Book Review
The AMS does not provide abstracts of book reviews.
You may download the entire review from the links below.
Full text of review:
PDF
This review is available free of charge.
Book Information:
Author:
Akio Kawauchi
Title:
A survey of knot theory
Additional book information:
Birkhäuser-Verlag,
Basel, Boston, and Berlin,
1996,
xxi+420 pp.,
ISBN 3-7643-5124-1,
$89.50$
1. J. W. Alexander, Note on Riemann spaces. Bull. Amer. Math. Soc., Vol. 26 (1920), pp. 370-372.
2. J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. USA, Vol. 9 (1923), pp. 539-549.
3. J.W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc., Vol. 30 (1928), pp. 275-306.
4. E. Artin, Theorie der zopfe, Hamburg Abh., Vol. 4 (1926), pp. 539-549.
5. C. W. Ashley, The Ashley Book of Knots, Doubleday and Co. (1944).
Marcia Ascher and Robert Ascher, Code of the quipu, University of Michigan Press, Ann Arbor, Mich., 1981. A study in media, mathematics, and culture. MR 646063
Michael Atiyah, The geometry and physics of knots, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1990. MR 1078014, DOI 10.1017/CBO9780511623868
Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
A. J. Casson and C. McA. Gordon, On slice knots in dimension three, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 39–53. MR 520521
J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 329–358. MR 0258014
J. H. Conway and C. McA. Gordon, Knots and links in spatial graphs, J. Graph Theory 7 (1983), no. 4, 445–453. MR 722061, DOI 10.1002/jgt.3190070410
Richard H. Crowell and Ralph H. Fox, Introduction to knot theory, Ginn and Company, Boston, Mass., 1963. Based upon lectures given at Haverford College under the Philips Lecture Program. MR 0146828
13. J. Elffers and M. Schuyt, Cat's Cradle and Other String Figures, Penguin Books (1978).
14. L. Euler, The Seven Bridges of Konigsburg, Presentation to the Russian Academy at St. Petersburg (1735).
Roger Fenn and Colin Rourke, On Kirby’s calculus of links, Topology 18 (1979), no. 1, 1–15. MR 528232, DOI 10.1016/0040-9383(79)90010-7
R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167. MR 0140099
Ralph H. Fox and John W. Milnor, Singularities of $2$-spheres in $4$-space and cobordism of knots, Osaka Math. J. 3 (1966), 257–267. MR 211392
18. K. Fulves, Self-Working Rope Magic, Dover Pub. (1990).
19. K. F. Gauss, Gauss Werke - 1833, Kniglichen Gesellschaft der Wissenshaften zu Gttingen (1877). p. 605.
Cole A. Giller, A family of links and the Conway calculus, Trans. Amer. Math. Soc. 270 (1982), no. 1, 75–109. MR 642331, DOI 10.1090/S0002-9947-1982-0642331-X
Richard Hartley, The Conway potential function for links, Comment. Math. Helv. 58 (1983), no. 3, 365–378. MR 727708, DOI 10.1007/BF02564642
Vaughan F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103–111. MR 766964, DOI 10.1090/S0273-0979-1985-15304-2
Louis H. Kauffman, Branched coverings, open books and knot periodicity, Topology 13 (1974), 143–160. MR 375337, DOI 10.1016/0040-9383(74)90005-6
Louis H. Kauffman and Walter D. Neumann, Products of knots, branched fibrations and sums of singularities, Topology 16 (1977), no. 4, 369–393. MR 488073, DOI 10.1016/0040-9383(77)90042-8
Louis H. Kauffman, The Conway polynomial, Topology 20 (1981), no. 1, 101–108. MR 592573, DOI 10.1016/0040-9383(81)90017-3
Louis H. Kauffman, Formal knot theory, Mathematical Notes, vol. 30, Princeton University Press, Princeton, NJ, 1983. MR 712133
Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. MR 899057, DOI 10.1016/0040-9383(87)90009-7
Louis H. Kauffman, On knots, Annals of Mathematics Studies, vol. 115, Princeton University Press, Princeton, NJ, 1987. MR 907872
Louis H. Kauffman (ed.), Knots and applications, Series on Knots and Everything, vol. 6, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR 1364187, DOI 10.1142/9789812796189
Robion Kirby, A calculus for framed links in $S^{3}$, Invent. Math. 45 (1978), no. 1, 35–56. MR 467753, DOI 10.1007/BF01406222
J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR 246314, DOI 10.1007/BF02564525
32. A. A. Markov, ber die freie aquivalenz geschlossener zopfe, Recueil Mathematique Moscou, Vol. 1, pp. 73-78.
William Menasco and Morwen Thistlethwaite, The classification of alternating links, Ann. of Math. (2) 138 (1993), no. 1, 113–171. MR 1230928, DOI 10.2307/2946636
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
36. K. Reidemeister, Knotentheorie, (1932) Ergeben. Math. Grenzgeb., Bd. 1; Berlin: Springer Verlag.
N. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597. MR 1091619, DOI 10.1007/BF01239527
38. A. Stasiak, V. Katritch and L. H. Kauffman, Ideal Knots, (1998), Vol. 19 in Series on K&E, World Sci. Pub.
39. H. Seifert, Über das geschlecht von knoten, Math. Ann., Vol. 110, (1934), pp. 539-549.
Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772
- 1.
- J. W. Alexander, Note on Riemann spaces. Bull. Amer. Math. Soc., Vol. 26 (1920), pp. 370-372.
- 2.
- J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. USA, Vol. 9 (1923), pp. 539-549.
- 3.
- J.W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc., Vol. 30 (1928), pp. 275-306.
- 4.
- E. Artin, Theorie der zopfe, Hamburg Abh., Vol. 4 (1926), pp. 539-549.
- 5.
- C. W. Ashley, The Ashley Book of Knots, Doubleday and Co. (1944).
- 6.
- M. Ascher and R. Ascher, Code of the Quipu, University of Michigan Press, Ann Arbor (1981). MR 0646063
- 7.
- M.F. Atiyah, The Geometry and Physics of Knots, (1990), Cambridge University Press. MR 1078014
- 8.
- J. Birman, Braids, Links and Mapping Class Groups, Annals of Math. Studies, Vol. 82 (1974). MR 0375281
- 9.
- A. J. Casson and C. McA. Gordon, On slice knots in dimension three, In: Algebraic and Geometric Topology (Stanford 1976), pp. 39-53, Proc. Sympos. Pure Math., 32-II, (edited by J. Milgram), Amer. Math. Soc. (1983). MR 0520521
- 10.
- J. H. Conway, An enumeration of knots and links and some of their algebraic properties, In: Computational Problems in Abstract Algebra, Proc. Conf. Oxford (1967) (edited by J. Leech), pp. 329-358; New York: Pergamon Press. MR 0258014
- 11.
- J. H. Conway and C. McA Gordon, Knots and links in spatial graphs, J. Graph Theory, Vol. 7 (1983), No. 4, pp. 445-453. MR 0722061
- 12.
- R. H. Crowell and R. H. Fox, Introduction to Knot Theory, New York: Ginn and Co. (1963), or: Grad. Texts Math. 57, Berlin-Heidelberg-New York: Springer Verlag (1977). MR 0146828; MR 56:3829
- 13.
- J. Elffers and M. Schuyt, Cat's Cradle and Other String Figures, Penguin Books (1978).
- 14.
- L. Euler, The Seven Bridges of Konigsburg, Presentation to the Russian Academy at St. Petersburg (1735).
- 15.
- R. Fenn and C. Rourke, On Kirby's calculus of links, Topology, Vol. 18 (1979), pp. 539-549. MR 0528232
- 16.
- R. H. Fox., A quick trip through knot theory, In Topology of Three Manifolds - Proceedings of 1961 Topology Institute at Univ. of Georgia, edited by M. K. Fort, pp. 120-167. Englewood Cliffs, N. J. : Prentice-Hall. MR 0140099
- 17.
