Algebraic & Geometric Topology Volume 25, issue 9 (2025)

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ISSN (electronic): 1472-2739

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On the topology of character varieties of once-punctured torus bundles

Stephan Tillmann and Youheng Yao

Algebraic & Geometric Topology 25 (2025) 5389–5437

DOI: 10.2140/agt.2025.25.5389

Bibliography

1	H F Baker, Examples of applications of Newton’s polygon to the theory of singular points of algebraic functions, Trans. Cambridge Phil. Soc. 15 (1893) 403
2	K L Baker, K L Petersen, Character varieties of once-punctured torus bundles with tunnel number one, Int. J. Math. 24 (2013) 1350048 MR3078072
3	P Beelen, R Pellikaan, The Newton polygon of plane curves with many rational points, Des. Codes Cryptogr. 21 (2000) 41 MR1801161
4	S Boyer, E Luft, X Zhang, On the algebraic components of the SL(2, ℂ) character varieties of knot exteriors, Topology 41 (2002) 667 MR1905834
5	W Castryck, J Voight, On nondegeneracy of curves, Algebra Number Theory 3 (2009) 255 MR2525551
6	D A Cox, J Little, D O’Shea, Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra, Springer (2015) MR3330490
7	M Culler, Lifting representations to covering groups, Adv. Math. 59 (1986) 64 MR825087
8	M Culler, W Jaco, H Rubinstein, Incompressible surfaces in once-punctured torus bundles, Proc. Lond. Math. Soc. 45 (1982) 385 MR675414
9	M Culler, P B Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. of Math. 117 (1983) 109 MR683804
10	W Decker, G M Greuel, G Pfister, H Schönemann, Singular 4-2-1: a computer algebra system for polynomial computations (2021)
11	E Falbel, A Guilloux, Dimension of character varieties for 3-manifolds, Proc. Amer. Math. Soc. 145 (2017) 2727 MR3626524
12	W Floyd, A Hatcher, Incompressible surfaces in punctured-torus bundles, Topology Appl. 13 (1982) 263 MR651509
13	R H Fox, Free differential calculus, I : Derivation in the free group ring, Ann. of Math. 57 (1953) 547 MR53938
14	R Fricke, F Klein, Vorlesungen über die Theorie der automorphen Funktionen, I : Die gruppentheoretischen Grundlagen, Teubner (1897)
15	R Fricke, F Klein, Vorlesungen über die Theorie der automorphen Funktionen, II : Die funktionentheoretischen Ausführungen und die Andwendungen, Teubner (1912)
16	W M Goldman, J J Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math. 67 (1988) 43 MR972343
17	F González-Acuña, J M Montesinos-Amilibia, On the character variety of group representations in SL(2, ℂ) and PSL(2, ℂ), Math. Z. 214 (1993) 627 MR1248117
18	F Guéritaud, On canonical triangulations of once-punctured torus bundles and two-bridge link complements, Geom. Topol. 10 (2006) 1239 MR2255497
19	M Heusener, J Porti, The variety of characters in PSL₂(ℂ), Bol. Soc. Mat. Mex. 10 (2004) 221 MR2199350
20	M Heusener, J Porti, Deformations of reducible representations of 3-manifold groups into PSL₂(ℂ), Algebr. Geom. Topol. 5 (2005) 965 MR2171800
21	A G Khovanskii, Newton polyhedra, and the genus of complete intersections, Funktsional. Anal. i Prilozhen. 12 (1978) 51 MR487230
22	J W Morgan, P B Shalen, Degenerations of hyperbolic structures, III : Actions of 3-manifold groups on trees and Thurston’s compactness theorem, Ann. of Math. 127 (1988) 457 MR942518
23	J Porti, Torsion de Reidemeister pour les variétés hyperboliques, 612, Amer. Math. Soc. (1997) MR1396960
24	J Porti, Dimension of representation and character varieties for two- and three-orbifolds, Algebr. Geom. Topol. 22 (2022) 1905 MR4495671
25	W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)
26	S Tillmann, Character varieties of mutative 3-manifolds, Algebr. Geom. Topol. 4 (2004) 133 MR2059186
27	S Tillmann, Tropical varieties associated to ideal triangulations : the Whitehead link complement, from: "Characters in low-dimensional topology", Contemp. Math. 760, Amer. Math. Soc. (2020) 293 MR4193931
28	H Vogt, Sur les invariants fondamentaux des équations différentielles linéaires du second ordre, Ann. Sci. École Norm. Sup. 6 (1889) 3 MR1508833