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The complex volumes of twist knots
Author(s):
Jinseok
Cho;
Jun
Murakami;
Yoshiyuki
Yokota
Journal:
Proc. Amer. Math. Soc.
137
(2009),
3533-3541.
MSC (2000):
Primary 57M27;
Secondary 51M25, 58J28
Posted:
May 19, 2009
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Abstract:
For a given hyperbolic knot, the third author defined a function whose imaginary part gives the hyperbolic volume of the knot complement. We show that, for a twist knot, the function actually gives the complex volume of the knot complement using Zickert's and Neumann's theory of the extended Bloch groups and the complex volumes.
References:
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- 1.
- R. Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys. 39 (1997), no. 3, 269-275. MR 1434238 (98b:57012)
- 2.
- H. Murakami, Kashaev's invariant and the volume of a hyperbolic knot after Y. Yokota, Physics and combinatorics 1999 (Nagoya), 244-272, World Sci. Publ., River Edge, NJ, 2001. MR 1865040 (2002k:57031)
- 3.
- H. Murakami, J. Murakami, M. Okamoto, T. Takata, and Y. Yokota, Kashaev's conjecture and the Chern-Simons invariants of knots and links, Experiment. Math. 11 (2002), no. 3, 427-435. MR 1959752 (2004a:57016)
- 4.
- W. Neumann, Extended Bloch group and the Cheeger-Chern-Simons class, Geom. Topol. 8 (2004), 413-474. MR 2033484 (2005e:57042)
- 5.
- Y. Yokota, From the Jones polynomial to the A-polynomial of hyperbolic knots, Interdiscip. Inform. Sci. 9 (2003), no. 1, 11-21. MR 2023102 (2004j:57014)
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- 7.
- C. Zickert, The Chern-Simons invariant of a representation, arXiv:0710.2049.
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Additional Information:
Jinseok
Cho
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-742, Korea
Email:
jindol@math.snu.ac.kr
Jun
Murakami
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
Email:
murakami@waseda.jp
Yoshiyuki
Yokota
Affiliation:
Department of Mathematics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
Email:
jojo@tmu.ac.jp
DOI:
10.1090/S0002-9939-09-09906-7
PII:
S 0002-9939(09)09906-7
Keywords:
Twist knot,
volume conjecture,
complex volume
Received by editor(s):
December 29, 2008
Posted:
May 19, 2009
Additional Notes:
This work was carried out while the first author was visiting Waseda University with the support of the Korea National Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2007-612-C00037)
Communicated by:
Daniel Ruberman
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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