## Algebraic & Geometric Topology

### 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 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 $−eh$, and then expand the powers $eh$ 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 1, Number 1 (2001), 39-55.

Dates
Received: 18 October 2000
Accepted: 30 November 2000
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882583

Digital Object Identifier
doi:10.2140/agt.2001.1.39

Mathematical Reviews number (MathSciNet)
MR1800115

Zentralblatt MATH identifier
0964.57016

#### Citation

Kasahara, Yasushi. An expansion of the Jones representation of genus 2 and the Torelli group. Algebr. Geom. Topol. 1 (2001), no. 1, 39--55. doi:10.2140/agt.2001.1.39. https://projecteuclid.org/euclid.agt/1513882583

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