Skip to main content

Introduction

  • 1005 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 2069)

Abstract

In the last 3 decades, there has been significant progress in 3-dimensional topology, due in large part to the application of new techniques from other areas of mathematics and from physics. On the one hand, ideas from geometry have led to geometric decompositions of 3-manifolds and to invariants such as the A-polynomial and hyperbolic volume. On the other hand, ideas from quantum physics have led to the development of invariants such as the Jones polynomial and colored Jones polynomials.

Keywords

  • State Graph
  • Hyperbolic Geometry
  • Solid Torus
  • Jones Polynomial
  • Link Diagram

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.99
Price excludes VAT (Canada)
  • Compact, lightweight 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

Learn about institutional subscriptions
Fig. 1.1

References

  1. Adams, C.C.: Noncompact Fuchsian and quasi-Fuchsian surfaces in hyperbolic 3-manifolds. Algebr. Geomet. Topology 7, 565–582 (2007)

    CrossRef  MATH  Google Scholar 

  2. Agol, I.: Lower bounds on volumes of hyperbolic Haken 3-manifolds. arXiv:math/9906182

    Google Scholar 

  3. Agol, I.: The virtual Haken conjecture. arXiv:1204.2810. With an appendix by Ian Agol, Daniel Groves, and Jason Manning

    Google Scholar 

  4. Agol, I.: Criteria for virtual fibering. J. Topology 1(2), 269–284 (2008). doi:10.1112/jtopol/jtn003

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Agol, I., Storm, P.A., Thurston, W.P.: 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 

  6. Andreev, E.M.: Convex polyhedra in Lobačevskiĭ spaces. Mat. Sb. (N.S.) 81(123), 445–478 (1970)

    Google Scholar 

  7. Andreev, E.M.: Convex polyhedra of finite volume in Lobačevskiĭ space. Mat. Sb. (N.S.) 83(125), 256–260 (1970)

    Google Scholar 

  8. Atiyah, M.: The Geometry and Physics of Knots. Lezioni Lincee [Lincei Lectures]. Cambridge University Press, Cambridge (1990). doi:10.1017/CBO9780511623868

    Google Scholar 

  9. Cha, J.C., Livingston, C.: Knotinfo: Table of knot invariants (2012) http://www.indiana.edu/~knotinfo

  10. Champanerkar, A., Kofman, I., Patterson, E.: The next simplest hyperbolic knots. J. Knot Theor. Ramifications 13(7), 965–987 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Cromwell, P.R.: Homogeneous links. J. Lond. Math. Soc. (2) 39(3), 535–552 (1989). doi:10.1112/jlms/s2-39.3.535

    Google Scholar 

  12. Culler, M., Shalen, P.B.: Volumes of hyperbolic Haken manifolds. I. Invent. Math. 118(2), 285–329 (1994). doi:10.1007/BF01231535

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Dasbach, O.T., Futer, D., Kalfagianni, E., Lin, X.S., Stoltzfus, N.W.: The Jones polynomial and graphs on surfaces. J. Combin. Theor. Ser. B 98(2), 384–399 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Dasbach, O.T., Lin, X.S.: On the head and the tail of the colored Jones polynomial. Compos. Math. 142(5), 1332–1342 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Dasbach, O.T., Lin, X.S.: A volume-ish theorem for the Jones polynomial of alternating knots. Pac. J. Math. 231(2), 279–291 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Dimofte, T., Gukov, S.: Quantum field theory and the volume conjecture. Contemp. Math. 541, 41–68 (2011)

    CrossRef  MathSciNet  Google Scholar 

  17. Freyd, P., Yetter, D.N., Hoste, J., Lickorish, W.B.R., Millett, K.C., Ocneanu, A.: A new polynomial invariant of knots and links. Bull. Am. Math. Soc. (N.S.) 12(2), 239–246 (1985). doi:10.1090/S0273-0979-1985-15361-3

    Google Scholar 

  18. Futer, D.: Fiber detection for state surfaces. arXiv:1201.1643 (2012)

    Google Scholar 

  19. Futer, D., Kalfagianni, E., Purcell, J.S.: Quasifuchsian state surfaces. ArXiv:1209.5719 (2012)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  23. Futer, D., Kalfagianni, E., Purcell, J.S.: On diagrammatic bounds of knot volumes and spectral invariants. Geometriae Dedicata 147, 115–130 (2010). doi:10.1007/s10711-009-9442-6

