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

Yasushi Kasahara

Algebraic & Geometric Topology 1 (2001) 39–55

arXiv: math.GT/0012216


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 [t,t1]. 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.

Jones representation, mapping class group, Torelli group, Johnson homomorphism
Mathematical Subject Classification 2000
Primary: 57N05
Secondary: 20F38, 20C08, 20F40
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Received: 18 October 2000
Accepted: 30 November 2000
Published: 9 December 2000
Yasushi Kasahara
Department of Electronic and Photonic System Engineering
Kochi University of Technology