Hyperbolic torsion polynomials of pretzel knots. (English) Zbl 1467.57006
Summary: In this paper, we explicitly calculate the highest degree term of the hyperbolic torsion polynomial of an infinite family of pretzel knots. This gives supporting evidence for a conjecture of Dunfield, Friedl and Jackson [N. M. Dunfield et al., Exp. Math. 21, No. 4, 329–352 (2012; Zbl 1266.57008)] that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. The verification of the genus part of the conjecture for this family of knots also follows from the work of I. Agol and N. M. Dunfield [Ann. Math. Stud. 205, 1–20 (2020; Zbl 1452.57011)] or J. Porti [in: A mathematical tribute to Professor José María Montesinos Amilibia on the occasion of his seventieth birthday. Madrid: Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Geometría y Topología. 547–558 (2016; Zbl 1346.57013)].
MSC:
| 57K10 | Knot theory |
| 57K14 | Knot polynomials |
| 57M05 | Fundamental group, presentations, free differential calculus |
Keywords:
hyperbolic torsion polynomial; twisted Alexander polynomial; canonical component; pretzel knotReferences:
| [1] | I. Agol, N. M. Dunfield, Certifying the Thurston norm via SL(2, ℂ)-twisted homology. In: What’s Next?: The Mathematical Legacy of William P. Thurston, Ann. of Math. Stud. 205 1-20, Princeton Univ. Press 2020. · Zbl 1452.57011 |
| [2] | A. Aso, Twisted Alexander polynomials of (−2, 3, 2n + 1)-pretzel knotsHiroshima Math. J. 50 (2020), 43-57. MR4074378 Zbl 1440.57014 · Zbl 1440.57014 |
| [3] | H. Chen, Character varieties of odd classical pretzel knots. Internat. J. Math. 29 (2018), 1850060, 15 pages. MR3845396 Zbl 06931090 · Zbl 1432.57010 |
| [4] | M. Culler, P. B. Shalen, Varieties of group representations and splittings of 3-manifolds. Ann. of Math. (2) 117 (1983), 109-146. MR683804 Zbl 0529.57005 · Zbl 0529.57005 |
| [5] | N. M. Dunfield, S. Friedl, N. Jackson, Twisted Alexander polynomials of hyperbolic knots. Exp. Math. 21 (2012), 329-352. MR3004250 Zbl 1266.57008 · Zbl 1266.57008 |
| [6] | S. Friedl, T. Kim, The Thurston norm, fibered manifolds and twisted Alexander polynomials. Topology 45 (2006), 929-953. MR2263219 Zbl 1105.57009 · Zbl 1105.57009 |
| [7] | S. Friedl, S. Vidussi, A survey of twisted Alexander polynomials. In: The mathematics of knots, volume 1 of Contrib. Math. Comput. Sci., 45-94, Springer 2011. MR2777847 Zbl 1223.57012 · Zbl 1223.57012 |
| [8] | T. Kim, T. Kitayama, T. Morifuji, Twisted Alexander polynomials on curves in character varieties of knot groups. Internat. J. Math. 24 (2013), 1350022, 16 pages. MR3048009 Zbl 1275.57021 · Zbl 1275.57021 |
| [9] | T. Kim, T. Morifuji, Twisted Alexander polynomials and character varieties of 2-bridge knot groups. Internat. J. Math. 23 (2012), 1250022, 24 pages. MR2925470 Zbl 1255.57013 · Zbl 1255.57013 |
| [10] | T. Kitano, T. Morifuji, Divisibility of twisted Alexander polynomials and fibered knots. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 179-186. MR2165406 Zbl 1117.57004 · Zbl 1117.57004 |
| [11] | X. S. Lin, Representations of knot groups and twisted Alexander polynomials. Acta Math. Sin. Engl. Ser.) 17 (2001), 361-380. MR1852950 Zbl 0986.57003 · Zbl 0986.57003 |
| [12] | T. Morifuji, Twisted Alexander polynomials of twist knots for nonabelian representations. Bull. Sci. Math. 132 (2008), 439-453. MR2426646 Zbl 1147.57018 · Zbl 1147.57018 |
| [13] | T. Morifuji, On a conjecture of Dunfield, Friedl and Jackson. C. R. Math. Acad. Sci. Paris 350 (2012), 921-924. MR2990904 Zbl 1253.57007 · Zbl 1253.57007 |
| [14] | T. Morifuji, Representations of knot groups into SL(2, ℂ) and twisted Alexander polynomials. In: Handbook of group actions. Vol. I, volume 31 of Adv. Lect. Math., 527-576, Int. Press, Somerville, MA 2015. MR3380341 |
| [15] | T. Morifuji, A calculation of the hyperbolic torsion polynomial of a pretzel knot. Tokyo J. Math. 42 (2019), 219-224. MR3982055 Zbl 07114906 · Zbl 1423.57031 |
| [16] | T. Morifuji, A. T. Tran, Twisted Alexander polynomials of 2-bridge knots for parabolic representations. Pacific J. Math. 269 (2014), 433-451. MR3238485 Zbl 1305.57027 · Zbl 1305.57027 |
| [17] | T. Morifuji, A. T. Tran, Twisted Alexander polynomials of hyperbolic links. Publ. Res. Inst. Math. Sci. 53 (2017), 395-418. MR3669741 Zbl 1379.57021 · Zbl 1379.57021 |
| [18] | Y. Ohashi, Fiberedness of pretzel knots and twisted Alexander invariantsJapanese). Master Thesis, Tohoku University, 2013. |
| [19] | J. Porti, Nontrivial twisted Alexander polynomials. In: A mathematical tribute to Professor José María Montesinos Amilibia, 547-558, Dep. Geom. Topol. Fac. Cien. Mat. UCM, Madrid 2016. MR3525955 Zbl 1346.57013 · Zbl 1346.57013 |
| [20] | A. T. Tran, Reidemeister torsion and Dehn surgery on twist knots. Tokyo J. Math. 39 (2016), 517-526. MR3599506 Zbl 1367.57013 · Zbl 1367.57013 |
| [21] | M. Wada, Twisted Alexander polynomial for finitely presentable groups. Topology 33 (1994), 241-256. MR1273784 Zbl 0822.57006 · Zbl 0822.57006 |
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