Skip to main content
SpringerLink
Log in

A Survey of Hyperbolic Knot Theory

  • Conference paper
  • First Online:
Knots, Low-Dimensional Topology and Applications (KNOTS16 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 284))

Included in the following conference series:

Abstract

We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and bounding its cusp shape and cusp area. We give sample applications and state some open questions and conjectures.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    WYSIWYG stands for “what you see is what you get”.

References

  1. C. Adams, Hyperbolic Structures on Link Complements (University of Wisconsin, Madison, 1983). Ph.D. thesis

    Google Scholar 

  2. C. Adams, Hyperbolic Knots, Handbook of knot theory (Elsevier B. V., Amsterdam, 2005), pp. 1–18. MR 2179259

    Google Scholar 

  3. C. Adams, Triple crossing number of knots and links. J. Knot Theory Ramif. 22(2), 1350006 (2013). MR 3037297

    Google Scholar 

  4. C. Adams, Bipyramids and bounds on volumes of hyperbolic links. Topology Appl. 222, 100–114 (2017). MR 3630197

    Google Scholar 

  5. C. Adams, H. Bennett, C. Davis, M. Jennings, J. Kloke, N. Perry, E. Schoenfeld, Totally geodesic Seifert surfaces in hyperbolic knot and link complements. II. J. Differ. Geom. 79(1), 1–23 (2008). MR 2414747 (2009b:57032)

    Google Scholar 

  6. C. Adams, A. Colestock, J. Fowler, W. Gillam, E. Katerman, Cusp size bounds from singular surfaces in hyperbolic 3-manifolds. Trans. Am. Math. Soc. 358(2), 727–741 (electronic) (2006). MR MR2177038 (2006k:57041)

    Google Scholar 

  7. C.C. Adams, Limit volumes of hyperbolic three-orbifolds. J. Differ. Geom. 34(1), 115–141 (1991). MR 1114455 (92d:57029)

    Google Scholar 

  8. C.C. Adams, Toroidally alternating knots and links. Topology 33(2), 353–369 (1994)

    Article  MathSciNet  Google Scholar 

  9. C.C. Adams, J.F. Brock, J. Bugbee, T.D. Comar, K.A. Faigin, A.M. Huston, A.M. Joseph, D. Pesikoff, Almost alternating links. Topology Appl. 46(2), 151–165 (1992)

    Article  MathSciNet  Google Scholar 

  10. I. Agol, Lower bounds on volumes of hyperbolic Haken 3-manifolds (1999). arXiv:math/9906182

  11. I. Agol, Bounds on exceptional Dehn filling. Geom. Topol. 4, 431–449 (2000). MR 1799796 (2001j:57019)

    Google Scholar 

  12. I. Agol, The minimal volume orientable hyperbolic 2-cusped 3-manifolds. Proc. Am. Math. Soc. 138(10), 3723–3732 (2010). MR 2661571

    Google Scholar 

  13. I. Agol, Ideal triangulations of pseudo-Anosov mapping tori. Topol. Geom. Dimens. Three, Contemp. Math. 560, Amer. Math. Soc. (Providence, 2011), pp. 1–17. MR 2866919

    Google Scholar 

  14. I. Agol, P.A. Storm, W.P. Thurston, Lower bounds on volumes of hyperbolic Haken 3-manifolds. J. Am. Math. Soc. 20(4), 1053–1077 (2007). with an appendix by Nathan Dunfield

    Google Scholar 

  15. C.K. Atkinson, D. Futer, Small volume link orbifolds. Math. Res. Lett. 20(6), 995–1016 (2013). MR 3228616

    Google Scholar 

  16. C.K. Atkinson, D. Futer, The lowest volume 3-orbifolds with high torsion. Trans. Am. Math. Soc. 369(8), 5809–5827 (2017). MR 3646779

    Google Scholar 

  17. D. Bachman, S. Schleimer, Distance and bridge position. Pac. J. Math. 219(2), 221–235 (2005). MR 2175113 (2007a:57028)

