no. 3
A multivariable Casson–Lin type invariant
[Un invariant de Casson–Lin multivarié]
Benard, Leo1 ; Conway, Anthony2
1 Institut Mathématique de Jussieu-Paris Rive Gauche, Sorbonne Université, France
2 Department of Mathematics, Durham University, United Kingdom
Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1029-1084.

Nous définissons un invariant de Casson–Lin multivarié. Cet invariant est défini comme un comptage signé de représentations irréductibles SU(2) du groupe de l’entrelacs, avec traces méridionales fixées. Pour les entrelacs à 2 composantes avec coefficient d’enlacement égal à un, nous montrons que l’invariant est égal à une somme de signatures multivariées. Nous obtenons également des résultats concernant les déformations de représentations SU(2) de groupes d’entrelacs.

We introduce a multivariable Casson–Lin type invariant for links in S 3 . This invariant is defined as a signed count of irreducible SU(2) representations of the link group with fixed meridional traces. For 2-component links with linking number one, the invariant is shown to be a sum of multivariable signatures. We also obtain some results concerning deformations of SU(2) representations of link groups.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3330
Classification : 57M25
Keywords: Knot, link, $\protect \operatorname{SU}(2)$-representation, Casson invariant, Casson–Lin invariant, multivariable signature, character variety, Alexander polynomial, Burau representation, Gassner representation
Mot clés : Noeud, entrelacs, représentation $\protect \operatorname{SU}(2)$, invariant de Casson, invariant de Casson–Lin, signature multivariée, variété de caractères, polynôme d’Alexander, représentation de Burau, représentation de Gassner

Benard, Leo 1 ; Conway, Anthony 2

1 Institut Mathématique de Jussieu-Paris Rive Gauche, Sorbonne Université, France
2 Department of Mathematics, Durham University, United Kingdom
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2020__70_3_1029_0,
     author = {Benard, Leo and Conway, Anthony},
     title = {A multivariable {Casson{\textendash}Lin} type invariant},
     journal = {Annales de l'Institut Fourier},
     pages = {1029--1084},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {3},
     year = {2020},
     doi = {10.5802/aif.3330},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3330/}
}
TY  - JOUR
AU  - Benard, Leo
AU  - Conway, Anthony
TI  - A multivariable Casson–Lin type invariant
JO  - Annales de l'Institut Fourier
PY  - 2020
SP  - 1029
EP  - 1084
VL  - 70
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3330/
DO  - 10.5802/aif.3330
LA  - en
ID  - AIF_2020__70_3_1029_0
ER  - 
%0 Journal Article
%A Benard, Leo
%A Conway, Anthony
%T A multivariable Casson–Lin type invariant
%J Annales de l'Institut Fourier
%D 2020
%P 1029-1084
%V 70
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3330/
%R 10.5802/aif.3330
%G en
%F AIF_2020__70_3_1029_0
Benard, Leo; Conway, Anthony. A multivariable Casson–Lin type invariant. Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1029-1084. doi : 10.5802/aif.3330. https://aif.centre-mersenne.org/articles/10.5802/aif.3330/

[1] Akbulut, Selman; McCarthy, John D. Casson’s invariant for oriented homology 3-spheres. An exposition, Mathematical Notes, 36, Princeton University Press, 1990, xviii+182 pages | DOI | MR | Zbl

[2] Ben Abdelghani, Leila Espace des représentations du groupe d’un nœud classique dans un groupe de Lie, Ann. Inst. Fourier, Volume 50 (2000) no. 4, pp. 1297-1321 | MR

[3] Birman, Joan S. Braids, links, and mapping class groups, Annals of Mathematics Studies, 82, Princeton University Press; University of Tokyo Press, 1974, ix+228 pages | MR

[4] Boden, Hans U.; Harper, Eric The SU(N) Casson-Lin invariants for links, Pac. J. Math., Volume 285 (2016) no. 2, pp. 257-282 | DOI | MR

[5] Boden, Hans U.; Herald, Christopher M. The SU(2) Casson-Lin invariant of the Hopf link, Pac. J. Math., Volume 285 (2016) no. 2, pp. 283-288 | DOI | MR

[6] Burde, Gerhard Darstellungen von Knotengruppen, Math. Ann., Volume 173 (1967), pp. 24-33 | DOI | Zbl

[7] Burde, Gerhard; Zieschang, Heiner; Heusener, Michael Knots 3rd fully revised and extented edition, De Gruyter Studies in Mathematics, 5, Walter de Gruyter, 2014, xiii+417 pages | MR

[8] Cimasoni, David A geometric construction of the Conway potential function, Comment. Math. Helv., Volume 79 (2004) no. 1, pp. 124-146 | DOI | MR

[9] Cimasoni, David; Florens, Vincent Generalized Seifert surfaces and signatures of colored links, Trans. Am. Math. Soc., Volume 360 (2008) no. 3, pp. 1223-1264

[10] Cimasoni, David; Turaev, Vladimir A Lagrangian representation of tangles, Topology, Volume 44 (2005) no. 4, pp. 747-767

[11] Collin, Olivier; Steer, Brian Instanton Floer homology for knots via 3-orbifolds, J. Differ. Geom., Volume 51 (1999) no. 1, pp. 149-202 | MR

[12] Conway, Anthony Burau maps and twisted Alexander polynomials (2017) (to appear in Proc. Edinb. Math. Soc., https://arxiv.org/abs/1510.06678)

