Geodesic
An expansion of the Jones representation of genus 2 and the Torelli group
Kasahara, Yasushi1
1Department of Electronic and Photonic System Engineering, Kochi University of Technology, Tosayamada-cho, Kagami-gun, Kochi, 782–8502, Japan
Algebraic and Geometric Topology, Tome 1 (2001) no. 1, pp. 39-55
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

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.

DOI : 10.2140/agt.2001.1.39
Keywords: Jones representation, mapping class group, Torelli group, Johnson homomorphism

Kasahara, Yasushi  1

1 Department of Electronic and Photonic System Engineering, Kochi University of Technology, Tosayamada-cho, Kagami-gun, Kochi, 782–8502, Japan 
@article{10_2140_agt_2001_1_39,
     author = {Kasahara, Yasushi},
     title = {An expansion of the {Jones} representation of genus 2 and the {Torelli} group},
     journal = {Algebraic and Geometric Topology},
     pages = {39--55},
     year = {2001},
     volume = {1},
     number = {1},
     doi = {10.2140/agt.2001.1.39},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2001.1.39/}
}
TY  - JOUR
AU  - Kasahara, Yasushi
TI  - An expansion of the Jones representation of genus 2 and the Torelli group
JO  - Algebraic and Geometric Topology
PY  - 2001
SP  - 39
EP  - 55
VL  - 1
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2001.1.39/
DO  - 10.2140/agt.2001.1.39
ID  - 10_2140_agt_2001_1_39
ER  - 
%0 Journal Article
%A Kasahara, Yasushi
%T An expansion of the Jones representation of genus 2 and the Torelli group
%J Algebraic and Geometric Topology
%D 2001
%P 39-55
%V 1
%N 1
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2001.1.39/
%R 10.2140/agt.2001.1.39
%F 10_2140_agt_2001_1_39
Kasahara, Yasushi. An expansion of the Jones representation of genus 2 and the Torelli group. Algebraic and Geometric Topology, Tome 1 (2001) no. 1, pp. 39-55. doi: 10.2140/agt.2001.1.39

[1] M Asada, H Nakamura, On graded quotient modules of mapping class groups of surfaces, Israel J. Math. 90 (1995) 93

[2] D Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423

[3] J S Birman, Mapping class groups of surfaces, from: "Braids (Santa Cruz, CA, 1986)", Contemp. Math. 78, Amer. Math. Soc. (1988) 13

[4] J S Birman, New points of view in knot theory, Bull. Amer. Math. Soc. $($N.S.$)$ 28 (1993) 253

[5] J S Birman, H M Hilden, On isotopies of homeomorphisms of Riemann surfaces, Ann. of Math. $(2)$ 97 (1973) 424

[6] J S Birman, X S Lin, Knot polynomials and Vassiliev's invariants, Invent. Math. 111 (1993) 225

[7] W Fulton, J Harris, Representation theory, Graduate Texts in Mathematics 129, Springer (1991)

[8] S P Humphries, Normal closures of powers of Dehn twists in mapping class groups, Glasgow Math. J. 34 (1992) 313

[9] V F R Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. $(2)$ 126 (1987) 335

[10] Y Kasahara, The Jones representation of genus 2 at the 4th root of unity and the Torelli group, Topology Appl. 124 (2002) 129

[11] M Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Ecole Norm. Sup. $(3)$ 71 (1954) 101

[12] W Magnus, A Karrass, D Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley Sons], New York-London-Sydney (1966)

[13] S Morita, Casson's invariant for homology 3–spheres and characteristic classes of surface bundles I, Topology 28 (1989) 305

[14] S Morita, On the structure of the Torelli group and the Casson invariant, Topology 30 (1991) 603

[15] S Morita, Casson invariant, signature defect of framed manifolds and the secondary characteristic classes of surface bundles, J. Differential Geom. 47 (1997) 560

[16] S Morita, Structure of the mapping class groups of surfaces: a survey and a prospect, from: "Proceedings of the Kirbyfest (Berkeley, CA, 1998)", Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry (1999) 349

[17] J Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978) 347

Cité par Sources :