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. Futer, D., Kalfagianni, E., Purcell, J.S.: Slopes and colored Jones polynomials of adequate knots. Proc. Am. Math. Soc. 139, 1889–1896 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. Futer, D., Kalfagianni, E., Purcell, J.S.: Jones polynomials, volume, and essential knot surfaces: a survey. In: Proceedings of Knots in Poland III. Banach Center Publications (to appear)

    Google Scholar 

  26. Garoufalidis, S.: The degree of a q-holonomic sequence is a quadratic quasi-polynomial. Electron. J. Combin. 18(2), Paper 4, 23 (2011)

    Google Scholar 

  27. Garoufalidis, S.: The Jones slopes of a knot. Quant. Topology 2(1), 43–69 (2011). doi:10.4171/QT/13

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. Garoufalidis, S., Lê, T.T.Q.: The colored Jones function is q-holonomic. Geom. Topology 9, 1253–1293 (electronic) (2005). doi:10.2140/gt.2005.9.1253

    Google Scholar 

  29. Gromov, M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. (56), 5–99 (1982/1983)

    Google Scholar 

  30. Hoste, J., Thistlethwaite, M.B.: Knotscape (2012) http://www.math.utk.edu/~morwen

  31. Jaco, W.H., Shalen, P.B.: Seifert fibered spaces in 3-manifolds. Mem. Am. Math. Soc. 21(220), viii+192 (1979)

    Google Scholar 

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

    Google Scholar 

  33. Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc. (N.S.) 12(1), 103–111 (1985). doi:10.1090/S0273-0979-1985-15304-2

    Google Scholar 

  34. Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. (2) 126(2), 335–388 (1987)

    Google Scholar 

  35. Jørgensen, T.: Compact 3-manifolds of constant negative curvature fibering over the circle. Ann. Math. (2) 106(1), 61–72 (1977)

    Google Scholar 

  36. Kashaev, R.M.: Quantum dilogarithm as a 6j-symbol. Mod. Phys. Lett. A 9(40), 3757–3768 (1994). doi:10.1142/S0217732394003610

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. Kashaev, R.M.: A link invariant from quantum dilogarithm. Mod. Phys. Lett. A 10(19), 1409–1418 (1995). doi:10.1142/S0217732395001526

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. Kashaev, R.M.: The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39(3), 269–275 (1997)

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. Kauffman, L.H.: State models and the Jones polynomial. Topology 26(3), 395–407 (1987). doi:10.1016/0040-9383(87)90009-7

    CrossRef  MathSciNet  MATH  Google Scholar 

  40. Kauffman, L.H.: An invariant of regular isotopy. Trans. Am. Math. Soc. 318(2), 417–471 (1990). doi:10.2307/2001315

    CrossRef  MathSciNet  MATH  Google Scholar 

  41. Kuessner, T.: Guts of surfaces in punctured-torus bundles. Arch. Math. (Basel) 86(2), 176–184 (2006). doi:10.1007/s00013-005-1097-4

    Google Scholar 

  42. Lackenby, M.: 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 

  43. Lickorish, W.B.R.: An Introduction to Knot Theory. Graduate Texts in Mathematics, vol. 175. Springer, New York (1997)

    Google Scholar 

  44. Lickorish, W.B.R., Thistlethwaite, M.B.: Some links with nontrivial polynomials and their crossing-numbers. Comment. Math. Helv. 63(4), 527–539 (1988)

    CrossRef  MathSciNet  MATH  Google Scholar 

  45. Manchón, P.M.G.: Extreme coefficients of Jones polynomials and graph theory. J. Knot Theor. Ramifications 13(2), 277–295 (2004). doi:10.1142/S0218216504003135

    CrossRef  MATH  Google Scholar 

  46. Menasco, W.W.: Polyhedra representation of link complements. In: Low-Dimensional Topology (San Francisco, CA, 1981). Contemporary Mathematics, vol. 20, pp. 305–325. American Mathematical Society, Providence (1983)