    Google Scholar 

  18. K.L. Baker, Surgery descriptions and volumes of Berge knots. II. Descriptions on the minimally twisted five chain link. J. Knot Theory Ramif. 17(9), 1099–1120 (2008). MR MR2457838

    Google Scholar 

  19. R. Benedetti, C. Petronio, Lectures on Hyperbolic Geometry (Universitext, Springer, Berlin, 1992). MR MR1219310 (94e:57015)

    Google Scholar 

  20. G. Besson, G. Courtois, S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5(5), 731–799 (1995). MR 1354289

    Google Scholar 

  21. S.A. Bleiler, C.D. Hodgson, Spherical space forms and Dehn filling. Topology 35(3), 809–833 (1996)

    Google Scholar 

  22. J. Boland, C. Connell, J. Souto, Volume rigidity for finite volume manifolds. Am. J. Math. 127(3), 535–550 (2005). MR 2141643

    Google Scholar 

  23. K. Böröczky, Packing of spheres in spaces of constant curvature. Acta Math. Acad. Sci. Hungar. 32(3–4), 243–261 (1978). MR MR512399 (80h:52014)

    Google Scholar 

  24. S.D. Burton, E. Kalfagianni, Geometric estimates from spanning surfaces. Bull. Lond. Math. Soc. 49(4), 694–708 (2017)

    Article  MathSciNet  Google Scholar 

  25. R.D. Canary, D.B.A. Epstein, P.L. Green, Notes on notes of Thurston, in Fundamentals of hyperbolic geometry: selected expositions, vol. 328, London Mathematical Society Lecture Note series (Cambridge University Press, Cambridge, 2006), pp. 1–115. With a new foreword by Canary. MR 2235710

    Google Scholar 

  26. A. Champanerkar, I. Kofman, J.S. Purcell, Geometrically and diagrammatically maximal knots. J. Lond. Math. Soc. (2) 94(3), 883–908 (2016). MR 3614933

    Google Scholar 

  27. A. Champanerkar, I. Kofman, J.S. Purcell, Volume bounds for weaving knots. Algebr. Geom. Topol. 16 (6), 3301–3323 (2016). MR 3584259

    Google Scholar 

  28. Y.-E. Choi, Positively oriented ideal triangulations on hyperbolic three-manifolds. Topology 43(6), 1345–1371 (2004). MR 2081429

    Google Scholar 

  29. M. Culler, N.M. Dunfield, M. Goerner, J.R. Weeks, SnapPy, a computer program for studying the geometry and topology of \(3\)-manifolds (2017). http://snappy.computop.org

  30. O. Dasbach, A. Tsvietkova, A refined upper bound for the hyperbolic volume of alternating links and the colored Jones polynomial. Math. Res. Lett. 22(4), 1047–1060 (2015). MR 3391876

    Google Scholar 

  31. O. Dasbach, A. Tsvietkova, Simplicial volume of links from link diagrams. Math. Proc. Camb. Phil. Soc. 166(1), 75–81 (2019). MR 3893305

    Google Scholar 

  32. M. Eudave-Muñoz, J. Luecke, Knots with bounded cusp volume yet large tunnel number. J. Knot Theory Ramif. 8(4), 437–446 (1999). MR 1697382 (2000g:57007)

    Google Scholar 

  33. K. Finlinson, J.S. Purcell, Volumes of Montesinos links. Pac. J. Math. 282(1), 63–105 (2016). MR 3463425

    Google Scholar 

  34. D. Futer, F. Guéritaud, Angled decompositions of arborescent link complements. Proc. Lond. Math. Soc. (3) 98(2), 325–364 (2009). MR 2481951

    Google Scholar 

  35. D. Futer, F. Guéritaud, From angled triangulations to hyperbolic structures, in Interactions between hyperbolic geometry, quantum topology and number theory, vol. 541, Contemporary Mathematics (American Mathematical Society, Providence, 2011), pp. 159–182. MR 2796632

