# ANNALES DE L'INSTITUT FOURIER

The leading coefficient of the ${L}^{2}$-Alexander torsion
Annales de l'Institut Fourier, Online first, 43 p.

We give upper and lower bounds on the leading coefficients of the ${L}^{2}$-Alexander torsions of a $3$-manifold $M$ in terms of hyperbolic volumes and of relative ${L}^{2}$-torsions of sutured manifolds obtained by cutting $M$ along certain surfaces.

We prove that for numerous families of knot exteriors the lower and upper bounds are equal, notably for exteriors of 2-bridge knots. In particular we compute the leading coefficient explicitly for 2-bridge knots.

Nous trouvons des bornes supérieure et inférieure des coefficients dominants des torsions d’Alexander ${L}^{2}$ d’une variété de dimension 3 $M$, en fonction de volumes hyperboliques et de torsions ${L}^{2}$ relatives de variétés suturées qu’on obtient en découpant $M$ le long de certaines surfaces.

Nous démontrons que pour de nombreuses familles d’extérieurs de nœuds, les bornes inférieure et supérieure sont égales, notamment pour les extérieurs des nœuds à deux ponts. En particulier, nous calculons explicitement le coefficient dominant pour les nœuds à deux ponts.

Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3509
Classification: 57M25,  57M27
Keywords: ${L}^{2}$-invariants, $3$-manifolds, Thurston norm
Ben Aribi, Fathi ; Friedl, Stefan 1; Herrmann, Gerrit 1

1 Fakultät für Mathematik Universität Regensburg (Germany)
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Ben Aribi, Fathi; Friedl, Stefan; Herrmann, Gerrit. The leading coefficient of the $L^2$-Alexander torsion. Annales de l'Institut Fourier, Online first, 43 p.

[1] Agol, Ian Criteria for virtual fibering, J. Topol., Volume 1 (2008) no. 2, pp. 269-284 | DOI | MR

[2] Agol, Ian The virtual Haken conjecture, Doc. Math., Volume 18 (2013), pp. 1045-1087 (with an appendix by Agol, Daniel Groves, and Jason Manning) | DOI | MR

[3] Agol, Ian; Dunfield, Nathan M. Certifying the Thurston Norm via $\mathrm{SL}\left(2,ℂ\right)$-twisted Homology, What’s Next?: The Mathematical Legacy of William P. Thurston (Annals of Mathematics Studies), Volume 205, Princeton University Press, 2020, pp. 1-20

[4] Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry 3-manifold groups, EMS Series of Lectures in Mathematics, European Mathematical Society, 2015, xiv+215 pages | DOI | MR

[5] Ben Aribi, Fathi The ${L}^{2}$-Alexander invariant detects the unknot, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 15 (2016), pp. 683-708 | MR

[6] Ben Aribi, Fathi Gluing formulas for the ${L}^{2}$-Alexander torsions, Commun. Contemp. Math., Volume 21 (2019) no. 3, 1850013, 31 pages | DOI | MR

[7] Ben Aribi, Fathi; Conway, Anthony ${L}^{2}$-Burau maps and ${L}^{2}$-Alexander torsions, Osaka J. Math., Volume 55 (2018) no. 3, pp. 529-545 | MR

[8] Cochran, Tim D. Noncommutative knot theory, Algebr. Geom. Topol., Volume 4 (2004), pp. 347-398 | DOI | MR

[9] Dubois, Jérôme; Friedl, Stefan; Lück, Wolfgang The ${L}^{2}$-Alexander torsion is symmetric, Algebr. Geom. Topol., Volume 15 (2015) no. 6, pp. 3599-3612 | DOI | MR

[10] Dubois, Jérôme; Friedl, Stefan; Lück, Wolfgang Three flavors of twisted invariants of knots, Introduction to modern mathematics (Advanced Lectures in Mathematics (ALM)), Volume 33, International Press, 2015, pp. 143-169 | MR

[11] Dubois, Jérôme; Friedl, Stefan; Lück, Wolfgang The ${L}^{2}$-Alexander torsion of 3-manifolds, J. Topol., Volume 9 (2016) no. 3, pp. 889-926 | DOI | MR

[12] Dubois, Jérôme; Wegner, Christian Weighted ${L}^{2}$-invariants and applications to knot theory, Commun. Contemp. Math., Volume 17 (2015) no. 1, 1450010, 29 pages | DOI | MR

[13] Eisenbud, David; Neumann, Walter Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies, 110, Princeton University Press, 1985, vii+173 pages | MR

[14] Friedl, Stefan; Lück, Wolfgang The ${L}^{2}$-torsion function and the Thurston norm of 3-manifolds, Comment. Math. Helv., Volume 94 (2019) no. 1, pp. 21-52 | DOI | MR

[15] Friedl, Stefan; Vidussi, Stefano A survey of twisted Alexander polynomials, The mathematics of knots (Contributions in Mathematical and Computational Sciences), Volume 1, Springer, 2011, pp. 45-94 | DOI | MR

