The SL(2,C) Casson invariant for Dehn surgeries on two-bridge knots

Boden, Hans¹ ; Curtis, Cynthia²

¹Mathematics & Statistics, McMaster University, 1280 Main St. W., Hamilton, Ontario L8S 4K1 Canada
²Mathematics & Statistics, The College of New Jersey, PO Box 7718, Ewing, NJ 08628 USA

Algebraic and Geometric Topology, Tome 12 (2012) no. 4, pp. 2095-2126

Résumé

We investigate the behavior of the SL(2, ℂ) Casson invariant for 3–manifolds obtained by Dehn surgery along two-bridge knots. Using the results of Hatcher and Thurston, and also results of Ohtsuki, we outline how to compute the Culler–Shalen seminorms, and we illustrate this approach by providing explicit computations for double twist knots. We then apply the surgery formula of Curtis [Topology 40 (2001), 773–787] to deduce the SL(2, ℂ) Casson invariant for the 3–manifolds obtained by (p∕q)–Dehn surgery on such knots. These results are applied to prove nontriviality of the SL(2, ℂ) Casson invariant for nearly all 3–manifolds obtained by nontrivial Dehn surgery on a hyperbolic two-bridge knot. We relate the formulas derived to degrees of A–polynomials and use this information to identify factors of higher multiplicity in the –polynomial, which is the A–polynomial with multiplicities as defined by Boyer–Zhang.

Détail

DOI : 10.2140/agt.2012.12.2095

Keywords: Casson invariant, character variety, two-bridge knot

Bibliographie

[1] H U Boden, Nontriviality of the M-degree of the A-polynomial, to appear in Proc. Amer. Math. Soc. (2012)

[2] H U Boden, C L Curtis, The SL2(C) Casson invariant for Seifert fibered homology spheres and surgeries on twist knots, J. Knot Theory Ramifications 15 (2006) 813

[3] H U Boden, C L Curtis, Splicing and the SL2(C) Casson invariant, Proc. Amer. Math. Soc. 136 (2008) 2615

[4] H U Boden, K L Petersen, Notes on the A–polynomial and –polynomial, in preparation

[5] M Boileau, B Leeb, J Porti, Geometrization of 3–dimensional orbifolds, Ann. of Math. 162 (2005) 195

[6] S Boyer, personal communication (2012)

[7] S Boyer, X Zhang, On Culler–Shalen seminorms and Dehn filling, Ann. of Math. 148 (1998) 737

[8] S Boyer, X Zhang, A proof of the finite filling conjecture, J. Differential Geom. 59 (2001) 87

[9] M Brittenham, Y Q Wu, The classification of exceptional Dehn surgeries on 2–bridge knots, Comm. Anal. Geom. 9 (2001) 97

[10] G Burde, H Zieschang, Knots, 5, Walter de Gruyter Co. (1985)

[11] J C Cha, C Livingston, KnotInfo: Table of Knot Invariants

[12] D Cooper, M Culler, H Gillet, D D Long, P B Shalen, Plane curves associated to character varieties of 3–manifolds, Invent. Math. 118 (1994) 47

[13] D Cooper, D D Long, Remarks on the A–polynomial of a knot, J. Knot Theory Ramifications 5 (1996) 609

[14] D Cooper, D D Long, The A–polynomial has ones in the corners, Bull. London Math. Soc. 29 (1997) 231

[15] M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987) 237

[16] M Culler, P B Shalen, Varieties of group representations and splittings of 3–manifolds, Ann. of Math. 117 (1983) 109

[17] M Culler, P B Shalen, Bounded, separating, incompressible surfaces in knot manifolds, Invent. Math. 75 (1984) 537

[18] C L Curtis, An intersection theory count of the SL2(C)–representations of the fundamental group of a 3–manifold, Topology 40 (2001) 773

[19] C L Curtis, Erratum to : “An intersection theory count of the SL2(C)–representations of the fundamental group of a 3–manifold” [Topology 40 (2001), no. 4, 773–787 ; MR1851563 (2002k :57022)], Topology 42 (2003) 929

[20] A Hatcher, W Thurston, Incompressible surfaces in 2–bridge knot complements, Invent. Math. 79 (1985) 225

[21] J Hoste, P D Shanahan, A formula for the A–polynomial of twist knots, J. Knot Theory Ramifications 13 (2004) 193

[22] M L Macasieb, K L Petersen, R M Van Luijk, On character varieties of two-bridge knot groups, Proc. Lond. Math. Soc. 103 (2011) 473

[23] T W Mattman, The Culler–Shalen seminorms of pretzel knots, ProQuest LLC, Ann Arbor, MI (2000) 118

[24] F Nagasato, On a behavior of a slice of the SL2(C)–character variety of a knot group under the connected sum, Topology Appl. 157 (2010) 182

[25] T Ohtsuki, Ideal points and incompressible surfaces in two-bridge knot complements, J. Math. Soc. Japan 46 (1994) 51

[26] T Ohtsuki, R Riley, M Sakuma, Epimorphisms between 2–bridge link groups, from: "The Zieschang Gedenkschrift" (editors M Boileau, M Scharlemann, R Weidmann), Geom. Topol. Monogr. 14 (2008) 417

[27] R Riley, Nonabelian representations of 2–bridge knot groups, Quart. J. Math. Oxford Ser. 35 (1984) 191

[28] P B Shalen, Representations of 3–manifold groups, from: "Handbook of geometric topology" (editors R J Daverman, R B Sher), North-Holland (2002) 955

[29] W A Stein, Sage Mathematics Software (Version 4.8), Software

[30] L T K Tkhang, Varieties of representations and their subvarieties of cohomology jumps for knot groups, Mat. Sb. 184 (1993) 57