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An expansion of the Jones representation of genus 2 and the Torelli group. II. (English) Zbl 1058.57015

J. Knot Theory Ramifications 13, No. 2, 297-306 (2004); addendum ibid. 20, No. 6, 939-941 (2011).
The author investigates the algebraic properties of the Jones representation \[ \rho : M_2 \to \text{GL}(5, \mathbb Z[t,t^{-1}]) \] of the mapping class group \(M_2\) of a closed oriented surface of genus 2. The genus 2 Torelli group \(I_2\) is the kernel of the symplectic representation \(M_2 \to \text{Sp}(4,\mathbb Z)\). By considering the perturbation of \(\rho\) at \(t=1\), that is, putting \(t=e^h\) and expanding the powers of \(e^h\) in their Taylor series at \(h=0\), the author obtains a filtration of \(I_2\). The main results of the paper involve the structures of the associated graded quotients and “may reinforce the suspicion” that the restriction of \(\rho\) to \(I_2\) is not faithful. The paper is a sequel to the author’s earlier work [Algebr. Geom. Topol. 1, 39–55 (2001; Zbl 0964.57016)].

MSC:

57M99 General low-dimensional topology
20C08 Hecke algebras and their representations
20C30 Representations of finite symmetric groups

Citations:

Zbl 0964.57016
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References:

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