    Google Scholar 

  36. D. Futer, F. Guéritaud, Explicit angle structures for veering triangulations. Algebr. Geom. Topol. 13(1), 205–235 (2013). MR 3031641

    Google Scholar 

  37. D. Futer, E. Kalfagianni, J.S. Purcell, Dehn filling, volume, and the Jones polynomial. J. Differ. Geom. 78(3), 429–464 (2008). MR 2396249

    Google Scholar 

  38. D. Futer, E. Kalfagianni, J.S. Purcell, Symmetric links and Conway sums: volume and Jones polynomial. Math. Res. Lett. 16(2), 233–253 (2009). MR 2496741

    Google Scholar 

  39. D. Futer, E. Kalfagianni, J.S. Purcell, Cusp areas of Farey manifolds and applications to knot theory. Int. Math. Res. Not. IMRN 2010(23), 4434–4497 (2010)

    MathSciNet  MATH  Google Scholar 

  40. D. Futer, E. Kalfagianni, J.S. Purcell, On diagrammatic bounds of knot volumes and spectral invariants. Geom. Dedicata 147, 115–130 (2010). MR 2660569

    Google Scholar 

  41. D. Futer, E. Kalfagianni, J.S. Purcell, Guts of Surfaces and the Colored Jones Polynomial, vol. 2069, Lecture Notes in Mathematics (Springer, Heidelberg, 2013)

    Book  Google Scholar 

  42. D. Futer, E. Kalfagianni, J.S. Purcell, Jones polynomials, volume and essential knot surfaces: a survey, in Knots in Poland. III. Part 1, Banach Center Publication, vol. 100. Polish Academy Science Institute of Mathematics, (Warsaw, 2014), pp. 51–77. MR 3220475

    Google Scholar 

  43. D. Futer, E. Kalfagianni, J.S. Purcell, Hyperbolic semi-adequate links. Commun. Anal. Geom. 23(5), 993–1030 (2015). MR 3458811

    Google Scholar 

  44. D. Futer, J.S. Purcell, Links with no exceptional surgeries. Comment. Math. Helv. 82(3), 629–664 (2007). MR 2314056

    Google Scholar 

  45. D. Futer, S. Schleimer, Cusp geometry of fibered 3-manifolds. Am. J. Math. 136(2), 309–356 (2014). MR 3188063

    Google Scholar 

  46. D. Gabai, R. Meyerhoff, P. Milley, Minimum volume cusped hyperbolic three-manifolds. J. Am. Math. Soc. 22(4), 1157–1215 (2009). MR 2525782

    Google Scholar 

  47. A. Giambrone, Combinatorics of link diagrams and volume. J. Knot Theory Ramif. 24(1), 1550001, 21 (2015). MR 3319678

    Google Scholar 

  48. M. Goerner, Regular tessellation links (2014). arXiv:1406.2827

  49. F. Guéritaud, On canonical triangulations of once-punctured torus bundles and two-bridge link complements. Geom. Topol. 10, 1239–1284 (2006). With an appendix by David Futer. MR 2255497

    Google Scholar 

  50. E. Hironaka, E. Kin, A family of pseudo-Anosov braids with small dilatation. Algebr. Geom. Topol. 6, 699–738 (2006). MR 2240913

    Google Scholar 

  51. C.D. Hodgson, S.P. Kerckhoff, Universal bounds for hyperbolic Dehn surgery. Ann. Math. (2) 162(1), 367–421 (2005). MR MR2178964

    Google Scholar 

  52. C.D. Hodgson, J.H. Rubinstein, H. Segerman, Triangulations of hyperbolic 3-manifolds admitting strict angle structures. J. Topol. 5(4), 887–908 (2012). MR 3001314

    Google Scholar 

  53. C.D. Hodgson, J.H. Rubinstein, H. Segerman, S. Tillmann, Veering triangulations admit strict angle structures. Geom. Topol. 15(4), 2073–2089 (2011). MR 2860987