[13] Conway, Anthony; Friedl, Stefan; Toffoli, Enrico The Blanchfield pairing of colored links (2017) (to appear in Indiana Univ. Math. J., https://arxiv.org/abs/1609.08057)

[14] Conway, Anthony; Solenn, Estier Conway’s potential function via the Gassner representation (2017) (https://arxiv.org/abs/1709.03479)

[15] Conway, John H. An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Oxford, 1967), Pergamon Press, 1970, pp. 329-358 | MR | Zbl

[16] Cooper, Daryl The universal abelian cover of a link, Low-dimensional topology (Bangor, 1979) (London Mathematical Society Lecture Note Series), Volume 48, Cambridge University Press, 1982, pp. 51-66 | MR

[17] Culler, Marc; Shalen, Peter B. Varieties of group representations and splittings of 3-manifolds, Ann. Math., Volume 117 (1983), pp. 109-146 | Zbl

[18] Ehresmann, Charles Les connexions infinitésimales dans un espace fibré différentiable, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone; Masson et Cie., 1951, pp. 29-55 | MR | Zbl

[19] Friedl, Stefan; Powell, Mark Links not concordant to the Hopf link, Math. Proc. Camb. Philos. Soc., Volume 156 (2014) no. 3, pp. 425-459 | DOI | MR

[20] Frohman, Charles D.; Klassen, Eric P. Deforming representations of knot groups in SU(2), Comment. Math. Helv., Volume 66 (1991) no. 1, pp. 340-361

[21] Garoufalidis, Stavros Does the Jones polynomial determine the signature of a knot? (2003) (https://arxiv.org/abs/math/0310203)

[22] Harper, Eric; Saveliev, Nikolai A Casson-Lin type invariant for links, Pac. J. Math., Volume 248 (2010) no. 1, pp. 139-154 | DOI | MR

[23] Harper, Eric; Saveliev, Nikolai Instanton Floer homology for two-component links, J. Knot Theory Ramifications, Volume 21 (2012) no. 5, 1250054, 8 pages | DOI | MR

[24] Hartley, Richard The Conway potential function for links, Comment. Math. Helv., Volume 58 (1983) no. 3, pp. 365-378 | DOI | MR

[25] Herald, Christopher M. Flat connections, the Alexander invariant, and Casson’s invariant, Commun. Anal. Geom., Volume 5 (1997) no. 1, pp. 93-120 | DOI | MR

[26] Heusener, Michael An orientation for the SU (2)-representation space of knot groups, Topology Appl., Volume 127 (2003) no. 1-2, pp. 175-197 | DOI | MR

[27] Heusener, Michael; Kroll, Jochen Deforming abelian SU (2)-representations of knot groups, Comment. Math. Helv., Volume 73 (1998) no. 3, pp. 480-498 | DOI | MR

[28] Heusener, Michael; Porti, Joan Representations of knot groups into SL n (ℂ) and twisted Alexander polynomials, Pac. J. Math., Volume 277 (2015) no. 2, pp. 313-354 | DOI | Zbl

[29] Heusener, Michael; Porti, Joan; Suárez-Peiró, Eva Deformations of reducible representations of 3-manifold groups into Sl 2 (C), J. Reine Angew. Math. (2001), pp. 191-228 | Zbl

[30] Hillman, Jonathan Algebraic invariants of links, Series on Knots and Everything, 52, World Scientific, 2012, xiv+353 pages

[31] Kawauchi, Akio A survey of knot theory, Birkhäuser, 1996, xxii+420 pages (Translated and revised from the 1990 Japanese original by the author) | MR

[32] Kirk, Paul; Livingston, Charles; Wang, Zhenghan The Gassner representation for string links, Commun. Contemp. Math., Volume 3 (2001) no. 1, pp. 87-136 | DOI | MR

[33] Klassen, Eric P. Representations of knot groups in SU (2), Trans. Am. Math. Soc., Volume 326 (1991) no. 2, pp. 795-828 | DOI | MR

[34] Lickorish, W. B. Raymond An introduction to knot theory, Graduate Texts in Mathematics, 175, Springer, 1997, x+201 pages | DOI | MR

[35] Lin, Xiao-Song A knot invariant via representation spaces, J. Differ. Geom., Volume 35 (1992) no. 2, pp. 337-357 | MR

[36] Long, Darren D. On the linear representation of braid groups, Trans. Am. Math. Soc., Volume 311 (1989) no. 2, pp. 535-560 | DOI | MR

[37] Lubotzky, Alexander; Magid, Andy R. Varieties of representations of finitely generated groups., 336, American Mathematical Society, 1985 | Zbl

[38] Milnor, John Link groups, Ann. Math., Volume 59 (1954), pp. 177-195 | DOI | MR

[39] Murakami, Jun A state model for the multivariable Alexander polynomial, Pac. J. Math., Volume 157 (1993) no. 1, pp. 109-135 | MR

[40] Nagel, Matthias; Owens, Brendan Unlinking information from 4-manifolds, Bull. Lond. Math. Soc., Volume 47 (2015) no. 6, pp. 964-979 | DOI | MR

[41] Nicolaescu, Liviu I. The Reidemeister torsion of 3-manifolds, De Gruyter Studies in Mathematics, 30, Walter de Gruyter, 2003, xiv+249 pages | DOI | MR

[42] Porti, Joan Torsion de Reidemeister pour les variétés hyperboliques., 612, American Mathematical Society, 1997 | Zbl

[43] de Rham, Georges Introduction aux polynômes d’un nœud., Enseign. Math., Volume 13 (1967) no. 2, pp. 187-194 | Zbl

Cité par Sources :