Nous définissons un invariant de Casson–Lin multivarié. Cet invariant est défini comme un comptage signé de représentations irréductibles 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 de groupes d’entrelacs.
We introduce a multivariable Casson–Lin type invariant for links in . This invariant is defined as a signed count of irreducible 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 representations of link groups.
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Classification : 57M25
Mots clés : Noeud, entrelacs, représentation , invariant de Casson, invariant de Casson–Lin, signature multivariée, variété de caractères, polynôme d’Alexander, représentation de Burau, représentation de Gassner
@article{AIF_2020__70_3_1029_0, author = {Benard, Leo and Conway, Anthony}, title = {A multivariable Casson--Lin type invariant}, journal = {Annales de l'Institut Fourier}, pages = {1029--1084}, publisher = {Association des Annales de l'institut Fourier}, volume = {70}, number = {3}, year = {2020}, doi = {10.5802/aif.3330}, language = {en}, url = {https://aif.centre-mersenne.org/item/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/item/AIF_2020__70_3_1029_0/
[1] Casson’s invariant for oriented homology -spheres. An exposition, Mathematical Notes, Volume 36, Princeton University Press, 1990, xviii+182 pages | Article | MR 1030042 | Zbl 0695.57011
[2] 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 1799747
[3] Braids, links, and mapping class groups, Annals of Mathematics Studies, Volume 82, Princeton University Press; University of Tokyo Press, 1974, ix+228 pages | MR 0375281
[4] The Casson-Lin invariants for links, Pac. J. Math., Volume 285 (2016) no. 2, pp. 257-282 | Article | MR 3575566
[5] The Casson-Lin invariant of the Hopf link, Pac. J. Math., Volume 285 (2016) no. 2, pp. 283-288 | Article | MR 3575567
[6] Darstellungen von Knotengruppen, Math. Ann., Volume 173 (1967), pp. 24-33 | Article | Zbl 0146.45602
[7] Knots 3rd fully revised and extented edition, De Gruyter Studies in Mathematics, Volume 5, Walter de Gruyter, 2014, xiii+417 pages | MR 3156509
[8] A geometric construction of the Conway potential function, Comment. Math. Helv., Volume 79 (2004) no. 1, pp. 124-146 | Article | MR 2031702
[9] Generalized Seifert surfaces and signatures of colored links, Trans. Am. Math. Soc., Volume 360 (2008) no. 3, pp. 1223-1264
[10] A Lagrangian representation of tangles, Topology, Volume 44 (2005) no. 4, pp. 747-767
[11] Instanton Floer homology for knots via -orbifolds, J. Differ. Geom., Volume 51 (1999) no. 1, pp. 149-202 | MR 1703606
[12] Burau maps and twisted Alexander polynomials (2017) (to appear in Proc. Edinb. Math. Soc., https://arxiv.org/abs/1510.06678)
[13] The Blanchfield pairing of colored links (2017) (to appear in Indiana Univ. Math. J., https://arxiv.org/abs/1609.08057)
[14] Conway’s potential function via the Gassner representation (2017) (https://arxiv.org/abs/1709.03479)
[15] 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 0258014 | Zbl 0202.54703
[16] 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 662427
[17] Varieties of group representations and splittings of 3-manifolds, Ann. Math., Volume 117 (1983), pp. 109-146 | Zbl 0529.57005
[18] 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 0042768 | Zbl 0054.07201
[19] Links not concordant to the Hopf link, Math. Proc. Camb. Philos. Soc., Volume 156 (2014) no. 3, pp. 425-459 | Article | MR 3181634
[20] Deforming representations of knot groups in SU(2), Comment. Math. Helv., Volume 66 (1991) no. 1, pp. 340-361
[21] Does the Jones polynomial determine the signature of a knot? (2003) (https://arxiv.org/abs/math/0310203)
[22] A Casson-Lin type invariant for links, Pac. J. Math., Volume 248 (2010) no. 1, pp. 139-154 | Article | MR 2734168
[23] Instanton Floer homology for two-component links, J. Knot Theory Ramifications, Volume 21 (2012) no. 5, 1250054, 8 pages | Article | MR 2902278
[24] The Conway potential function for links, Comment. Math. Helv., Volume 58 (1983) no. 3, pp. 365-378 | Article | MR 727708
[25] Flat connections, the Alexander invariant, and Casson’s invariant, Commun. Anal. Geom., Volume 5 (1997) no. 1, pp. 93-120 | Article | MR 1456309
[26] An orientation for the -representation space of knot groups, Topology Appl., Volume 127 (2003) no. 1-2, pp. 175-197 | Article | MR 1953326
[27] Deforming abelian -representations of knot groups, Comment. Math. Helv., Volume 73 (1998) no. 3, pp. 480-498 | Article | MR 1633375
[28] Representations of knot groups into and twisted Alexander polynomials, Pac. J. Math., Volume 277 (2015) no. 2, pp. 313-354 | Article | Zbl 1323.57005
[29] Deformations of reducible representations of 3-manifold groups into , J. Reine Angew. Math. (2001), pp. 191-228 | Zbl 0964.57006
[30] Algebraic invariants of links, Series on Knots and Everything, Volume 52, World Scientific, 2012, xiv+353 pages
[31] A survey of knot theory, Birkhäuser, 1996, xxii+420 pages (Translated and revised from the 1990 Japanese original by the author) | MR 1417494
[32] The Gassner representation for string links, Commun. Contemp. Math., Volume 3 (2001) no. 1, pp. 87-136 | Article | MR 1820015
[33] Representations of knot groups in , Trans. Am. Math. Soc., Volume 326 (1991) no. 2, pp. 795-828 | Article | MR 1008696
[34] An introduction to knot theory, Graduate Texts in Mathematics, Volume 175, Springer, 1997, x+201 pages | Article | MR 1472978
[35] A knot invariant via representation spaces, J. Differ. Geom., Volume 35 (1992) no. 2, pp. 337-357 | MR 1158339
[36] On the linear representation of braid groups, Trans. Am. Math. Soc., Volume 311 (1989) no. 2, pp. 535-560 | Article | MR 943606
[37] Varieties of representations of finitely generated groups. Volume 336, American Mathematical Society, 1985 | Zbl 0598.14042
[38] Link groups, Ann. Math., Volume 59 (1954), pp. 177-195 | Article | MR 0071020
[39] A state model for the multivariable Alexander polynomial, Pac. J. Math., Volume 157 (1993) no. 1, pp. 109-135 | MR 1197048
[40] Unlinking information from 4-manifolds, Bull. Lond. Math. Soc., Volume 47 (2015) no. 6, pp. 964-979 | Article | MR 3431577
[41] The Reidemeister torsion of 3-manifolds, De Gruyter Studies in Mathematics, Volume 30, Walter de Gruyter, 2003, xiv+249 pages | Article | MR 1968575
[42] Torsion de Reidemeister pour les variétés hyperboliques. Volume 612, American Mathematical Society, 1997 | Zbl 0881.57020
[43] Introduction aux polynômes d’un nœud., Enseign. Math., Volume 13 (1967) no. 2, pp. 187-194 | Zbl 0157.54803