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The Torelli group and the Kauffman bracket skein module

Published online by Cambridge University Press:  23 March 2017

SHUNSUKE TSUJI
SHUNSUKE TSUJI*
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo 3-8-1 Komaba, Meguro, Tokyo, 153-8914Japan e-mail: [email protected]
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Abstract

We introduce an embedding of the Torelli group of a compact connected oriented surface with non-empty connected boundary into the completed Kauffman bracket skein algebra of the surface, which gives a new construction of the first Johnson homomorphism.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 1 , July 2018 , pp. 163 - 178
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Johnson, D. An abelian quotient of the mapping class group ${\Ncal I}_g$. Math. Ann. 249 (1980), 225–242.Google Scholar
[2] Johnson, D. The structure of the Torelli group II: a characterisation of the group generated by twists on bounding curves. Topology 24 no. 2 (1985), 113–126.Google Scholar
[3] Kawazumi, N. and Kuno, Y. The logarithms of Dehn twists. Quantum Topology 5 (3) (2014), 347–423Google Scholar
[4] Kawazumi, N. and Kuno, Y. Groupoid-theoretical methods in the mapping class groups of surfaces. arXiv: 1109.6479 (2011), UTMS preprint: 2011–28.Google Scholar
[5] Massuyeau, G. and Turaev, V. Fox pairings and generalised Dehn twists. Ann. Inst. Fourier 63 (2013), 2403–2456.Google Scholar
[6] Morita, S. On the structure of the Torelli group and the Casson invariant. Topology 30 (1991), 603–621.Google Scholar
[7] Muller, G. Skein algebra and cluster algebras of marked surfaces. arXiv: 1104.0020 (2012).Google Scholar
[8] Putman, A. An infinite presentation of the Torelli group. Geom. Funct. Anal. 19 (2009), no. 2, 591–643.Google Scholar
[9] Tsuji, S. Dehn twists on Kauffman bracket skein algebras. Preprint, arXiv:1510.05139 (2015).Google Scholar
[10] Tsuji, S. The quotient of a Kauffman bracket skein algebra by the square of an augmentation ideal. In preparation.Google Scholar
[11] Tsuji, S. In preparation.Google Scholar
[12] Turaev, V. G. Skein quantisation of Poisson algebras of loops on surfaces. Ann. Sci. Ecole Norm. Sup. (4) 24 (1991), no. 6.Google Scholar