The SL(2, C) Casson Invariant for Knots and the Â-polynomial

Boden, Hans U. ; Curtis, Cynthia L.

Canadian journal of mathematics, Tome 68 (2016) no. 1, pp. 3-23

Résumé

In this paper, we extend the definition of the $SL\left( 2,\,\mathbb{C} \right)$ Casson invariant to arbitrary knots $K$ in integral homology 3-spheres and relate it to the $m$ -degree of the $\widehat{A}$ -polynomial of $K$ . We prove a product formula for the $\widehat{A}$ -polynomial of the connected sum ${{K}_{1}}\#{{K}_{2}}$ of two knots in ${{S}^{3}}$ and deduce additivity of the $SL\left( 2,\,\mathbb{C} \right)$ Casson knot invariant under connected sums for a large class of knots in ${{S}^{3}}$ . We also present an example of a nontrivial knot $K$ in ${{S}^{3}}$ with trivial $\widehat{A}$ -polynomial and trivial $SL\left( 2,\,\mathbb{C} \right)$ Casson knot invariant, showing that neither of these invariants detect the unknot.

Détail

DOI : 10.4153/CJM-2015-025-5

Mots-clés : 57M27, 57M25, 57M05, knots, 3-manifolds, character variety, Casson invariant, A-polynomial

Bibliographie

[1] [1] Boden, H. U., Nontriviality of the M-degree of the A-polynomial. Proc. Amer. Math. Soc. 142(2014), no. 6, 2173–2177. Google Scholar | DOI

[2] [2] Boden, H. U. and Curtis, C. L., The SL2(ℂ) Casson invariant for Seifert fibered homology spheres and surgeries on twist knots. J. Knot Theory Ramific. 15(2006), no. 7, 813–837. Google Scholar | DOI

[3] [3] Boden, H.U and Curtis, C.L, Splicing and the SL2(ℂ) Casson invariant. Proc. Amer. Math. Soc. 136(2008), no. 7, 2615–2623. Google Scholar | DOI

[4] [4] Boden, H. U. and Curtis, C. L., The SL(2, ℂ) Casson invariant for for Dehn surgeries on two-bridge knots. Algebr. Geom. Topol. 12(2012), no. 4, 2095–2126. Google Scholar | DOI

[5] [5] Boyer, S. and Zhang, X., A proof of the finite filling conjecture. J. Differential Geom. 59(2001), 87–176. Google Scholar

[6] [6] Chesebro, E., Closed surfaces and character varieties. Algebr. Geom. Topol. 13(2013), no. 4, 2001–2037. Google Scholar | DOI

[7] [7] Cooper, D., Culler, M., Gillet, H., Long, D. D., and Shalen, P. B., Plane curves associated to character varieties of 3-manifolds. Invent. Math. 118(1994), no. 1, 47–84. Google Scholar | DOI

[8] [8] Cooper, D. and Long, D. D., Remarks on the A-polynomial of a knot. J. Knot Theory Ramifications 5(1996), no. 5, 609–628. Google Scholar | DOI

[9] [9] Cooper, D. and Long, D. D., The A-polynomial has ones in the corners. Bull. Lond. Math. Soc. 29(1997), no. 2, 231–238. http://dx.doi.Org/10.1112/SOO24609396002251 Google Scholar

[10] [10] Culler, M., C.|McA. Gordon, Luecke, J., and Shalen, P. B., Dehn surgery on knots. Ann. of Math. 125(1987), no. 2, 237–300. http://dx.doi.Org/10.2307/1971311 Google Scholar

[11] [11] Culler, M. and Shalen, P. B., Varieties of group representations and splittings of 3-manifolds. Ann. of Math. 117(1983), no. 1, 109–146. http://dx.doi.Org/10.2307/2006973 Google Scholar

[12] [12] Curtis, C. L., An intersection theory count of the SL(2,ℂ) -representations of the fundamental group of a 3-manifold. Topology 40(2001), 773–787. http://dx.doi.Org/10.1016/S0040-9383(99)00083-X Google Scholar

[13] Curtis, C. L., Erratum to “An intersection theory count of the SL(2,ℂ) - representations of the fundamental group of a 3-manifold”. Topology 42(2003), no. 4, 929. http://dx.doi.Org/10.1016/S0040-9383(02)00101-5 Google Scholar

[14] [14] Fulton, W., Algebraic curves. An introduction to algebraic geometry. Notes written with the collaboration of Richard Weiss, Mathematics Lecture Notes Series, W. A. Benjamin, New York, 1969. Google Scholar

[15] [15] Johnson, D. and Millson, J., Deformation spaces associated to compact hyperbolic manifolds. In: Discrete groups in geometry and analysis (New Haven, Conn., 1984), Progr. Math., 67, Birkhäuser Boston, 1987, pp. 48–106. Google Scholar

[16] [16] Kapovich, M. and Millson, J., On representation varieties of 3-manifold groups. arxiv:1303.2347 Google Scholar

[17] [17] Klassen, E. P., Representations in SU (2) of the fundamental groups of the Whitehead link and of doubled knots. Forum Math. 5(1993), 93–109. http://dx.doi.Org/10.1515/form.1993.5.93 Google Scholar

[18] [18] Le, T. L. K, Varieties of representations and their cohomology-jump subvarieties for knot groups. (Russian) Mat. Sb. 184(1993), no. 2, 57–82; translation in: Russian Acad. Sci. Sb. Math.) 78(1994), no. 1, 187–209. Google Scholar | DOI

[19] [19] Le, T. K. T. and Tran, A. T., On the A] conjecture for knots. http://arxiv:1111.5258v4 Google Scholar

[20] [20] Marché, J., The skein module of torus knots. Quantum Topol. 1(2010), no. 4, 413–421. http://dx.doi.Org/10.4171/QT/11 Google Scholar

[21] [21] Shafarevich, I. R., Basic algebraic geometry I. In: Varieties in protective space. Second edition. Springer-Verlag, Berlin, 1994. Google Scholar

[22] [22] Sikora, A. S., Character varieties. Trans. Amer. Math. Soc. 364(2012), no. 10, 5173–5208. Google Scholar | DOI

[23] [23] Tran, A. T., The universal character ring of the (−2, 2m + 1, 2n)-pretzel link. Internat. J. Math. 24(2013), no. 8, 1350063. Google Scholar | DOI