    Google Scholar 

  54. N.R. Hoffman, J.S. Purcell, Geometry of planar surfaces and exceptional fillings. Bull. Lond. Math. Soc. 49(2), 185–201 (2017). MR 3656288

    Google Scholar 

  55. J. Hoste, M. Thistlethwaite, J. Weeks, The first 1,701,936 knots. Math. Intell. 20(4), 33–48 (1998). MR MR1646740 (99i:57015)

    Google Scholar 

  56. J.A. Howie, Surface-alternating knots and links, Ph.D. thesis, University of Melbourne, 2015

    Google Scholar 

  57. J.A. Howie, J.S. Purcell, Geometry of alternating links on surfaces (2017). arXiv:math/1712.01373

  58. W.H. Jaco, P.B. Shalen, Seifert fibered spaces in \(3\)-manifolds. Mem. Am. Math. Soc. 21(220), viii+192 (1979). MR MR539411 (81c:57010)

    Google Scholar 

  59. K. Johannson, Homotopy Equivalences of  \(3\)-Manifolds with Boundaries, vol. 761, Lecture Notes in Mathematics (Springer, Berlin, 1979)

    Google Scholar 

  60. J. Johnson, WYSIWYG Hyperbolic knots, Low Dimensional Topology Blog. https://ldtopology.wordpress.com/2007/11/18/temporary/

  61. J. Johnson, Y. Moriah, Bridge distance and plat projections. Algebr. Geom. Topol. 16(6), 3361–3384 (2016). MR 3584261

    Google Scholar 

  62. M. Lackenby, Word hyperbolic Dehn surgery. Invent. Math. 140(2), 243–282 (2000)

    Article  MathSciNet  Google Scholar 

  63. M. Lackenby, The volume of hyperbolic alternating link complements. Proc. Lond. Math. Soc. (3) 88(1), 204–224 (2004). With an appendix by Ian Agol and Dylan Thurston

    Google Scholar 

  64. M. Lackenby, J.S. Purcell, Cusp volumes of alternating knots. Geom. Topol. 20(4), 2053–2078 (2016). MR 3548463

    Google Scholar 

  65. M. Lackenby, J.S. Purcell, Essential twisted surfaces in alternating link complements. Algebr. Geom. Topol. 16(6), 3209–3270 (2016). MR 3584257

    Google Scholar 

  66. F. Luo, S. Tillmann, Angle structures and normal surfaces. Trans. Am. Math. Soc. 360(6), 2849–2866 (2008). MR 2379778

    Google Scholar 

  67. W.W. Menasco, Polyhedra representation of link complements, in Low-dimensional Topology (San Francisco, Calif., 1981), vol. 20, Contemporary Mathematics (American Mathematical Society, Providence, 1983), pp. 305–325. MR MR718149 (85e:57006)

    Google Scholar 

  68. W.W. Menasco, Menasco, closed incompressible surfaces in alternating knot and link complements. Topology 23(1), 37–44 (1984)

    Article  MathSciNet  Google Scholar 

  69. J. Milnor, Groups which act on \(S^n\) without fixed points. Am.. J. Math. 79, 623–630 (1957). MR 0090056

    Google Scholar 

  70. Y. Miyamoto, Volumes of hyperbolic manifolds with geodesic boundary. Topology 33(4), 613–629 (1994). MR 1293303

    Google Scholar 

  71. W.D. Neumann, D. Zagier, Volumes of hyperbolic three-manifolds. Topology 24(3), 307–332 (1985). MR 815482

    Google Scholar 

  72. M. Ozawa, Non-triviality of generalized alternating knots. J. Knot Theory Ramif. 15(3), 351–360 (2006)

    Article  MathSciNet  Google Scholar 

  73. M. Ozawa, Essential state surfaces for knots and links. J. Aust. Math. Soc. 91(3), 391–404 (2011)

    Article  MathSciNet  Google Scholar 

  74. J.S. Purcell, Cusp shapes under cone deformation. J. Differ. Geom. 80(3), 453–500 (2008). MR 2472480

    Google Scholar 

  75. J.S. Purcell, An introduction to fully augmented links, in Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory, Contemporary Mathematics (American Mathematical Society, Providence, 2011), pp. 205–220. MR 2796634

