Incompressibility criteria for spun-normal surfaces

Dunfield, Nathan; Garoufalidis, Stavros

doi:10.1090/S0002-9947-2012-05663-7

Incompressibility criteria for spun-normal surfaces
HTML articles powered by AMS MathViewer

by Nathan M. Dunfield and Stavros Garoufalidis PDF

Trans. Amer. Math. Soc. 364 (2012), 6109-6137 Request permission

Abstract:

We give a simple sufficient condition for a spun-normal surface in an ideal triangulation to be incompressible, namely that it is a vertex surface with nonempty boundary which has a quadrilateral in each tetrahedron. While this condition is far from being necessary, it is powerful enough to give two new results: the existence of alternating knots with noninteger boundary slopes, and a proof of the Slope Conjecture for a large class of 2-fusion knots.

While the condition and conclusion are purely topological, the proof uses the Culler-Shalen theory of essential surfaces arising from ideal points of the character variety, as reinterpreted by Thurston and Yoshida. The criterion itself comes from the work of Kabaya, which we place into the language of normal surface theory. This allows the criterion to be easily applied, and gives the framework for proving that the surface is incompressible.

We also explore which spun-normal surfaces arise from ideal points of the deformation variety. In particular, we give an example where no vertex or fundamental surface arises in this way.

References

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.
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. MR 1288467, DOI 10.1007/BF01231526
M. Culler. A table of $A$-polynomials computed via numerical methods. http://www.math.uic.edu/~culler/Apolynomials/
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/
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. MR 881270, DOI 10.2307/1971311
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. MR 683804, DOI 10.2307/2006973
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. MR 1695208, DOI 10.1007/s002220050321
Nathan M. Dunfield, A table of boundary slopes of Montesinos knots, Topology 40 (2001), no. 2, 309–315. MR 1808223, DOI 10.1016/S0040-9383(99)00064-6
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.
N. M. Dunfield and S. Garoufalidis. Ancillary files stored with the arXiv version of this paper. http://arxiv.org/src/1102.4588/anc
W. Floyd and A. Hatcher, Incompressible surfaces in punctured-torus bundles, Topology Appl. 13 (1982), no. 3, 263–282. MR 651509, DOI 10.1016/0166-8641(82)90035-9
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. MR 2763776, DOI 10.1090/S0002-9939-2010-10617-2
Stavros Garoufalidis, The Jones slopes of a knot, Quantum Topol. 2 (2011), no. 1, 43–69. MR 2763086, DOI 10.4171/QT/13
S. Garoufalidis. Tropicalization and the Slope Conjecture for 2-fusion knots. Preprint 2011.
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. MR 2153123, DOI 10.2140/agt.2005.5.379
Wolfgang Haken, Theorie der Normalflächen, Acta Math. 105 (1961), 245–375 (German). MR 141106, DOI 10.1007/BF02559591
A. E. Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 (1982), no. 2, 373–377. MR 658066, DOI 10.2140/pjm.1982.99.373
A. Hatcher and U. Oertel, Boundary slopes for Montesinos knots, Topology 28 (1989), no. 4, 453–480. MR 1030987, DOI 10.1016/0040-9383(89)90005-0
A. Hatcher and W. Thurston, Incompressible surfaces in $2$-bridge knot complements, Invent. Math. 79 (1985), no. 2, 225–246. MR 778125, DOI 10.1007/BF01388971
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
J. Hoste and M. Thistlethwaite. Knotscape, 1999. http://www.math.utk.edu/~morwen
William Jaco and Eric Sedgwick, Decision problems in the space of Dehn fillings, Topology 42 (2003), no. 4, 845–906. MR 1958532, DOI 10.1016/S0040-9383(02)00083-6
Yuichi Kabaya, A method to find ideal points from ideal triangulations, J. Knot Theory Ramifications 19 (2010), no. 4, 509–524. MR 2646644, DOI 10.1142/S0218216510007929
Ensil Kang, Normal surfaces in non-compact 3-manifolds, J. Aust. Math. Soc. 78 (2005), no. 3, 305–321. MR 2142159, DOI 10.1017/S1446788700008557
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. MR 2172486, DOI 10.2140/gtm.2004.7.235
Sergei Matveev, Algorithmic topology and classification of 3-manifolds, 2nd ed., Algorithms and Computation in Mathematics, vol. 9, Springer, Berlin, 2007. MR 2341532
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. MR 1184415
Walter D. Neumann and Don Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), no. 3, 307–332. MR 815482, DOI 10.1016/0040-9383(85)90004-7
Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. MR 1277811
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. MR 1470718, DOI 10.1090/amsip/002.1/01
Henry Segerman, On spun-normal and twisted squares surfaces, Proc. Amer. Math. Soc. 137 (2009), no. 12, 4259–4273. MR 2538587, DOI 10.1090/S0002-9939-09-09960-2
Henry Segerman, Detection of incompressible surfaces in hyperbolic punctured torus bundles, Geom. Dedicata 150 (2011), 181–232. MR 2753704, DOI 10.1007/s10711-010-9501-z
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
W. Stein et al. SAGE Mathematics Software (Version 4.6), 2011. http://www.sagemath.org
W. P. Thurston. A geometric approach to Dehn surgery on 3-manifolds. Lecture notes taken by Craig Hodgson, Princeton 1981–2.
W. P. Thurston. The Geometry and Topology of Three-Manifolds. http://www.msri.org/publications/books/gt3m
Stephan Tillmann, Normal surfaces in topologically finite 3-manifolds, Enseign. Math. (2) 54 (2008), no. 3-4, 329–380. MR 2478091
S. Tillmann. Degenerations of ideal hyperbolic triangulations. Math. Zeit. To appear, 31 pages. arXiv:math.GT/0508295.
Genevieve S. Walsh, Incompressible surfaces and spunnormal form, Geom. Dedicata 151 (2011), 221–231. MR 2780747, DOI 10.1007/s10711-010-9529-0
Tomoyoshi Yoshida, On ideal points of deformation curves of hyperbolic $3$-manifolds with one cusp, Topology 30 (1991), no. 2, 155–170. MR 1098911, DOI 10.1016/0040-9383(91)90003-M