[16] Futer, David; Kalfagianni, Efstratia; Purcell, Jessica Guts of surfaces and the colored Jones polynomial, Lecture Notes in Mathematics, 2069, Springer, 2013, x+170 pages | DOI | MR

[17] Gabai, David Foliations and the topology of $3$-manifolds, J. Differ. Geom., Volume 18 (1983) no. 3, pp. 445-503 | MR

[18] Garoufalidis, Stavros; Le, Thang T.Q. An analytic version of the melvin-morton-rozansky conjecture (2005) | arXiv

[19] Hempel, John Residual finiteness for $3$-manifolds, Combinatorial group theory and topology (Alta, Utah, 1984) (Annals of Mathematics Studies), Volume 111, Princeton University Press, 1987, pp. 379-396 | MR

[20] Herrmann, Gerrit The ${L}^{2}$-Alexander torsion for Seifert fiber spaces, Arch. Math., Volume 109 (2017) no. 3, pp. 273-283 | DOI | MR

[21] Herrmann, Gerrit Sutured manifolds and ${L}^{2}$-Betti numbers (2018) | arXiv

[22] Jaco, William Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, 43, American Mathematical Society, 1980, xii+251 pages | MR

[23] Li, Weiping; Zhang, Weiping An ${L}^{2}$-Alexander invariant for knots, Commun. Contemp. Math., Volume 8 (2006) no. 2, pp. 167-187 | DOI | MR

[24] Lin, Xiao-Song ${L}^{2}$-Alexander invariants, slides for a talk at the conference AMS-IMS-SIAM Joint Meeting, Quantum Topology: Contemporary Issues and Perspectives (2005), http://web.archive.org/web/20061215073957/https://math.ucr.edu/~xl/snowbird.pdf (Accessed: 2006-12-15)

[25] Liu, Yi Degree of ${L}^{2}$-Alexander torsion for 3-manifolds, Invent. Math., Volume 207 (2017) no. 3, pp. 981-1030 | DOI | MR

[26] Lück, Wolfgang ${L}^{2}$-invariants: theory and applications to geometry and $K$-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 44, Springer, 2002, xvi+595 pages | DOI | MR

[27] Lück, Wolfgang Twisting ${L}^{2}$-invariants with finite-dimensional representations, J. Topol. Anal., Volume 10 (2018) no. 4, pp. 723-816 | DOI | MR

[28] Lück, Wolfgang; Schick, Thomas ${L}^{2}$-torsion of hyperbolic manifolds of finite volume, Geom. Funct. Anal., Volume 9 (1999) no. 3, pp. 518-567 | DOI | MR

[29] McMullen, Curtis T. The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology, Ann. Sci. Éc. Norm. Supér., Volume 35 (2002) no. 2, pp. 153-171 | DOI | MR

[30] Menasco, William Closed incompressible surfaces in alternating knot and link complements, Topology, Volume 23 (1984) no. 1, pp. 37-44 | DOI | MR

[31] Morton, Hugh R.; Traczyk, Paweł The Jones polynomial of satellite links around mutants, Braids (Santa Cruz, CA, 1986) (Contemporary Mathematics), Volume 78, American Mathematical Society, 1988, pp. 587-592 | DOI | MR

[32] Murakami, Hitoshi An introduction to the volume conjecture, Interactions between hyperbolic geometry, quantum topology and number theory (Contemporary Mathematics), Volume 541, American Mathematical Society, 2011, pp. 1-40 | DOI | MR

[33] Murakami, Hitoshi; Murakami, Jun The colored Jones polynomials and the simplicial volume of a knot, Acta Math., Volume 186 (2001) no. 1, pp. 85-104 | DOI | MR

[34] Neuwirth, Lee A note on torus knots and links determined by their groups, Duke Math. J., Volume 28 (1961), pp. 545-551

[35] Przytycki, Piotr; Wise, Daniel T. Mixed 3-manifolds are virtually special, J. Am. Math. Soc., Volume 31 (2018) no. 2, pp. 319-347 | DOI | MR

[36] Serre, Jean-Pierre Trees, Springer Monographs in Mathematics, Springer, 2003, x+142 pages (translated from the French original by John Stillwell, corrected 2nd printing of the 1980 English translation) | MR

[37] Thurston, William P. A norm for the homology of $3$-manifolds, Mem. Am. Math. Soc., Volume 59 (1986) no. 339, p. i-vi and 99–130 | MR

[38] Thurston, William P. The geometry and topology of three-manifolds (1980), 2002 (Princeton University Lecture Notes)

[39] Turaev, Vladimir Introduction to combinatorial torsions, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2001, viii+123 pages (notes taken by Felix Schlenk) | DOI | MR

[40] Turaev, Vladimir A homological estimate for the Thurston norm (2002) | arXiv

[41] Wada, Masaaki Twisted Alexander polynomial for finitely presentable groups, Topology, Volume 33 (1994) no. 2, pp. 241-256 | DOI | MR

[42] Wise, Daniel T. From riches to raags: 3-manifolds, right-angled Artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics, 117, American Mathematical Society, Providence, RI, 2012, xiv+141 pages | DOI | MR

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