Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Mobile Device Pairing
 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Strong integrality of quantum invariants of 3-manifolds

Author(s): Thang T. Q. Lê
Journal: Trans. Amer. Math. Soc. 360 (2008), 2941-2963.
MSC (2000): Primary 57M27; Secondary 57M25
Posted: December 11, 2007
MathSciNet review: 2379782
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We prove that the quantum $ SO(3)$-invariant of an arbitrary 3-manifold $ M$ is always an algebraic integer if the order of the quantum parameter is co-prime with the order of the torsion part of $ H_1(M,\mathbb{Z})$. An even stronger integrality, known as cyclotomic integrality, was established by Habiro for integral homology 3-spheres. Here we also generalize Habiro's result to all rational homology 3-spheres.

References:

[An]
G. Andrews, q-series: their development and applications in analysis, number theory, combinatorics, physics, and computer algebra, regional conference series in mathematics, number 66, Amer. Math. Soc., 1985. MR 858826 (88b:11063)

[BBL]
A. Beliakova, C. Blanchet, and T. T. Q. Le, Laplace transform and universal sl(2) invariants, preprint math.QA/0509394.

[GM]
P. Gilmer and G. Masbaum, Integral lattices in TQFTs, preprint math.QA/0411029.

[Ha1]
K. Habiro, On the quantum $ \mathfrak{sl}_2$ invariants of knots and integral homology spheres, Geom. Topol. Monogr. 4 (2002) 55-68. MR 2002603 (2004g:57023)

[Ha2]
K. Habiro, Cyclotomic completions of polynomial rings, Publ. Res. Inst. Math. Sci. 40 (2004), 1127-1146. MR 2105705 (2005j:13010)

[Ha3]
K. Habiro, An integral form of the quantized enveloping algebra of $ {\rm sl}\sb2$ and its completions, J. Pure Appl. Algebra 211 (2007), 265-292. MR 2333771

[HL]
K. Habiro and T. T. Q. Lê, in preparation.

[KM]
R. Kirby and P. Melvin, The $ 3$-manifold invariants of Witten and Reshetikhin-Turaev for $ {\rm sl}(2,C)$, Invent. Math. 105 (1991), 473-545. MR 1117149 (92e:57011)

[KK]
A. Kawauchi and S. Kojima, Algebraic classification of linking pairings on $ 3$-manifolds, Math. Ann. 253 (1980), 29-42. MR 594531 (82b:57007)

[Jo]
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), 335-388. MR 908150 (89c:46092)

[Le1]
T. T. Q. Lê, An invariant of integral homology 3-spheres which is universal for all finite type invariants, AMS translation series 2, Eds. V. Buchtaber and S. Novikov, 179 (1997), 75-100. MR 1437158 (99m:57008)

[Le2]
T. T. Q. Lê, Integrality and symmetry of quantum link invariants, Duke Math. J. 102 (2000) 273-306. MR 1749439 (2001c:57015)

[Le3]
T. T. Q. Lê, Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion, Topology Appl. 127 (2003), 125-152. MR 1953323 (2005b:57026)

[LL]
B.-H. Li and T.-J. Li, Generalized Gaussian sums: Chern-Simons-Witten-Jones invariants of lens-spaces, J. Knot Theory Ramifications 5 (1996), 183-224. MR 1395779 (97e:57018)

[Ma]
G. Masbaum, Skein-theoretical derivation of some formulas of Habiro, Algebraic and Geometric Topology 3 (2003), 537-556. MR 1997328 (2004f:57013)

[MR]
G. Masbaum and J. Roberts, A simple proof of integrality of quantum invariants at prime roots of unity, Math. Proc. Camb. Phil. Soc. 121 (1997), 443-454. MR 1434653 (98h:57037)

[MW]
G. Masbaum and H. Wenzl, Integral modular categories and integrality of quantum invariants at roots of unity of prime order, J. Reine Angew. Math. 505 (1998), 209-235. MR 1662260 (2000a:57043)

[Mu]
H. Murakami, Quantum $ SO(3)$-invariants dominate the $ SU(2)$-invariant of Casson and Walker, Math. Proc. Camb. Phil. Soc. 117 (1995), 237-249. MR 1307078 (95k:57027)

[LMO]
T. T. Q. Lê, J. Murakami, and T. Ohtsuki, On a universal perturbative invariant of $ 3$-manifolds, Topology 37 (1998), 539-574. MR 1604883 (99d:57004)

[Oht1]
T. Ohtsuki, A polynomial invariant of rational homology $ 3$-spheres, Invent. Math. 123 (1996), 241-257. MR 1374199 (97i:57018)

[Oht2]
T. Ohtsuki, Quantum invariants. A study of knots, 3-manifolds, and their sets, Series on Knots and Everything, 29, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. MR 1881401 (2003f:57027)

[RG]
H. Rademacher and E. Grosswald, Dedekind Sums, Amer. Math. Soc., Washington D. C., 1972. MR 0357299 (50:9767)

[Ros]
L. Rozansky, On p-adic properties of the Witten-Reshetikhin-Turaev invariant, preprint math.QA/9806075, 1998.

[TY]
T. Takata and Y. Yokota, The $ PSU(n)$ invariants of 3-manifolds are polynomials, J. Knot Theory Ramifications 8 (1999), no. 4, 521-532. MR 1697388 (2000k:57016)

[Tur]
V. G. Turaev, Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter, Berlin, New York, 1994. MR 1292673 (95k:57014)

[Wa]
C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 1963, 281-298 MR 0156890 (28:133)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57M27, 57M25

Retrieve articles in all Journals with MSC (2000): 57M27, 57M25

Additional Information:

Thang T. Q. Lê
Affiliation: Department of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: letu@math.gatech.edu

DOI: 10.1090/S0002-9947-07-04359-0
PII: S 0002-9947(07)04359-0
Received by editor(s): March 2, 2006
Posted: December 11, 2007
Additional Notes: The author was supported in part by the National Science Foundation
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia