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The Torelli group and the Kauffman bracket skein module

  • SHUNSUKE TSUJI (a1)
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.

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References
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[1] Johnson, D. An abelian quotient of the mapping class group . Math. Ann. 249 (1980), 225242.
[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), 113126.
[3] Kawazumi, N. and Kuno, Y. The logarithms of Dehn twists. Quantum Topology 5 (3) (2014), 347423
[4] Kawazumi, N. and Kuno, Y. Groupoid-theoretical methods in the mapping class groups of surfaces. arXiv: 1109.6479 (2011), UTMS preprint: 2011–28.
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[6] Morita, S. On the structure of the Torelli group and the Casson invariant. Topology 30 (1991), 603621.
[7] Muller, G. Skein algebra and cluster algebras of marked surfaces. arXiv: 1104.0020 (2012).
[8] Putman, A. An infinite presentation of the Torelli group. Geom. Funct. Anal. 19 (2009), no. 2, 591643.
[9] Tsuji, S. Dehn twists on Kauffman bracket skein algebras. Preprint, arXiv:1510.05139 (2015).
[10] Tsuji, S. The quotient of a Kauffman bracket skein algebra by the square of an augmentation ideal. In preparation.
[11] Tsuji, S. In preparation.
[12] Turaev, V. G. Skein quantisation of Poisson algebras of loops on surfaces. Ann. Sci. Ecole Norm. Sup. (4) 24 (1991), no. 6.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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