Book Review
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MathSciNet review:
3090428
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Book Information:
Authors:
S. Chmutov,
S. Duzhin and
J. Mostovoy
Title:
Introduction to Vassiliev knot invariants
Additional book information:
Cambridge University Press,
Cambridge,
2012,
xvi+504 pp.,
ISBN 978-1-107-02083-2,
US $70.00.,
hardcover
Dror Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), no. 2, 423–472. MR 1318886, DOI 10.1016/0040-9383(95)93237-2
D. Bar-Natan, Finite Type Invariants, in Encyclopedia of Mathematical Physics, (J.-P. Francoise, G. L. Naber and Tsou S. T., eds.) Elsevier, Oxford, 2006 (vol. 2 p. 340).
S. Chmutov, S. Duzhin, and J. Mostovoy, Introduction to Vassiliev knot invariants, Cambridge University Press, Cambridge, 2012. MR 2962302, DOI 10.1017/CBO9781139107846
Joan S. Birman and Xiao-Song Lin, Knot polynomials and Vassiliev’s invariants, Invent. Math. 111 (1993), no. 2, 225–270. MR 1198809, DOI 10.1007/BF01231287
Predrag Cvitanović, Group theory, Princeton University Press, Princeton, NJ, 2008. Birdtracks, Lie’s, and exceptional groups. MR 2418111, DOI 10.1515/9781400837670
M. N. Gusarov, A new form of the Conway-Jones polynomial of oriented links, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 193 (1991), no. Geom. i Topol. 1, 4–9, 161 (Russian, with English summary). MR 1157140
M. Gusarov, On $n$-equivalence of knots and invariants of finite degree, Topology of manifolds and varieties, Adv. Soviet Math., vol. 18, Amer. Math. Soc., Providence, RI, 1994, pp. 173–192. MR 1296895, DOI 10.1007/bf01130282
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
Maxim Kontsevich, Vassiliev’s knot invariants, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 137–150. MR 1237836
Maxim Kontsevich, Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, Vol. II (Paris, 1992) Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 97–121. MR 1341841
Roger Penrose, Applications of negative dimensional tensors, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) Academic Press, London, 1971, pp. 221–244. MR 0281657
N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), no. 1, 1–26. MR 1036112
V. A. Vassiliev, Cohomology of knot spaces, Theory of singularities and its applications, Adv. Soviet Math., vol. 1, Amer. Math. Soc., Providence, RI, 1990, pp. 23–69. MR 1089670
V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, Translations of Mathematical Monographs, vol. 98, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by B. Goldfarb. MR 1168473, DOI 10.1090/conm/478
Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772
References
- Dror Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), no. 2, 423–472. MR 1318886 (97d:57004), DOI 10.1016/0040-9383(95)93237-2
- D. Bar-Natan, Finite Type Invariants, in Encyclopedia of Mathematical Physics, (J.-P. Francoise, G. L. Naber and Tsou S. T., eds.) Elsevier, Oxford, 2006 (vol. 2 p. 340).
- S. Chmutov, S. Duzhin, and J. Mostovoy, Introduction to Vassiliev knot invariants, Cambridge University Press, Cambridge, 2012. MR 2962302
- Joan S. Birman and Xiao-Song Lin, Knot polynomials and Vassiliev’s invariants, Invent. Math. 111 (1993), no. 2, 225–270. MR 1198809 (94d:57010), DOI 10.1007/BF01231287
- Predrag Cvitanović, Group theory, Princeton University Press, Princeton, NJ, 2008. Birdtracks, Lie’s, and exceptional groups. MR 2418111 (2010b:22001)
- M. N. Gusarov, A new form of the Conway-Jones polynomial of oriented links, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 193 (1991), no. Geom. i Topol. 1, 4–9, 161 (Russian, with English summary). MR 1157140 (93b:57007)
- M. Gusarov, On $n$-equivalence of knots and invariants of finite degree, Topology of manifolds and varieties, Adv. Soviet Math., vol. 18, Amer. Math. Soc., Providence, RI, 1994, pp. 173–192. MR 1296895 (96i:57005)
- 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 (86e:57006), DOI 10.1090/S0273-0979-1985-15304-2
- Maxim Kontsevich, Vassiliev’s knot invariants, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 137–150. MR 1237836 (94k:57014)
- Maxim Kontsevich, Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, Vol. II (Paris, 1992) Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 97–121. MR 1341841 (96h:57027)
- Roger Penrose, Applications of negative dimensional tensors, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), Academic Press, London, 1971, pp. 221–244. MR 0281657 (43 \#7372)
- N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), no. 1, 1–26. MR 1036112 (91c:57016)
- V. A. Vassiliev, Cohomology of knot spaces, Theory of singularities and its applications, Adv. Soviet Math., vol. 1, Amer. Math. Soc., Providence, RI, 1990, pp. 23–69. MR 1089670 (92a:57016)
- V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, Translations of Mathematical Monographs, vol. 98, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by B. Goldfarb. MR 1168473 (94i:57020)
- Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772 (90h:57009)
Review Information:
Reviewer:
Dror Bar-Natan
Affiliation:
University of Toronto, Canada
Journal:
Bull. Amer. Math. Soc.
50 (2013), 685-690
DOI:
https://doi.org/10.1090/S0273-0979-2013-01413-7
Published electronically:
April 17, 2013
Additional Notes:
Picture credits: Rope from “The Project Gutenberg eBook, Knots, Splices and Rope Work, by A. Hyatt Verrill”, http://www.gutenberg.org/files/13510/13510-h/13510-h.htm. Plane from NASA, http://www.grc.nasa.gov/WWW/k-12/airplane/rotations.html.
TeX at http://drorbn.net/AcademicPensieve/2013-01/CDMReview/. This review was written while I was a guest at the Newton Institute, in Cambridge, UK. I wish to thank N. Bar-Natan, I. Halacheva, and P. Lee for comments and suggestions.
Review copyright:
© Copyright 2013
by the author under Creative Commons Attribution-NonCommercial 3.0 Unported License
