×

Incompressibility criteria for spun-normal surfaces. (English) Zbl 1281.57012

Let \(M\) be a compact oriented \(3\)-manifold with torus boundary, and \(S\) be an essential surface with boundary properly embedded in \(M\), then the boundary slope of \(S\) can be parametrized by the primitive homology class in \(H_1(\partial M; \mathbb{Z})/(\pm 1)\).
In the paper under review, the authors study the set \(bs(M)\) of all boundary slopes of essential surfaces in \(M\). The set \(bs(M)\) is an important invariant of \(M\), and is a very useful tool for the study of other topics, e.g. the exceptional Dehn fillings. A natural question is how to compute \(bs(M)\)? A general algorithm was given by Jaco and Sedgwick based on normal surface theory, but this algorithm seems impractical in some sense. Some progress has been made by several authors, but there are still some examples remaining where \(bs(M)\) is unknown.
One of the main results of this paper gives us a simple and sufficient condition for a normal surface to be essential. This is one of the first results obtained by directly applying normal surface algorithms.
{ Theorem 1.1.} Let \(M\) be a \(3\)-manifold with an ideal triangulation \(\mathcal{T}\). Suppose \(S\) is a vertex spun-normal surface in \(\mathcal{T}\) with nontrivial boundary. If \(S\) has a quadrilateral in every tetrahedron of \(\mathcal{T}\), then \(S\) is essential.
Kabaya got a weaker version of this theorem using a different technique. The authors of this paper apply the language of normal surface theory. This makes the result easier to apply.
Although the condition in the theorem is far from being necessary, the authors use it to answer a question of Hatcher and Oertel. Precisely, they show the following:
{ Theorem 1.3.} There are alternating knots with nonintegral boundary slopes.
The authors also give a proof of the Slope Conjecture of Garoufalidis using these results.
Reviewer: Yu Zhang (Harbin)

