (2013) 1253 | DOI

[13] D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67

[14] C Frohman, J Kania-Bartoszynska, The quantum content of the normal surfaces in a three-manifold, J. Knot Theory Ramifications 17 (2008) 1005 | DOI

[15] S Garoufalidis, 3D index data

[16] S Garoufalidis, The 3D index of an ideal triangulation and angle structures,

[17] S Garoufalidis, R Kashaev, From state-integrals to q–series,

[18] S Garoufalidis, T T Lê, Nahm sums, stability and the colored Jones polynomial, Res. Math. Sci. 2 (2015) | DOI

[19] S Garoufalidis, T Vuong, Alternating knots, planar graphs, and q–series, Ramanujan J. 36 (2015) 501 | DOI

[20] S Garoufalidis, D Zagier, Empirical relations between q–series and Kashaev’s invariant of knots (2013)

[21] I M Gel′Fand, M M Kapranov, A V Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhäuser (1994) | DOI

[22] F Guéritaud, S Schleimer, Canonical triangulations of Dehn fillings, Geom. Topol. 14 (2010) 193 | DOI

[23] A Hatcher, Notes on basic 3–manifold topology (2007)

[24] C D Hodgson, J H Rubinstein, H Segerman, Triangulations of hyperbolic 3–manifolds admitting strict angle structures, J. Topol. 5 (2012) 887 | DOI

[25] W Jaco, J H Rubinstein, 0–efficient triangulations of 3–manifolds, J. Differential Geom. 65 (2003) 61

[26] V F R Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987) 335 | DOI

[27] E Kang, J H Rubinstein, Ideal triangulations of 3–manifolds, I : Spun normal surface theory, from: "Proceedings of the Casson Fest" (editors C Gordon, Y Rieck), Geom. Topol. Monogr. 7 (2004) 235 | DOI

[28] E Kang, J H Rubinstein, Ideal triangulations of 3–manifolds, II : Taut and angle structures, Algebr. Geom. Topol. 5 (2005) 1505 | DOI

[29] R M Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys. 39 (1997) 269 | DOI

[30] R Kashaev, F Luo, G Vartanov, A TQFT of Turaev–Viro type on shaped triangulations,

[31] M Lackenby, Taut ideal triangulations of 3–manifolds, Geom. Topol. 4 (2000) 369 | DOI

[32] M Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243 | DOI

[33] M Lackenby, An algorithm to determine the Heegaard genus of simple 3–manifolds with nonempty boundary, Alg. Geom. Topol. 8 (2008) 911 | DOI

[34] J L Loday, Realization of the Stasheff polytope, Arch. Math. Basel 83 (2004) 267 | DOI

[35] F Luo, S Tillmann, Angle structures and normal surfaces, Trans. Amer. Math. Soc. 360 (2008) 2849 | DOI

[36] S Maclane, Categories for the working mathematician, 5, Springer (1971)

[37] S V Matveev, Transformations of special spines, and the Zeeman conjecture, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987) 1104, 1119

[38] S Matveev, Algorithmic topology and classification of 3–manifolds, 9, Springer (2007) | DOI

[39] W D Neumann, Combinatorics of triangulations and the Chern–Simons invariant for hyperbolic 3–manifolds, from: "Topology ’90" (editors B Apanasov, W D Neumann, A W Reid, L Siebenmann), Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 243 | DOI

[40] W D Neumann, D Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985) 307 | DOI

[41] C Petronio, Standard spines and 3–manifolds, PhD thesis, Scuola Normale Superiore, Pisa (1995)

[42] R Piergallini, Standard moves for standard polyhedra and spines, Rend. Circ. Mat. Palermo Suppl. (1988) 391

[43] J H Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3–dimensional manifolds, from: "Geometric topology" (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 1

[44] F Santos, Geometric bistellar flips: The setting, the context and a construction, from: "International Congress of Mathematicians, Vol. III" (editors M Sanz-Solé, J Soria, J L Varona, J Verdera), Eur. Math. Soc. (2006) 931

[45] J D Stasheff, Homotopy associativity of H–spaces, I, II, Trans. Amer. Math. Soc. 108, 275-292; ibid. 108 (1963) 293 | DOI

[46] J Stasheff, From operads to “physically” inspired theories, from: "Operads : Proceedings of Renaissance Conferences" (editors J L Loday, J D Stasheff, A A Voronov), Contemp. Math. 202, Amer. Math. Soc. (1997) 53 | DOI

[47] M Stocking, Almost normal surfaces in 3–manifolds, Trans. Amer. Math. Soc. 352 (2000) 171 | DOI

[48] W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)

[49] J Weeks, Hyperbolic structures on three-manifolds, PhD thesis, Princeton Univ. (1985)