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On the universal sl2 invariant of Brunnian bottom tangles

Published online by Cambridge University Press:  01 October 2012

SAKIE SUZUKI
SAKIE SUZUKI*
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan. e-mail: sakie@kurims.kyoto-u.ac.jp
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Abstract

The universal sl2 invariant is an invariant of bottom tangles from which one can recover the colored Jones polynomial of links. We are interested in the relationship between topological properties of bottom tangles and algebraic properties of the universal sl2 invariant. A bottom tangle T is called Brunnian if every proper subtangle of T is trivial. In this paper, we prove that the universal sl2 invariant of n-component Brunnian bottom tangles takes values in a small subalgebra of the n-fold completed tensor power of the quantized enveloping algebra Uh(sl2). As an application, we give a divisibility property of the colored Jones polynomial of Brunnian links.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 1 , January 2013 , pp. 127 - 143
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

REFERENCES

[1]De Concini, C. and Procesi, C.Quantum groups. D-Modules, Representation Theory and Quantum Groups (Venice, 1992). Lecture Notes in Math. vol. 1565 (Springer, Berlin, 1993), pp. 31–140.CrossRefGoogle Scholar
[2]Habiro, K.Bottom tangles and universal invariants. Alg. Geom. Topol. 6 (2006), 1113–1214.CrossRefGoogle Scholar
[3]Habiro, K.A unified Witten-Reshetikhin-Turaev invariants for integral homology spheres. Invent. Math. 171 (2008), no. 1, 1–81.CrossRefGoogle Scholar
[4]Habiro, K.Brunnian links, claspers and Goussarov–Vassiliev finite type invariants. Math. Proc. Camb. Phil. Soc. 142 (2007), no. 3, 459–468.CrossRefGoogle Scholar
[5]Hennings, M.Invariants of links and 3-manifolds obtained from Hopf algebras. J. London Math. Soc. (2) 54 (1996), no. 3, 594–624.CrossRefGoogle Scholar
[6]Kauffman, L. H.Gauss codes, quantum groups and ribbon Hopf algebras. Rev. Math. Phys. 5 (1993), no. 4, 735–773.CrossRefGoogle Scholar
[7]Kauffman, L. and Radford, D. E.Oriented quantum algebras, categories and invariants of knots and links. J. Knot Theory Ramifications 10 (2001), no. 7, 1047–1084.CrossRefGoogle Scholar
[8]Kerler, T.Genealogy of non-perturbative quantum-invariants of 3-manifolds: the surgical family. (English summary) Geometry and physics (Aarhus, 1995), 503–547.Google Scholar
[9]Lawrence, R. J.A universal link invariant. in: The Interface of Mathematics and Particle Physics (Oxford, 1988). Inst. Math. Appl. Conf. Ser. New Ser., vol. 24. Oxford University Press, New York, 1990, pp. 151–156.Google Scholar
[10]Lawrence, R. J.A universal link invariant using quantum groups. in: Differential Geometric Methods in Theoretical Physics (Chester, 1989). (World Sci. Publishing, Teaneck, NJ, 1989), pp. 55–63.Google Scholar
[11]Lusztig, G.Introduction to quantum groups. Progr. Math. 110 (1993).Google Scholar
[12]Ohtsuki, T.Colored ribbon Hopf algebras and universal invariants of framed links. J. Knot Theory Ramifications 2 (1993), no. 2, 211–232.CrossRefGoogle Scholar
[13]Reshetikhin, N. Y. and Turaev, V. G.Ribbon graphs and their invariants derived from quantum groups. Comm. Math. Phys. 127 (1990), no. 1, 1–26.CrossRefGoogle Scholar
[14]Suzuki, S.On the universal sl 2 invariant of ribbon bottom tangles. Algebr. Geom. Topol. 10 (2010), no. 2, 1027–1061.CrossRefGoogle Scholar
[15]Suzuki, S.On the universal sl 2 invariant of boundary bottom tangles. Algebr. Geom. Topol. 12 (2012), 997–1057.CrossRefGoogle Scholar
[16]Suzuki, S. On the colored Jones polynomials of ribbon links, boundary links and Brunnian links. arXiv:1111.6408.Google Scholar