# zbMATH — the first resource for mathematics

Espace des représentations du groupe d’un noeud classique dans un groupe de Lie. (Representation space of classical knot groups in a Lie group). (French) Zbl 0956.57006
Ann. Inst. Fourier 50, No. 4, 1297-1321 (2000); addendum ibid. 51, No. 4, 1151-1152 (2001).
Let $$X$$ be the complement of a knot in a rational homology 3-sphere, let $$\Pi$$ be the fundamental group of $$X$$ and let $$\rho_0$$ be an abelian representation of $$\Pi$$ in a connected complex reductive Lie group $$G$$ which factorizes through the infinite cyclic group $$H_1(X,Z)/\text{tors}(H_1(X,Z))$$. In this paper, the author proposes a general method for the study of the local structure of the variety of representations of $$\Pi$$ in a neighborhood of the representation $$\rho_0$$. In particular, she proves the following
Theorem. Suppose that the group $$\Pi$$ is equipped with an involution $$k$$ such that $$k(\mu)=\mu^{-1}$$, where $$\mu$$ is a meridian of the knot, and suppose that $$\rho_0(\mu)=h_0$$, where $$h_0$$ is in the Cartan subalgebra $$\mathcal H$$ of $$\mathcal G$$. Suppose also that there exists a root $$\alpha_{i_{0}}$$ of the Lie algebra $$\mathcal G$$ with respect to the Cartan subalgebra $$\mathcal H$$ such that $$w_{i_{0}}= e^{{\alpha_{i_{0}}}(h_{0})}$$ is a simple root of the Alexander polynomial of the knot. Then, there exists an arc of non-metabelian representations $$\rho_t:\Pi\to G$$ having $$\rho_0$$ as an endpoint.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 20C99 Representation theory of groups 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
Lie groups; group cohomology; representation variety
Full Text:
##### References:
 [1] M. ARTIN, On the solutions of analytic equations, Inventiones Math., 5 (1968), 277-291. · Zbl 0172.05301 [2] L. BEN ABDELGHANI, Arcs de représentations du groupe d’un nœud dans un groupe de Lie, Thèse de Doctorat de l’Université de Bourgogne, (1998). · Zbl 0923.57003 [3] E. BRIESKORN, H. KNOERRER, Plane algebraic curves, Birkhaeuser Verlag, Basel, 1986. translated from the German by John Stillwell. · Zbl 0588.14019 [4] C.M. BOILEAU, Nœuds rigidement inversibles, Knot Theory and Manifolds (Proc. Vancouver, 1983) ed. D. Rolfsen, New York, Springer-Verlag, LNM 1144, (1985). · Zbl 0571.57006 [5] N. BOURBAKI, Groupes et algèbres de Lie, Chapitres 2 et 3, (1972) Hermann, Paris. · Zbl 0244.22007 [6] K.S. BROWN, Cohomology of groups, Springer-Verlag, New York-Heidelberg-Berlin, 1982. · Zbl 0584.20036 [7] D. COX, J. LITTLE, D. O’SHEA, Ideals, varieties, and algorithms second edition, Undergraduate Texts in Mathematics, 1997, 1992 Springer-Verlag, New York, Inc.. [8] M. CULLER, P.B. SHALEN, Varieties of group representations and splittings of 3-manifolds, Annals of Mathematics, 117 (1983), 109-146. · Zbl 0529.57005 [9] C. FROHMAN, E. KLASSEN, Deforming representations of knot groups in SU(2), Comment. Math. Helv., 66 (1991), 340-361. · Zbl 0738.57001 [10] C. McA.GORDON, Some aspects of classical knot theory, Knot Theory Proceedings, Plans sur Bex, Switzerland 1977 Springer, LNM 685 (1978). · Zbl 0386.57002 [11] L. GUILLOU, A. MARIN, Notes sur l’invariant de Casson des sphères d’homologie de dimension trois, L’Enseignement Mathématique, tome 38 (1992), 233-290. · Zbl 0776.57008 [12] C.M. HERALD, -existence of irreducible representations for knot complements with nonconstant equivariant signature, Math. Ann., 309, Number 1, (1997). · Zbl 0887.57013 [13] M. HEUSENER, J. KROLL, Deforming abelian SU(2)-representations of knot groups, Comentari Math. Helv., 73 (1998), 480-498. · Zbl 0910.57004 [14] E.P. KLASSEN, Representations of knot groups in SU(2), Transactions of The American Mathematical Society, Volume 326, Number 2, August 1991. · Zbl 0743.57003 [15] A. LUBOTZKY, A. MAGID, Varieties of representations of finitely generated groups, Memoirs of the A.M.S., (58) 336 (1985). · Zbl 0598.14042 [16] A.L. ONISHCHIK, E.B. VINBERG, Lie groups and Lie algebras III, Encyclopaedia of Mathematical Sciences Volume 41, Springer-Verlag (1991). · Zbl 0797.22001 [17] J. PORTI, Torsion de Reidemeister pour LES variétés hyperboliques, Mémoirs AMS, Vol. 128, n° 612, Am. Math. Soc., Providence (1997). · Zbl 0881.57020 [18] V.L. POPOV, E.B. VINBERG, Invariant theory, Algebraic Geometry IV, A.N. Parshin, I.R. Shafarevich (Eds.) Encyclopaedia of Mathematical Sciences Volume 55, Springer-Verlag Berlin Heidelberg (1994). · Zbl 0789.14008 [19] R. STEINBERG, Conjugacy classes in algebraic groups, LNM 366, Berlin-Heidelberg-New York, Springer-Verlag (1974). · Zbl 0281.20037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.