- R. H. Fox and J. Milnor, Singularities of 2- spheres in 4 - space and cobordism of knots. Osaka J. Math., Vol. 3 (1966), pp. 539-549. MR 0211392
- 18.
- K. Fulves, Self-Working Rope Magic, Dover Pub. (1990).
- 19.
- K. F. Gauss, Gauss Werke - 1833, Kniglichen Gesellschaft der Wissenshaften zu Gttingen (1877). p. 605.
- 20.
- C. A. Giller, A family of links and the Conway calculus, Trans. Amer. Math. Soc., Vol. 270, (1982), pp. 539-549. MR 0642331
- 21.
- R. Hartley, The Conway potential function for links, Comment. Math. Helv., Vol. 58, (1983), pp. 539-549. MR 0727708
- 22.
- V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc., Vol. 12 (1985), pp. 539-549. MR 0766964
- 23.
- L. H. Kauffman, Branched coverings, open books and knot periodicity, Topology, Vol. 13 (1974), pp. 539-549. MR 0375337
- 24.
- L. H. Kauffman and W. D. Neumann, Products of knots, branched fibrations and sums of singularities. Topology, Vol. 16 (1977), pp. 539-549. MR 0488073
- 25.
- L. H. Kauffman, The Conway Polynomial, Topology, Vol. 20 (1981), pp. 539-549. MR 0592573
- 26.
- L. H. Kauffman, Formal Knot Theory, Mathematical Notes No. 30 (1983), Princeton University Press. MR 0712133
- 27.
- L. H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), pp. 539-549. MR 0899057
- 28.
- L. H. Kauffman On Knots, Ann. of Math. Studies, Vol. 115, Princeton University Press (1987). MR 0907872
- 29.
- L. H. Kauffman (editor), Knots and Applications, Vol. 6 in Series on K&E, World Scientific Pub. Co. (1995). MR 1364187
- 30.
- R. Kirby, A calculus for framed links in , Invent. Math., Vol. 45 (1978), pp. 539-549. MR 0467753
- 31.
- J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv., Vol. 44 (1969), pp. 539-549. MR 0246314
- 32.
- A. A. Markov, ber die freie aquivalenz geschlossener zopfe, Recueil Mathematique Moscou, Vol. 1, pp. 73-78.
- 33.
- W. Menasco and M. Thistlethwaite, The classification of alternating links, Ann. of Math., Vol. 138, (1993), pp. 539-549. MR 1230928
- 34.
- J. Milnor, On manifolds homeomorphic to the 7-sphere, Annals of Math., Vol. 64, No. 2, Sept. 1956, pp. 399-405. MR 18:498d
- 35.
- J. Milnor, Singular Points of Complex Hypersurfaces, Ann. of Math. Studies Vol. 61 (1968), Princeton University Press, Princeton, New Jersey. MR 0239612
- 36.
- K. Reidemeister, Knotentheorie, (1932) Ergeben. Math. Grenzgeb., Bd. 1; Berlin: Springer Verlag.
- 37.
- N. Y. Reshetikhin and V. Turaev, Invariants of three manifolds via link polynomials and quantum groups, Invent. Math., Vol. 103 (1991), pp. 539-549. MR 1091619
- 38.
- A. Stasiak, V. Katritch and L. H. Kauffman, Ideal Knots, (1998), Vol. 19 in Series on K&E, World Sci. Pub.
- 39.
- H. Seifert, Über das geschlecht von knoten, Math. Ann., Vol. 110, (1934), pp. 539-549.
- 40.
- E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys., Vol. 121 (1989), pp. 539-549. MR 0990772
Review Information:
Reviewer:
Louis H. Kauffman
Affiliation:
University of Illinois at Chicago
Email:
kauffman@uic.edu
Journal:
Bull. Amer. Math. Soc.
36 (1999), 539-549
DOI:
https://doi.org/10.1090/S0273-0979-99-00797-1
Published electronically:
July 29, 1999
Review copyright:
© Copyright 1999
American Mathematical Society