    Google Scholar 

  47. Menasco, W.W.: Closed incompressible surfaces in alternating knot and link complements. Topology 23(1), 37–44 (1984). doi:10.1016/0040-9383(84)90023-5

    CrossRef  MathSciNet  MATH  Google Scholar 

  48. Miyamoto, Y.: Volumes of hyperbolic manifolds with geodesic boundary. Topology 33(4), 613–629 (1994). doi:10.1016/0040-9383(94)90001-9

    CrossRef  MathSciNet  MATH  Google Scholar 

  49. Morgan, J., Tian, G.: Ricci Flow and the Poincaré Conjecture. Clay Mathematics Monographs, vol. 3. American Mathematical Society, Providence (2007)

    Google Scholar 

  50. Mostow, G.D.: Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms. Inst. Hautes Études Sci. Publ. Math. (34), 53–104 (1968)

    CrossRef  MathSciNet  MATH  Google Scholar 

  51. Murakami, H.: An introduction to the volume conjecture. In: Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory. Contemporary Mathematics, vol. 541, pp. 1–40. American Mathematical Society, Providence (2011). doi:10.1090/conm/541/10677

    Google Scholar 

  52. Murakami, H., Murakami, J.: The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186(1), 85–104 (2001)

    CrossRef  MathSciNet  MATH  Google Scholar 

  53. Ni, Y.: Knot Floer homology detects fibred knots. Invent. Math. 170(3), 577–608 (2007). doi:10.1007/s00222-007-0075-9

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  55. Ozsváth, P., Szabó, Z.: Holomorphic disks and genus bounds. Geom. Topology 8, 311–334 (2004). doi:10.2140/gt.2004.8.311

    CrossRef  MATH  Google Scholar 

  56. Ozsváth, P., Szabó, Z.: Link Floer homology and the Thurston norm. J. Am. Math. Soc. 21(3), 671–709 (2008). doi:10.1090/S0894-0347-08-00586-9

    CrossRef  MATH  Google Scholar 

  57. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159 (2002)

    Google Scholar 

  58. Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109 (2003)

    Google Scholar 

  59. Prasad, G.: Strong rigidity of Q-rank 1 lattices. Invent. Math. 21, 255–286 (1973)

    CrossRef  MathSciNet  MATH  Google Scholar 

  60. Reshetikhin, N., Turaev, V.G.: Ribbon graphs and their invariants derived from quantum groups. Comm. Math. Phys. 127(1), 1–26 (1990)

    CrossRef  MathSciNet  MATH  Google Scholar 

  61. Reshetikhin, N., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103(3), 547–597 (1991). doi:10.1007/BF01239527

    CrossRef  MathSciNet  MATH  Google Scholar 

  62. Riley, R.: Discrete parabolic representations of link groups. Mathematika 22(2), 141–150 (1975)

    CrossRef  MathSciNet  MATH  Google Scholar 

  63. Riley, R.: A quadratic parabolic group. Math. Proc. Camb. Phil. Soc. 77, 281–288 (1975)

    CrossRef  MathSciNet  MATH  Google Scholar 

  64. Stoimenow, A.: Coefficients and non-triviality of the Jones polynomial. J. Reine Angew. Math. 657, 1–55 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

  65. Stoimenow, A.: On the crossing number of semi-adequate links. Forum Math. (in press). doi:10.1515/forum-2011-0121

    Google Scholar 

  66. Thistlethwaite, M.B.: On the Kauffman polynomial of an adequate link. Invent. Math. 93(2), 285–296 (1988). doi:10.1007/BF01394334

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  68. Thurston, W.P.: A norm for the homology of 3-manifolds. Mem. Am. Math. Soc. 59(339), i–vi and 99–130 (1986)

    Google Scholar 

  69. Witten, E.: 2 + 1-dimensional gravity as an exactly soluble system. Nucl. Phys. B 311(1), 46–78 (1988/1989). doi:10.1016/0550-3213(88)90143-5

    Google Scholar 

  70. Witten, E.: Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121(3), 351–399 (1989)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Futer, D., Kalfagianni, E., Purcell, J. (2013). Introduction. In: Guts of Surfaces and the Colored Jones Polynomial. Lecture Notes in Mathematics, vol 2069. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33302-6_1

Download citation