We give upper and lower bounds on the leading coefficients of the -Alexander torsions of a -manifold in terms of hyperbolic volumes and of relative -torsions of sutured manifolds obtained by cutting 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 d’une variété de dimension 3 , en fonction de volumes hyperboliques et de torsions relatives de variétés suturées qu’on obtient en découpant 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:
Published online:
Keywords: $L^2$-invariants, $3$-manifolds, Thurston norm
Mot clés : invariants $L^2$, variétés de dimension 3, norme de Thurston
Ben Aribi, Fathi ; Friedl, Stefan 1; Herrmann, Gerrit 1
@article{AIF_2022__72_5_1993_0, author = {Ben Aribi, Fathi and Friedl, Stefan and Herrmann, Gerrit}, title = {The leading coefficient of the $L^2${-Alexander} torsion}, journal = {Annales de l'Institut Fourier}, pages = {1993--2035}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {72}, number = {5}, year = {2022}, doi = {10.5802/aif.3509}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3509/} }
TY - JOUR AU - Ben Aribi, Fathi AU - Friedl, Stefan AU - Herrmann, Gerrit TI - The leading coefficient of the $L^2$-Alexander torsion JO - Annales de l'Institut Fourier PY - 2022 SP - 1993 EP - 2035 VL - 72 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3509/ DO - 10.5802/aif.3509 LA - en ID - AIF_2022__72_5_1993_0 ER -
%0 Journal Article %A Ben Aribi, Fathi %A Friedl, Stefan %A Herrmann, Gerrit %T The leading coefficient of the $L^2$-Alexander torsion %J Annales de l'Institut Fourier %D 2022 %P 1993-2035 %V 72 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3509/ %R 10.5802/aif.3509 %G en %F AIF_2022__72_5_1993_0
Ben Aribi, Fathi; Friedl, Stefan; Herrmann, Gerrit. The leading coefficient of the $L^2$-Alexander torsion. Annales de l'Institut Fourier, Volume 72 (2022) no. 5, pp. 1993-2035. doi : 10.5802/aif.3509. https://aif.centre-mersenne.org/articles/10.5802/aif.3509/
[1] Criteria for virtual fibering, J. Topol., Volume 1 (2008) no. 2, pp. 269-284 | DOI | MR | Zbl
[2] The virtual Haken conjecture, Doc. Math., Volume 18 (2013), pp. 1045-1087 (with an appendix by Agol, Daniel Groves, and Jason Manning) | DOI | MR | Zbl
[3] Certifying the Thurston Norm via -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 | MR | Zbl
[4] 3-manifold groups, EMS Series of Lectures in Mathematics, European Mathematical Society, 2015, xiv+215 pages | DOI | MR
[5] The -Alexander invariant detects the unknot, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 15 (2016), pp. 683-708 | MR | Zbl
[6] Gluing formulas for the -Alexander torsions, Commun. Contemp. Math., Volume 21 (2019) no. 3, 1850013, 31 pages | DOI | MR | Zbl
[7] -Burau maps and -Alexander torsions, Osaka J. Math., Volume 55 (2018) no. 3, pp. 529-545 | MR | Zbl
[8] Noncommutative knot theory, Algebr. Geom. Topol., Volume 4 (2004), pp. 347-398 | DOI | MR | Zbl
[9] The -Alexander torsion is symmetric, Algebr. Geom. Topol., Volume 15 (2015) no. 6, pp. 3599-3612 | DOI | MR | Zbl
[10] 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 | Zbl
[11] The -Alexander torsion of 3-manifolds, J. Topol., Volume 9 (2016) no. 3, pp. 889-926 | DOI | MR | Zbl
[12] Weighted -invariants and applications to knot theory, Commun. Contemp. Math., Volume 17 (2015) no. 1, 1450010, 29 pages | DOI | MR | Zbl
[13] Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies, 110, Princeton University Press, 1985, vii+173 pages | MR
[14] The -torsion function and the Thurston norm of 3-manifolds, Comment. Math. Helv., Volume 94 (2019) no. 1, pp. 21-52 | DOI | MR | Zbl
[15] 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 | Zbl
[16] Guts of surfaces and the colored Jones polynomial, Lecture Notes in Mathematics, 2069, Springer, 2013, x+170 pages | DOI | MR
[17] Foliations and the topology of -manifolds, J. Differ. Geom., Volume 18 (1983) no. 3, pp. 445-503 | MR | Zbl
[18] An analytic version of the melvin-morton-rozansky conjecture (2005) | arXiv
[19] Residual finiteness for -manifolds, Combinatorial group theory and topology (Alta, Utah, 1984) (Annals of Mathematics Studies), Volume 111, Princeton University Press, 1987, pp. 379-396 | DOI | MR | Zbl
[20] The -Alexander torsion for Seifert fiber spaces, Arch. Math., Volume 109 (2017) no. 3, pp. 273-283 | DOI | MR | Zbl
[21] Sutured manifolds and -Betti numbers (2018) | arXiv
[22] Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, 43, American Mathematical Society, 1980, xii+251 pages | DOI | MR
[23] An -Alexander invariant for knots, Commun. Contemp. Math., Volume 8 (2006) no. 2, pp. 167-187 | DOI | MR | Zbl
[24] -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] Degree of -Alexander torsion for 3-manifolds, Invent. Math., Volume 207 (2017) no. 3, pp. 981-1030 | DOI | MR | Zbl
[26] -invariants: theory and applications to geometry and -theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 44, Springer, 2002, xvi+595 pages | DOI | MR
[27] Twisting -invariants with finite-dimensional representations, J. Topol. Anal., Volume 10 (2018) no. 4, pp. 723-816 | DOI | MR | Zbl
[28] -torsion of hyperbolic manifolds of finite volume, Geom. Funct. Anal., Volume 9 (1999) no. 3, pp. 518-567 | DOI | MR | Zbl
[29] 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 | Numdam | MR | Zbl
[30] Closed incompressible surfaces in alternating knot and link complements, Topology, Volume 23 (1984) no. 1, pp. 37-44 | DOI | MR | Zbl
[31] 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 | Zbl
[32] 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 | Zbl
[33] The colored Jones polynomials and the simplicial volume of a knot, Acta Math., Volume 186 (2001) no. 1, pp. 85-104 | DOI | MR | Zbl
[34] A note on torus knots and links determined by their groups, Duke Math. J., Volume 28 (1961), pp. 545-551 | MR | Zbl
[35] Mixed 3-manifolds are virtually special, J. Am. Math. Soc., Volume 31 (2018) no. 2, pp. 319-347 | DOI | MR | Zbl
[36] 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] A norm for the homology of -manifolds, Mem. Am. Math. Soc., Volume 59 (1986) no. 339, p. i-vi and 99–130 | MR | Zbl
[38] The geometry and topology of three-manifolds (1980), 2002 (Princeton University Lecture Notes)
[39] Introduction to combinatorial torsions, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2001, viii+123 pages (notes taken by Felix Schlenk) | DOI | MR
[40] A homological estimate for the Thurston norm (2002) | arXiv
[41] Twisted Alexander polynomial for finitely presentable groups, Topology, Volume 33 (1994) no. 2, pp. 241-256 | DOI | MR | Zbl
[42] 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
Cited by Sources: