Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-24T03:34:37.456Z Has data issue: false hasContentIssue false
  • English
  • Français

Contact structures and reducible surgeries

Part of: Differential topology Low-dimensional topology

Published online by Cambridge University Press:  24 September 2015

Tye Lidman  and
Steven Sivek
Tye Lidman
Affiliation:
Mathematics Department, The University of Texas at Austin, RLM 8.100, 2515 Speedway Stop C1200, Austin, TX 78712-1202, USA email tlid@math.utexas.edu
Steven Sivek
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA email ssivek@math.princeton.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus $g$ must have slope $2g-1$, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston–Bennequin numbers of cables.

Keywords

cabling conjectureDehn surgerycontact structuresStein fillings

MSC classification

Primary: 57M25: Knots and links in $S^3$ 57R17: Symplectic and contact topology
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 1 , January 2016 , pp. 152 - 186
Copyright
© The Authors 2015 

References

Akbulut, S. and Kirby, R., Mazur manifolds, Michigan Math. J. 26 (1979), 259–284; MR 544597 (80h:57004).CrossRefGoogle Scholar
Akbulut, S. and McCarthy, J. D., Casson’s invariant for oriented homology 3-spheres: An exposition, Mathematical Notes, vol. 36 (Princeton University Press, Princeton, NJ, 1990); MR 1030042 (90k:57017).CrossRefGoogle Scholar
Baker, K., Knots on once-punctured torus fibers, PhD thesis, University of Texas at Austin (2004).Google Scholar
Bennequin, D., Entrelacements et équations de Pfaff, in Third Schnepfenried Geometry Conference, Vol. 1 (Schnepfenried, 1982), Astérisque, vol. 107 (Société Mathématique de France, Paris, 1983), 87–161; MR 753131 (86e:58070).Google Scholar
Berge, J., The knots in D 2 × S 1 which have nontrivial Dehn surgeries that yield D 2 × S 1, Topology Appl. 38 (1991), 1–19; MR 1093862 (92d:57005).CrossRefGoogle Scholar
Berge, J., Some knots with surgeries yielding lens spaces, unpublished.Google Scholar
Bhupal, M. and Ono, K., Symplectic fillings of links of quotient surface singularities, Nagoya Math. J. 207 (2012), 1–45; MR 2957141.CrossRefGoogle Scholar
Bhupal, M. and Ozbagci, B., Milnor open books of links of some rational surface singularities, Pacific J. Math. 254 (2011), 47–65; MR 2900660.CrossRefGoogle Scholar
F. Bourgeois, T. Ekholm and Y. Eliashberg, Effect of Legendrian surgery, Geom. Topol. 16 (2012), 301–389; with an appendix by S. Ganatra and M. Maydanskiy; MR 2916289.Google Scholar
Boyer, S. and Lines, D., Surgery formulae for Casson’s invariant and extensions to homology lens spaces, J. Reine Angew. Math. 405 (1990), 181–220; MR 1041002 (91e:57026).Google Scholar
Boyer, S. and Zhang, X., Reducing Dehn filling and toroidal Dehn filling, Topology Appl. 68 (1996), 285–303; MR 1377050 (97f:57018).CrossRefGoogle Scholar
Cha, J. C. and Livingston, C., KnotInfo: Table of knot invariants, http://www.indiana.edu/∼knotinfo.Google Scholar
Chantraine, B., Lagrangian concordance of Legendrian knots, Algebr. Geom. Topol. 10 (2010), 63–85; MR 2580429 (2011f:57049).CrossRefGoogle Scholar
Cieliebak, K., Handle attaching in symplectic homology and the chord conjecture, J. Eur. Math. Soc. (JEMS) 4 (2002), 115–142; MR 1911873 (2003d:53153).CrossRefGoogle Scholar
Cieliebak, K. and Eliashberg, Y., From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds, American Mathematical Society Colloquium Publications, vol. 59 (American Mathematical Society, Providence, RI, 2012); MR 3012475.Google Scholar
Cromwell, P. R., Homogeneous links, J. Lond. Math. Soc. (2) 39 (1989), 535–552; MR 1002465 (90f:57001).CrossRefGoogle Scholar
Eliashberg, Y., Filling by holomorphic discs and its applications, in Geometry of low-dimensional manifolds, 2 (Durham, 1989), London Mathematical Society Lecture Note Series, vol. 151 (Cambridge University Press, Cambridge, 1990), 45–67; MR 1171908 (93g:53060).Google Scholar
Etnyre, J. B., Legendrian and transversal knots, in Handbook of knot theory (Elsevier, Amsterdam, 2005), 105–185; MR 2179261 (2006j:57050).CrossRefGoogle Scholar
Etnyre, J. B. and Honda, K., Knots and contact geometry. I. Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001), 63–120; MR 1959579 (2004d:57032).CrossRefGoogle Scholar
Etnyre, J. B. and Honda, K., Cabling and transverse simplicity, Ann. of Math. (2) 162 (2005), 1305–1333; MR 2179731 (2006j:57051).CrossRefGoogle Scholar
Etnyre, J. B., LaFountain, D. J. and Tosun, B., Legendrian and transverse cables of positive torus knots, Geom. Topol. 16 (2012), 1639–1689; MR 2967060.CrossRefGoogle Scholar
Etnyre, J. B. and Van Horn-Morris, J., Fibered transverse knots and the Bennequin bound, Int. Math. Res. Not. IMRN 2011 1483–1509; MR 2806512 (2012k:57019).Google Scholar
Eudave-Muñoz, M., Band sums of links which yield composite links. The cabling conjecture for strongly invertible knots, Trans. Amer. Math. Soc. 330 (1992), 463–501; MR 1112545 (92m:57009).CrossRefGoogle Scholar
Fuchs, D., Chekanov–Eliashberg invariant of Legendrian knots: existence of augmentations, J. Geom. Phys. 47 (2003), 43–65; MR 1985483 (2004h:57007).CrossRefGoogle Scholar
Gabai, D., Foliations and the topology of 3-manifolds. III, J. Differential Geom. 26 (1987), 479–536; MR 910018 (89a:57014b).Google Scholar
Gabai, D., Surgery on knots in solid tori, Topology 28 (1989), 1–6; MR 991095 (90h:57005).CrossRefGoogle Scholar
Giroux, E., Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math. 141 (2000), 615–689; MR 1779622 (2001i:53147).CrossRefGoogle Scholar
Gompf, R. E., Handlebody construction of Stein surfaces, Ann. of Math. (2) 148 (1998), 619–693; MR 1668563 (2000a:57070).CrossRefGoogle Scholar
González-Acuña, F. and Short, H., Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc. 99 (1986), 89–102; MR 809502 (87c:57003).CrossRefGoogle Scholar
Gordon, C. M. and Luecke, J., Only integral Dehn surgeries can yield reducible manifolds, Math. Proc. Cambridge Philos. Soc. 102 (1987), 97–101; MR 886439 (89a:57003).CrossRefGoogle Scholar
Gordon, C. M. and Luecke, J., Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371–415; MR 965210 (90a:57006a).CrossRefGoogle Scholar
Gromov, M., Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347; MR 809718 (87j:53053).CrossRefGoogle Scholar
Hayden, K. and Sabloff, J. M., Positive knots and Lagrangian fillability, Proc. Amer. Math. Soc. 143 (2015), 1813–1821; MR 3314092.CrossRefGoogle Scholar
Hedden, M., On knot Floer homology and cabling. II, Int. Math. Res. Not. IMRN 2009 2248–2274; MR 2511910 (2011f:57015).Google Scholar
Hedden, M., Notions of positivity and the Ozsváth–Szabó concordance invariant, J. Knot Theory Ramifications 19 (2010), 617–629; MR 2646650 (2011j:57020).CrossRefGoogle Scholar
Hind, R., Stein fillings of lens spaces, Commun. Contemp. Math. 5 (2003), 967–982; MR 2030565 (2005e:53141).CrossRefGoogle Scholar
Hirasawa, M., Murasugi, K. and Silver, D. S., When does a satellite knot fiber? Hiroshima Math. J. 38 (2008), 411–423; MR 2477750 (2010i:57016).Google Scholar
Hom, J., A note on cabling and L-space surgeries, Algebr. Geom. Topol. 11 (2011), 219–223; MR 2764041 (2012i:57020).CrossRefGoogle Scholar
Hom, J., Lidman, T. and Zufelt, N., Reducible surgeries and Heegaard Floer homology, Math. Res. Lett. 22 (2015), 763–788.CrossRefGoogle Scholar
Honda, K., On the classification of tight contact structures. I, Geom. Topol. 4 (2000), 309–368; MR 1786111 (2001i:53148).CrossRefGoogle Scholar
Howie, J., A proof of the Scott-Wiegold conjecture on free products of cyclic groups, J. Pure Appl. Algebra 173 (2002), 167–176; MR 1915093 (2003h:20048).CrossRefGoogle Scholar
Ishikawa, M., On the Thurston–Bennequin invariant of graph divide links, Math. Proc. Cambridge Philos. Soc. 139 (2005), 487–495; MR 2177173 (2006k:57035).CrossRefGoogle Scholar
Jong, I. D. and Kishimoto, K., On positive knots of genus two, Kobe J. Math. 30 (2013), 1–18; MR 3157050.Google Scholar
Kaloti, A., Stein fillings of planar open books, Preprint (2013), arXiv:1311.0208.Google Scholar
Leverson, C., Augmentations and rulings of Legendrian knots, Preprint (2014), arXiv:1403.4982.Google Scholar
Lisca, P., Symplectic fillings and positive scalar curvature, Geom. Topol. 2 (1998), 103–116 (electronic). MR 1633282 (99f:57038).CrossRefGoogle Scholar
Lisca, P., On lens spaces and their symplectic fillings, Math. Res. Lett. 11 (2004), 13–22; MR 2046195 (2004m:57058).CrossRefGoogle Scholar
Lisca, P., On symplectic fillings of lens spaces, Trans. Amer. Math. Soc. 360 (2008), 765–799 (electronic). MR 2346471 (2008h:57039).CrossRefGoogle Scholar
Livingston, C., Computations of the Ozsváth–Szabó knot concordance invariant, Geom. Topol. 8 (2004), 735–742 (electronic); MR 2057779 (2005d:57019).CrossRefGoogle Scholar
Luft, E. and Zhang, X., Symmetric knots and the cabling conjecture, Math. Ann. 298 (1994), 489–496; MR 1262772 (95f:57014).CrossRefGoogle Scholar
Manolescu, C. and Ozsváth, P., On the Khovanov and knot Floer homologies of quasi-alternating links, in Proceedings of Gökova Geometry-Topology Conference 2007 (GGT and International Press, Somerville, MA, 2008), 60–81; MR 2509750 (2010k:57029).Google Scholar
Matignon, D. and Sayari, N., Longitudinal slope and Dehn fillings, Hiroshima Math. J. 33 (2003), 127–136; MR 1966655 (2004c:57029).CrossRefGoogle Scholar
McDuff, D., The structure of rational and ruled symplectic 4-manifolds, J. Amer. Math. Soc. 3 (1990), 679–712; MR 1049697 (91k:58042).CrossRefGoogle Scholar
McLean, M., Lefschetz fibrations and symplectic homology, Geom. Topol. 13 (2009), 1877–1944; MR 2497314 (2011d:53224).CrossRefGoogle Scholar
Moser, L., Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737–745; MR 0383406 (52 #4287).CrossRefGoogle Scholar
Ni, Y., Knot Floer homology detects fibred knots, Invent. Math. 170 (2007), 577–608; MR 2357503 (2008j:57053).CrossRefGoogle Scholar
Ozbagci, B., Surgery diagrams for horizontal contact structures, Acta Math. Hungar. 120 (2008), 193–208; MR 2431368 (2009e:57047).CrossRefGoogle Scholar
Ozbagci, B. and Stipsicz, A. I., Surgery on contact 3-manifolds and Stein surfaces, Bolyai Society Mathematical Studies, vol. 13 (Springer and János Bolyai Mathematical Society, Berlin and Budapest, 2004); MR 2114165 (2005k:53171).CrossRefGoogle Scholar
Ozsváth, P. and Szabó, Z., Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003), 225–254 (electronic). MR 1988285 (2004f:57040).CrossRefGoogle Scholar
Ozsváth, P. and Szabó, Z., Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615–639; MR 2026543 (2004i:57036).CrossRefGoogle Scholar
Ozsváth, P. and Szabó, Z., Holomorphic disks and knot invariants, Adv. Math. 186 (2004), 58–116; MR 2065507 (2005e:57044).CrossRefGoogle Scholar
Ozsváth, P. and Szabó, Z., Holomorphic disks and three-manifold invariants: Properties and applications, Ann. of Math. (2) 159 (2004), 1159–1245; MR 2113020 (2006b:57017).CrossRefGoogle Scholar
Ozsváth, P. and Szabó, Z., On knot Floer homology and lens space surgeries, Topology 44 (2005), 1281–1300; MR 2168576 (2006f:57034).CrossRefGoogle Scholar
Plamenevskaya, O., Bounds for the Thurston–Bennequin number from Floer homology, Algebr. Geom. Topol. 4 (2004), 399–406; MR 2077671 (2005d:57039).CrossRefGoogle Scholar
Plamenevskaya, O., Transverse knots and Khovanov homology, Math. Res. Lett. 13 (2006), 571–586; MR 2250492 (2007d:57043).CrossRefGoogle Scholar
Plamenevskaya, O. and Van Horn-Morris, J., Planar open books, monodromy factorizations and symplectic fillings, Geom. Topol. 14 (2010), 2077–2101; MR 2740642 (2012c:57047).CrossRefGoogle Scholar
Przytycki, J. H. and Taniyama, K., Almost positive links have negative signature, J. Knot Theory Ramifications 19 (2010), 187–289; MR 2647054 (2011d:57041).CrossRefGoogle Scholar
Rasmussen, J., Khovanov homology and the slice genus, Invent. Math. 182 (2010), 419–447; MR 2729272 (2011k:57020).CrossRefGoogle Scholar
Rudolph, L., A congruence between link polynomials, Math. Proc. Cambridge Philos. Soc. 107 (1990), 319–327; MR 1027784 (90k:57010).CrossRefGoogle Scholar
Rudolph, L., Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. (N.S.) 29 (1993), 51–59; MR 1193540 (94d:57028).CrossRefGoogle Scholar
Rudolph, L., An obstruction to sliceness via contact geometry and ‘classical’ gauge theory, Invent. Math. 119 (1995), 155–163; MR 1309974 (95k:57013).CrossRefGoogle Scholar
Scharlemann, M., Producing reducible 3-manifolds by surgery on a knot, Topology 29 (1990), 481–500; MR 1071370 (91i:57003).CrossRefGoogle Scholar
Shibuya, T., Genus of torus links and cable links, Kobe J. Math. 6 (1989), 37–42; MR 1023523 (91b:57007).Google Scholar
Shumakovitch, A. N., Rasmussen invariant, slice-Bennequin inequality, and sliceness of knots, J. Knot Theory Ramifications 16 (2007), 1403–1412; MR 2384833 (2008m:57034).CrossRefGoogle Scholar
Stoimenow, A., Properties of closed 3-braids and other link braid representations, Preprint (2006), arXiv:math.GT/0606435.Google Scholar
Tosun, B., On the Legendrian and transverse classification of cablings, Math. Res. Lett. 20 (2013), 787–803; MR 3188033.CrossRefGoogle Scholar
Walker, K., An extension of Casson’s invariant, Annals of Mathematics Studies, vol. 126 (Princeton University Press, Princeton, NJ, 1992); MR 1154798 (93e:57031).CrossRefGoogle Scholar
Yamada, Y., Divide knot presentation of sporadic knots of Berge’s lens space surgery, Preprint (2012), arXiv:1210.2905.Google Scholar