#### Algebraic and Geometric Topology 1 (2001),
paper no. 3, pages 39-55.

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

### Yasushi Kasahara

**Abstract**.
We study the algebraic property of the representation of the mapping
class group of a closed oriented surface of genus 2 constructed by VFR
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

**AMS subject classification**.
Primary: 57N05.
Secondary: 20F38, 20C08, 20F40.

**DOI:** 10.2140/agt.2001.1.39

**E-print:** `arXiv:math.GT/0012216`

Submitted: 18 October 2000.
Accepted: 30 November 2000.
Published: 9 December 2000.

Notes on file formats
Yasushi Kasahara

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

Email: kasahara@ele.kochi-tech.ac.jp

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