An expansion of the Jones representation of genus 2 and the Torelli group

We study the algebraic property of the representation of the mapping class group of a closed oriented surface of genus $2$ constructed by V F R Jones [Annals of Math. 126 (1987) 335-388]. It arises from the Iwahori–Hecke algebra representations of Artin’s braid group of $6$ strings, and is defined over integral Laurent polynomials $ℤ [t, t^{- 1}]$ . We substitute the parameter $t$ with $- e^{h}$ , and then expand the powers $e^{h}$ in their Taylor series. This expansion naturally induces a filtration on the Torelli group which is coarser than its lower central series. We present some results on the structure of the associated graded quotients, which include that the second Johnson homomorphism factors through the representation. As an application, we also discuss the relation with the Casson invariant of homology $3$ –spheres.

Keywords

Jones representation, mapping class group, Torelli group, Johnson homomorphism

Mathematical Subject Classification 2000

Primary: 57N05

Secondary: 20F38, 20C08, 20F40

References

Forward citations

Publication

Received: 18 October 2000

Accepted: 30 November 2000

Published: 9 December 2000

Authors

Yasushi Kasahara
Department of Electronic and Photonic System Engineering Kochi University of Technology Tosayamada-cho Kagami-gun Kochi 782–8502 Japan

Volume 1, issue 1 (2001)

Yasushi Kasahara