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Twisted Alexander polynomials and ideal points giving Seifert surfaces. (English) Zbl 1310.57026

Summary: The coefficients of twisted Alexander polynomials of a knot induce regular functions of the \(SL_{2}(\mathbb {C})\)-character variety. We prove that the function of the highest degree has a finite value at an ideal point which gives a minimal genus Seifert surface by Culler-Shalen theory. It implies a partial affirmative answer to a conjecture by Dunfield, Friedl, and Jackson.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
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[1] Cha, J.C.: Fibred knots and twisted Alexander invariants. Trans. Am. Math. Soc. 355(10), 4187-4200 (2003) · Zbl 1028.57004 · doi:10.1090/S0002-9947-03-03348-8
[2] Culler, M., Shalen, P.B.: Varieties of group representations and splittings of 3-manifolds. Ann. Math. (2) 117(1), 109-146 (1983) · Zbl 0529.57005 · doi:10.2307/2006973
[3] Dunfield, N.M., Friedl, S., Jackson, N.: Twisted Alexander polynomials of hyperbolic knots. Experiment. Math. 21, 329-352 (2012) · Zbl 1266.57008 · doi:10.1080/10586458.2012.669268
[4] Friedl, S., Kim, T.: The Thurston norm, fibered manifolds and twisted Alexander polynomials. Topology 45(6), 929-953 (2006) · Zbl 1105.57009 · doi:10.1016/j.top.2006.06.003
[5] Friedl, S., Kim, T.: Twisted Alexander norms give lower bounds on the Thurston norm. Trans. Am. Math. Soc. 360(9), 4597-4618 (2008) · Zbl 1152.57011 · doi:10.1090/S0002-9947-08-04455-3
[6] Friedl, S., Kim, T., Kitayama, T.: Poincaré duality and degrees of twisted Alexander polynomials. Indiana Univ. Math. J. 61, 147-192 (2012) · Zbl 1273.57009 · doi:10.1512/iumj.2012.61.4779
[7] Friedl, S., Vidussi, S.: Twisted Alexander polynomials detect fibered 3-manifolds. Ann. Math. (2) 173(3), 1587-1643 (2011) · Zbl 1231.57020 · doi:10.4007/annals.2011.173.3.8
[8] Friedl, S., Vidussi, S.: A survey of twisted Alexander polynomials. The mathematics of knots, 45-94, Contrib. Math. Comput. Sci., 1. Springer, Heidelberg (2011) · Zbl 1223.57012
[9] Friedl, S., Vidussi, S.: The Thurston norm and twisted Alexander polynomials. to appear in J. Reine Angew. Math. arXiv:1204.6456 · Zbl 1331.57017
[10] Goda, H., Kitano, T., Morifuji, T.: Reidemeister torsion, twisted Alexander polynomial and fibered knots. Comment. Math. Helv. 80(1), 51-61 (2005) · Zbl 1066.57008 · doi:10.4171/CMH/3
[11] Kim, T., Kitayama, T., Morifuji, T.: Twisted Alexander polynomials on curves in character varieties of knot groups. Int. J. Math. 24(3), 1350022 (2013). 16 pp · Zbl 1275.57021 · doi:10.1142/S0129167X13500225
[12] Kim, T., Morifuji, T.: Twisted Alexander polynomials and character varieties of 2-bridge knot groups. Int. J. Math. 23(6), 1250022 (2012). 24 pp · Zbl 1255.57013 · doi:10.1142/S0129167X11007653
[13] Kirk, P., Livingston, C.: Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants. Topology 38(3), 635-661 (1999) · Zbl 0928.57005 · doi:10.1016/S0040-9383(98)00039-1
[14] Kitano, T.: Twisted Alexander polynomial and Reidemeister torsion. Pacific J. Math. 174(2), 431-442 (1996) · Zbl 0863.57001
[15] Kitayama, T.: Twisted Alexander polynomials and incompressible surfaces given by ideal points. in preparation · Zbl 1409.57015
[16] Lin, X.S.: Representations of knot groups and twisted Alexander polynomials. Acta Math. Sin. (Engl. Ser.) 17(3), 361-380 (2001) · Zbl 0986.57003 · doi:10.1007/s101140100122
[17] Lubotzky, A., Magid, A.R.: Varieties of representations of finitely generated groups. Mem. Am. Math. Soc. 58(336), xi+117 (1985) · Zbl 0598.14042
[18] Lyon, H.C.: Simple knots with unique spanning surfaces. Topology 13, 275-279 (1974) · Zbl 0321.55002 · doi:10.1016/0040-9383(74)90020-2
[19] Milnor, J.: Whitehead torsion. Bull. Am. Math. Soc. 72, 358-426 (1966) · Zbl 0147.23104 · doi:10.1090/S0002-9904-1966-11484-2
[20] Morifuji, T.: On a conjecture of Dunfield, Friedl and Jackson. C. R. Math. Acad. Sci. Paris 350(19-20), 921-924 (2012) · Zbl 1253.57007 · doi:10.1016/j.crma.2012.10.013
[21] Nicolaescu, L.I.: The Reidemeister Torsion of 3-Manifolds. de Gruyter Studies in Mathematics, 30, p xiv + 249. Walter de Gruyter & Co., Berlin (2003) · Zbl 1024.57002
[22] Serre, J.-P.: Arbres, amalgames, SL2. (French, avec un sommaire anglais), Rédigé avec la collaboration de Hyman Bass, Astérisque, No. 46, Société Mathematique de France, Paris, 1977, pp. 189 (1 plate) · Zbl 0369.20013
[23] Serre, J.-P.: Trees. Translated from the French by John Stillwell, p ix+142. Springer-Verlag, Berlin-New York (1980) · Zbl 1013.20001
[24] Shalen, P.B.: Representations of 3-Manifold Groups. Handbook of geometric topology, 955-1044, North-Holland, Amsterdam (2002) · Zbl 1012.57003
[25] Turaev, V.: Introduction to Combinatorial Torsions. Notes taken by Felix Schlenk, Lectures in Mathematics ETH Zurich, viii+123 (2001) · Zbl 0970.57001
[26] Turaev, V.: Torsions of 3-Dimensional Manifolds. Progress in Mathematics, 208, p x+196. Birkhauser Verlag, Basel (2002) · Zbl 1012.57002 · doi:10.1007/978-3-0348-7999-6
[27] Wada, M.: Twisted Alexander polynomial for finitely presentable groups. Topology 33(2), 241-256 (1994) · Zbl 0822.57006 · doi:10.1016/0040-9383(94)90013-2
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