An orientation for the SU(2)-representation space of knot groups. (English) Zbl 1019.57002

Heusener, Michael

Topology Appl. 127, No. 1-2, 175-197 (2003).

Let \(K\subset S^3\) be a knot and denote by \(\hat R(K)\) the space of equivalence classes of irreducible representations of the knot group \(G=\pi_1(S^3\setminus K)\) in \(SU(2)\). Let \(Reg(K)\subset \hat R(K)\) be the space of regular representations, i.e irreducible representations \(\rho:G\rightarrow SU(2)\) such that \(H_\rho^1(G)\cong rb\) where \(H_\rho^*(G)=H^*(G,Ad\rho)\) denotes the twisted cohomology group of \(G\) with coefficients in \(su(2)\).
It was proved by M. Heusener and E. P. Klassen in [Proc. Am. Math. Soc. 125, 3039-3047 (1997; Zbl 0883.57001)] that \(Reg(K)\subset \hat R(K)\) is a real one dimensional manifold. In this paper, the author proves that \(Reg(K)\) also carries a canonical orientation. A first consequence of this result is that there exists a large class of knots which have at least a one dimensional representation space \(\hat R(K)\). As a further application, the author explains a generalization of a result of X. S. Lin [J. Differential Geometry 35, 337-357 (1992; Zbl 0774.57007)] M. Heusener and J. Kroll [Comment. Math. Helv. 73, 480-498 (1998; Zbl 0910.57004)]. Using the orientation of the representation space, he proves that the generalized Lin invariant and the signature of the knot are the same quantities as they turn out to be a weighted sum of zeros of the Alexander polynomial on the unit circle.

Reviewer: Leila Ben Abdelghani (Monastir)

MSC:

57M25	Knots and links in the \(3\)-sphere (MSC2010)
57M05	Fundamental group, presentations, free differential calculus
57M27	Invariants of knots and \(3\)-manifolds (MSC2010)

Keywords:

knot group; Casson invariant; representation spaces

Citations:

Zbl 0883.57001; Zbl 0774.57007; Zbl 0910.57004

References:

[1]	Akbulut, S.; McCarthy, J., Casson’s Invariant for Homology 3-Spheres—An Exposition. Casson’s Invariant for Homology 3-Spheres—An Exposition, Math. Notes, 36 (1990), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0695.57011
[2]	Bochnak, J.; Coste, M.; Roy, M.-F., Géométrie Algébrique Réelle. Géométrie Algébrique Réelle, Ergebnisse der Mathematik und ihrer Grenzgebiete (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0633.14016
[3]	Birman, J. S., Braids, Links and Mapping Class Groups. Braids, Links and Mapping Class Groups, Ann. of Math. Stud., 82 (1976), Princeton University Press: Princeton University Press Princeton, NJ
[4]	Birman, J. S., On the stable equivalence of plat representations of knots and links, Canad. J. Math., 28, 264-290 (1976) · Zbl 0339.55005
[5]	Burde, G., \( SU(2)\)-representation spaces for two-bridge knot groups, Math. Ann., 288, 103-119 (1990) · Zbl 0694.57003
[6]	Burde, G.; Zieschang, H., Knots (1985), de Gruyter: de Gruyter Berlin · Zbl 0568.57001
[7]	Frohman, C. D.; Long, D. D., Casson’s invariant and surgery on knots, Proc. Edinburgh Math. Soc. (2), 35, 3, 383-395 (1992) · Zbl 0764.57010
[8]	Guillou, L.; Marin, A., Notes sur l’invariant de Casson des sphères d’homologie de dimension trois, Enseign. Math. (2), 38, 233-290 (1992) · Zbl 0776.57008
[9]	Heusener, M., \( SO(3)\)-representation curves for two-bridge knot groups, Math. Ann., 298, 327-348 (1994) · Zbl 0795.57003
[10]	Hilton, H. H., Generators for two subgroups related to the braid groups, Pacific J. Math., 59, 475-486 (1975) · Zbl 0317.57005
[11]	Hirsch, M. W., Differential Topology. Differential Topology, Graduate Texts in Math., 33 (1976), Springer-Verlag: Springer-Verlag Berlin · Zbl 0356.57001
[12]	Heusener, M.; Klassen, E. P., Deformations of dihedral representations, Proc. Amer. Math. Soc., 125, 3039-3047 (1997) · Zbl 0883.57001
[13]	Heusener, M.; Kroll, J., Deforming abelian \(SU(2)\)-representations of knot groups, Comment. Math. Helv., 73, 480-498 (1998) · Zbl 0910.57004
[14]	Kauffman, L. H., On Knots. On Knots, Ann. of Math. Stud., 115 (1987), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0627.57002
[15]	Kirby, R., Problems in low-dimensional topology, (Geometric Topology (Athens, GA, 1993) (1997), American Mathematical Society: American Mathematical Society Providence, RI), 35-473 · Zbl 0888.57014
[16]	Klassen, E. P., Representations of knot groups in \(SU(2)\), Trans. Amer. Math. Soc., 326, 2, 795-828 (August 1991) · Zbl 0743.57003
[17]	Lin, X.-S., A knot invariant via representation spaces, J. Differential Geometry, 35, 337-357 (1992) · Zbl 0774.57007
[18]	Lubotzky, A.; Magid, A. R., Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc., 58, 336, xi+117 (1985) · Zbl 0598.14042
[19]	Porti, J., Torsion de Reidemeister pour les variétés hyperboliques, Mem. Amer. Math. Soc., 128, 612, x+139 (1997) · Zbl 0881.57020
[20]	Reidemeister, K., Knoten und Geflechte, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl., 5, 105-115 (1960) · Zbl 0118.39203
[21]	Weil, A., Remarks on the cohomology of groups, Ann. of Math., 80, 149-157 (1964) · Zbl 0192.12802