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] B. A. Burton, J. H. Rubinstein, and S. Tillmann. The Weber-Seifert dodecahedral space is non-Haken. Trans. Amer. Math. Soc. (2011). 22 pages, to appear. arXiv:0909.4625. · Zbl 1250.57033
[2] D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen, Plane curves associated to character varieties of 3-manifolds, Invent. Math. 118 (1994), no. 1, 47 – 84. · Zbl 0842.57013 · doi:10.1007/BF01231526
[3] M. Culler. A table of \( A\)-polynomials computed via numerical methods. http://www.math.uic.edu/ culler/Apolynomials/
[4] M. Culler, N. M. Dunfield, and J. R. Weeks. SnapPy, a computer program for studying the geometry and topology of 3-manifolds. http://snappy.computop.org/
[5] Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237 – 300. · Zbl 0633.57006 · doi:10.2307/1971311
[6] Marc Culler and Peter B. Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. of Math. (2) 117 (1983), no. 1, 109 – 146. · Zbl 0529.57005 · doi:10.2307/2006973
[7] Nathan M. Dunfield, Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. 136 (1999), no. 3, 623 – 657. · Zbl 0928.57012 · doi:10.1007/s002220050321
[8] Nathan M. Dunfield, A table of boundary slopes of Montesinos knots, Topology 40 (2001), no. 2, 309 – 315. · Zbl 0967.57014 · doi:10.1016/S0040-9383(99)00064-6
[9] N. M. Dunfield. The Mahler measure of the \( A\)-polynomial of \( m129(0,3)\). Appendix to D. Boyd and F. Rodriguez Villegas, Mahler’s Measure and the Dilogarithm II, Preprint, 2003. arXiv:math.NT/0308041.
[10] N. M. Dunfield and S. Garoufalidis. Ancillary files stored with the arXiv version of this paper. http://arxiv.org/src/1102.4588/anc
[11] W. Floyd and A. Hatcher, Incompressible surfaces in punctured-torus bundles, Topology Appl. 13 (1982), no. 3, 263 – 282. · Zbl 0493.57004 · doi:10.1016/0166-8641(82)90035-9
[12] David Futer, Efstratia Kalfagianni, and Jessica S. Purcell, Slopes and colored Jones polynomials of adequate knots, Proc. Amer. Math. Soc. 139 (2011), no. 5, 1889 – 1896. · Zbl 1232.57007
[13] Stavros Garoufalidis, The Jones slopes of a knot, Quantum Topol. 2 (2011), no. 1, 43 – 69. · Zbl 1228.57004 · doi:10.4171/QT/13
[14] S. Garoufalidis. Tropicalization and the Slope Conjecture for 2-fusion knots. Preprint 2011.
[15] Stavros Garoufalidis and Yueheng Lan, Experimental evidence for the volume conjecture for the simplest hyperbolic non-2-bridge knot, Algebr. Geom. Topol. 5 (2005), 379 – 403. · Zbl 1092.57005 · doi:10.2140/agt.2005.5.379
[16] Wolfgang Haken, Theorie der Normalflächen, Acta Math. 105 (1961), 245 – 375 (German). , https://doi.org/10.1007/BF02559591 Horst Schubert, Bestimmung der Primfaktorzerlegung von Verkettungen, Math. Z. 76 (1961), 116 – 148 (German). , https://doi.org/10.1007/BF01210965 Wolfgang Haken, Ein Verfahren zur Aufspaltung einer 3-Mannigfaltigkeit in irreduzible 3-Mannigfaltigkeiten, Math. Z. 76 (1961), 427 – 467 (German). · Zbl 0111.18803 · doi:10.1007/BF01210988
[17] A. E. Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 (1982), no. 2, 373 – 377. · Zbl 0502.57005
[18] A. Hatcher and U. Oertel, Boundary slopes for Montesinos knots, Topology 28 (1989), no. 4, 453 – 480. · Zbl 0686.57006 · doi:10.1016/0040-9383(89)90005-0
[19] A. Hatcher and W. Thurston, Incompressible surfaces in 2-bridge knot complements, Invent. Math. 79 (1985), no. 2, 225 – 246. · Zbl 0602.57002 · doi:10.1007/BF01388971
[20] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. · Zbl 1044.55001
[21] J. Hoste and M. Thistlethwaite. Knotscape, 1999. http://www.math.utk.edu/ morwen
[22] William Jaco and Eric Sedgwick, Decision problems in the space of Dehn fillings, Topology 42 (2003), no. 4, 845 – 906. · Zbl 1013.57013 · doi:10.1016/S0040-9383(02)00083-6
[23] Yuichi Kabaya, A method to find ideal points from ideal triangulations, J. Knot Theory Ramifications 19 (2010), no. 4, 509 – 524. · Zbl 1196.57011 · doi:10.1142/S0218216510007929
[24] Ensil Kang, Normal surfaces in non-compact 3-manifolds, J. Aust. Math. Soc. 78 (2005), no. 3, 305 – 321. · Zbl 1077.57016 · doi:10.1017/S1446788700008557
[25] Ensil Kang and J. Hyam Rubinstein, Ideal triangulations of 3-manifolds. I. Spun normal surface theory, Proceedings of the Casson Fest, Geom. Topol. Monogr., vol. 7, Geom. Topol. Publ., Coventry, 2004, pp. 235 – 265. · Zbl 1085.57016 · doi:10.2140/gtm.2004.7.235
[26] Sergei Matveev, Algorithmic topology and classification of 3-manifolds, 2nd ed., Algorithms and Computation in Mathematics, vol. 9, Springer, Berlin, 2007. · Zbl 1128.57001
[27] Walter D. Neumann, Combinatorics of triangulations and the Chern-Simons invariant for hyperbolic 3-manifolds, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 243 – 271. · Zbl 0768.57006
[28] Walter D. Neumann and Don Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), no. 3, 307 – 332. · Zbl 0589.57015 · doi:10.1016/0040-9383(85)90004-7
[29] Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. · Zbl 0854.57002
[30] J. H. Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3-dimensional manifolds, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 1 – 20. · Zbl 0889.57021
[31] Henry Segerman, On spun-normal and twisted squares surfaces, Proc. Amer. Math. Soc. 137 (2009), no. 12, 4259 – 4273. · Zbl 1204.57016
[32] Henry Segerman, Detection of incompressible surfaces in hyperbolic punctured torus bundles, Geom. Dedicata 150 (2011), 181 – 232. · Zbl 1236.57023 · doi:10.1007/s10711-010-9501-z
[33] H. Segerman and S. Tillmann. Pseudo-Developing Maps for Ideal Triangulations I: Essential Edges and Generalised Hyperbolic Gluing Equations. Preprint 2010. http://www.ms.unimelb.edu.au/ segerman · Zbl 1333.57017
[34] W. Stein et al. SAGE Mathematics Software (Version 4.6), 2011. http://www.sagemath.org
[35] W. P. Thurston. A geometric approach to Dehn surgery on 3-manifolds. Lecture notes taken by Craig Hodgson, Princeton 1981-2.
[36] W. P. Thurston. The Geometry and Topology of Three-Manifolds. http://www.msri.org/publications/books/gt3m
[37] Stephan Tillmann, Normal surfaces in topologically finite 3-manifolds, Enseign. Math. (2) 54 (2008), no. 3-4, 329 – 380. · Zbl 1214.57022
[38] S. Tillmann. Degenerations of ideal hyperbolic triangulations. Math. Zeit. To appear, 31 pages. arXiv:math.GT/0508295. · Zbl 1271.57029
[39] Genevieve S. Walsh, Incompressible surfaces and spunnormal form, Geom. Dedicata 151 (2011), 221 – 231. · Zbl 1236.57026 · doi:10.1007/s10711-010-9529-0
[40] Tomoyoshi Yoshida, On ideal points of deformation curves of hyperbolic 3-manifolds with one cusp, Topology 30 (1991), no. 2, 155 – 170. · Zbl 0726.57011 · doi:10.1016/0040-9383(91)90003-M
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.