\(SU(2)\)-representation spaces for two-bridge knots. (English) Zbl 0694.57003
Representation spaces of the fundamental groups of 2-bridge knot and link complements into SO(3) and SU(2) are described by real plane algebraic curves. The curves are investigated via their singularities and in certain cases proved to be rational. A simple proof is given for the fact that the fundamental group of any manifold resulting from \(S^ 3\) by Dehn surgery along a two-bridge knot possesses non-abelian homomorphic images in SO(3) and SU(2).
Reviewer: G.Burde
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
| 20C32 | Representations of infinite symmetric groups |
| 20F34 | Fundamental groups and their automorphisms (group-theoretic aspects) |
| 14H99 | Curves in algebraic geometry |
Keywords:
representations of knot groups in SO(3) and SU(2); real plane algebraic curves; singularities; Dehn surgery along a two-bridge knotReferences:
| [1] | Akbulut, S., McCarthy, J.: Casson’s invariant for homology 2-spheres ? an exposition. Preprint 1986 · Zbl 0695.57011 |
| [2] | Boyer, S., Nicas, A.: Varieties of group representations and Casson’s invariant for rational homology 3-spheres. Preprint 1986 · Zbl 0707.57009 |
| [3] | Blaschke, W.: Kinematik und Quaternionen. Berlin: Deutscher Verlag der Wissenschaften 1960 · Zbl 0098.34701 |
| [4] | Burde, G.: Darstellung von Knotengruppen. Math. Ann.73, 24-33 (1967) · Zbl 0146.45602 · doi:10.1007/BF01351516 |
| [5] | Burde, G.: Verschlingsinvariante von Knoten und Verkettungen mit zwei Brücken. Math. Z.145, 235-242 (1975) · Zbl 0312.55002 · doi:10.1007/BF01215288 |
| [6] | Burde, G., Zieschang, H.: Knots. Studies in Mathematics 5. Berlin New York: de Gruyter 1985 · Zbl 0568.57001 |
| [7] | Conway, J.: An enumeration of knots and links and some of their related properties. Computational problems in abstract algebra, pp. 329-358. New York: Pergamon Press 1970 |
| [8] | Klassen, E.: Representation of knots groups inSU(2). Dissertation, Cornell University, 1987 |
| [9] | Levine, J.J.: Knot cobordism groups in codimension 2. Comment. Math. Helv.44, 229-244 (1969) · Zbl 0176.22101 · doi:10.1007/BF02564525 |
| [10] | Murasugi, K.: On periodic knots. Comment. Math. Helv.46, 162-174 (1971) · doi:10.1007/BF02566836 |
| [11] | De Rham, G.: Introduction aux polynomes d’un noeud. Enseign. Math., II. Sér.5, 263-266 (1972) |
| [12] | Riley, R.: A finiteness theorem for alternating links. J. Lond. Math. Soc., II. Ser.24, 217-242 (1972) · Zbl 0229.55002 · doi:10.1112/plms/s3-24.2.217 |
| [13] | Riley, R.: Nonabelian representations of 2-bridge knot groups. Q. J. Math. Oxf., II. Ser.35, 191-208 (1984) · Zbl 0549.57005 · doi:10.1093/qmath/35.2.191 |
| [14] | Schubert, H.: Knoten mit zwei Brücken. Math. Z.65, 133-170 (1956) · Zbl 0071.39002 · doi:10.1007/BF01473875 |
| [15] | Siebenman, L.: Exercises sur les noeuds rationnels. Polycopie, Orsay, 1975 |
| [16] | Takahashi, M.: Two-bridge knots have property P. Mem. Am. Math. Soc.29, No. 239 (1981) · Zbl 0451.57003 |
| [17] | Walker, R.J.: Algebraic curves. New York: Dover 1962 · Zbl 0103.38202 |
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