    Google Scholar 

  76. J.S. Purcell, Hyperbolic knot theory (2017). http://users.monash.edu/~jpurcell/hypknottheory.html

  77. I. Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. Math. (2) 139(3), 553–580 (1994). MR 1283870

    Google Scholar 

  78. M. Sakuma, J. Weeks, Examples of canonical decompositions of hyperbolic link complements. Jpn. J. Math. (N.S.) 212, 393–439 (1995). MR 1364387

    Google Scholar 

  79. M. Thistlethwaite, A. Tsvietkova, An alternative approach to hyperbolic structures on link complements. Algebr. Geom. Topol. 14(3), 1307–1337 (2014). MR 3190595

    Google Scholar 

  80. W.P. Thurston, The Geometry and Topology of Three-manifolds, Princeton Univ. Math. Dept. Notes (1979). http://www.msri.org/communications/books/gt3m

  81. W.P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Am. Math. Soc. (N.S.) 6(3), 357–381 (1982)

    Google Scholar 

  82. W.P. Thurston, Hyperbolic structures on 3-manifolds II: surface groups and 3-manifolds which fiber over the circle (1998). arXiv:math/9801045

  83. A. Tsvietkova, Determining isotopy classes of crossing arcs in alternating links. Asian J. Math. 22(6), 1005–1024 (2018)

    Google Scholar 

  84. J. Weeks, Computation of hyperbolic structures in knot theory, Handbook of knot theory (Elsevier B. V., Amsterdam, 2005), pp. 461–480. MR 2179268

    Google Scholar 

  85. W. Worden, Experimental statistics of veering triangulations. Exp. Math. (to appear). arXiv:1710.01198. https://doi.org/10.1080/10586458.2018.1437850

  86. K. Yoshida, The minimal volume orientable hyperbolic 3-manifold with 4 cusps. Pac. J. Math. 266(2), 457–476 (2013). MR 3130632

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, PA, 19122, USA

    David Futer

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Efstratia Kalfagianni

  3. School of Mathematical Sciences, Monash University, Clayton, VIC, 3800, Australia

    Jessica S. Purcell

Authors
  1. David Futer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Efstratia Kalfagianni
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jessica S. Purcell
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Efstratia Kalfagianni .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Williams College, Williamstown, MA, USA

    Colin C. Adams

  2. Department of Mathematics, University of Texas at Austin, Austin, TX, USA

    Cameron McA. Gordon

  3. Department of Mathematics, Vanderbilt University, Nashville, TN, USA

    Vaughan F.R. Jones

  4. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, USA

    Louis H. Kauffman

  5. School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece

    Sofia Lambropoulou

  6. Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, USA

    Kenneth C. Millett

  7. Department of Mathematics, Columbian College of Arts & Sciences, George Washington University, Washington, DC, USA

    Jozef H. Przytycki

  8. Department of Mathematics and Applications, University of Milano-Bicocca, Milano, Italy

    Renzo Ricca

  9. Department of Mathematics, North Carolina State University, Raleigh, NC, USA

    Radmila Sazdanovic

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Futer, D., Kalfagianni, E., Purcell, J.S. (2019). A Survey of Hyperbolic Knot Theory. In: Adams, C., et al. Knots, Low-Dimensional Topology and Applications. KNOTS16 2016. Springer Proceedings in Mathematics & Statistics, vol 284. Springer, Cham. https://doi.org/10.1007/978-3-030-16031-9_1

Download citation

Publish with us

Policies and ethics

Search

Navigation