No Access

ASYMPTOTIC BEHAVIORS OF THE COLORED JONES POLYNOMIALS OF A TORUS KNOT

HITOSHI MURAKAMI

Department of Mathematics, Tokyo Institute of Technology, Oh-Okayama, Meguro, Tokyo 152-8551, Japan

https://doi.org/10.1142/S0129167X04002454Cited by:11 (Source: Crossref)

Abstract

We study the asymptotic behaviors of the colored Jones polynomials of torus knots. Contrary to the works by R. Kashaev, O. Tirkkonen, Y. Yokota, and the author, they do not seem to give the volumes or the Chern–Simons invariants of the three-manifolds obtained by Dehn surgeries. On the other hand it is proved that in some cases the limits give the inverse of the Alexander polynomial.

Keywords:

AMSC: Primary 57M27, Primary 57M25

References

D. Bar-Natan and S. Garoufalidis, Invent. Math. 125(1), 103 (1996), DOI: 10.1007/s002220050070. Crossref, Web of Science, Google Scholar
S. Garoufalidis and T. T. Q. Le, The colored Jones function is q-holonomic , arXiv:math.GT/0309214 . Google Scholar
R. Gelca, Proc. Amer. Math. Soc. 130(4), 1235 (2002), DOI: 10.1090/S0002-9939-01-06157-3. Crossref, Web of Science, Google Scholar
R. Gelca and J. Sain, J. Knot Theory Ramifications 12(2), 187 (2003), DOI: 10.1142/S021821650300238X. Link, Web of Science, Google Scholar
M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Pub. Math. 56 (1982) (1983) 5–99 . Google Scholar
K. Hikami, Difference equation of the colored Jones polynomial for torus knot , arXiv:math.GT/0403224 . Google Scholar
K. Hikami and A. N. Kirillov, Phys. Lett. B 575, 343 (2003). Crossref, Web of Science, Google Scholar
V. F. R. Jones, Bull. Amer. Math. Soc. (N.S.) 12(1), 103 (1985), DOI: 10.1090/S0273-0979-1985-15304-2. Crossref, Web of Science, Google Scholar
R. M. Kashaev, Modern Phys. Lett. A 10(19), 1409 (1995). Link, Web of Science, Google Scholar
R. M. Kashaev, Lett. Math. Phys. 39(3), 269 (1997), DOI: 10.1023/A:1007364912784. Crossref, Web of Science, Google Scholar
R. M. Kashaev and O. Tirkkonen, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 269, 262 (2000). Google Scholar
A. N. Kirillov and N. Yu. Reshetikhin , Infinite Dimensional Lie Algebras and Groups , Advanced Series in Mathematical Physics 7 , ed. V. G. Kac ( World Scientifics , Singapore , 1989 ) . Google Scholar
W. B. R. Lickorish , An Introduction to Knot Theory , Graduate Texts in Mathematics 175 ( Springer-Verlag , New York , 1997 ) . Crossref, Google Scholar
J. E. Marsden and M. J. Hoffman , Basic Complex Analysis ( W. H. Freeman and Company , New York , 1987 ) . Google Scholar
P. M. Melvin and H. R. Morton, Commun. Math. Phys. 169(3), 501 (1995), DOI: 10.1007/BF02099310. Crossref, Web of Science, Google Scholar
H. R. Morton, Math. Proc. Cambridge Philos. Soc. 117(1), 129 (1995), DOI: 10.1017/S0305004100072959. Crossref, Web of Science, Google Scholar
L. Moser, Pacific J. Math. 38, 737 (1971). Crossref, Web of Science, Google Scholar
H. Murakami, The asymptotic behavior of the colored Jones function of a knot and its volume, Proceedings of "Art of Low Dimensional Topology VI", ed. T. Kohno (2000) pp. 87–96, arXiv:math.GT/0004036. Google Scholar
H. Murakami and J. Murakami, Acta Math. 186(1), 85 (2001), DOI: 10.1007/BF02392716. Crossref, Web of Science, Google Scholar
H. Murakami and Y. Yokota, The colored Jones polynomials of the figure-eight knot and its Dehn surgery spaces , arXiv:math.GT/0401084 . Google Scholar
W. D. Neumann and D. Zagier, Topology 24(3), 307 (1985), DOI: 10.1016/0040-9383(85)90004-7. Crossref, Web of Science, Google Scholar
L. Rozansky, Commun. Math. Phys. 175(2), 275 (1996), DOI: 10.1007/BF02102409. Crossref, Web of Science, Google Scholar
T. Yoshida, Invent. Math. 81(3), 473 (1985), DOI: 10.1007/BF01388583. Crossref, Web of Science, Google Scholar