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

AGT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to