A quantum categorification of the Alexander polynomial

Robert, Louis-Hadrien¹ ; Wagner, Emmanuel²

¹ RMATH, Université de Luxembourg, Esch-sur-Alzette, Luxembourg
² Institut de Mathématiques de Jussieu - Paris Rive Gauche, Université de Paris, Sorbonne Université - Campus Pierre et Marie Curie, Paris, France, Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Université Bourgogne, Dijon, France

Geometry & topology, Tome 26 (2022) no. 5, pp. 1985-2064.

Résumé

Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov and Rozansky.

Détail

DOI : 10.2140/gt.2022.26.1985

Classification : 57M27, 57R56, 17B10, 17B35, 17B37
Keywords: knot homology, Alexander polynomial, foams, Soergel bimodules

Bibliographie

[1] D Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443 | DOI

[2] C Blanchet, N Habegger, G Masbaum, P Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995) 883 | DOI

[3] S Cautis, Remarks on coloured triply graded link invariants, Algebr. Geom. Topol. 17 (2017) 3811 | DOI

[4] J H Conway, An enumeration of knots and links, and some of their algebraic properties, from: "Computational problems in abstract algebra" (editor J Leech), Pergamon (1970) 329 | DOI

[5] N Dowlin, A categorification of the HOMFLY-PT polynomial with a spectral sequence to knot Floer homology, preprint (2017)

[6] N Dowlin, Knot Floer homology and Khovanov–Rozansky homology for singular links, Algebr. Geom. Topol. 18 (2018) 3839 | DOI

[7] N M Dunfield, S Gukov, J Rasmussen, The superpolynomial for knot homologies, Exp. Math. 15 (2006) 129 | DOI

[8] M Ehrig, D Tubbenhauer, P Wedrich, Functoriality of colored link homologies, Proc. Lond. Math. Soc. 117 (2018) 996 | DOI

[9] A P Ellis, I Petkova, V Vértesi, Quantum gl1|1 and tangle Floer homology, Adv. Math. 350 (2019) 130 | DOI

[10] P Freyd, D Yetter, J Hoste, W B R Lickorish, K Millett, A Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985) 239 | DOI

[11] N Geer, B Patureau-Mirand, Multivariable link invariants arising from sl(2|1) and the Alexander polynomial, J. Pure Appl. Algebra 210 (2007) 283 | DOI

[12] A Gilmore, Invariance and the knot Floer cube of resolutions, Quantum Topol. 7 (2016) 107 | DOI

[13] C Kassel, V Turaev, Braid groups, 247, Springer (2008) | DOI

[14] L H Kauffman, H Saleur, Free fermions and the Alexander–Conway polynomial, Comm. Math. Phys. 141 (1991) 293 | DOI

[15] L H Kauffman, P Vogel, Link polynomials and a graphical calculus, J. Knot Theory Ramifications 1 (1992) 59 | DOI

[16] M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359 | DOI

[17] M Khovanov, sl(3) link homology, Algebr. Geom. Topol. 4 (2004) 1045 | DOI

[18] M Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, Int. J. Math. 18 (2007) 869 | DOI

[19] M Khovanov, L H Robert, Foam evaluation and Kronheimer–Mrowka theories, Adv. Math. 376 (2021) | DOI

[20] M Khovanov, L Rozansky, Matrix factorizations and link homology, II, Geom. Topol. 12 (2008) 1387 | DOI

[21] P Kronheimer, T Mrowka, Knots, sutures, and excision, J. Differential Geom. 84 (2010) 301

[22] M Mackaay, M Stošić, P Vaz, sl(N)–link homology (N ≥ 4) using foams and the Kapustin–Li formula, Geom. Topol. 13 (2009) 1075 | DOI

[23] C Manolescu, An untwisted cube of resolutions for knot Floer homology, Quantum Topol. 5 (2014) 185 | DOI

[24] H R Morton, S G Lukac, The HOMFLY polynomial of the decorated Hopf link, J. Knot Theory Ramifications 12 (2003) 395 | DOI

[25] H Murakami, T Ohtsuki, S Yamada, HOMFLY polynomial via an invariant of colored plane graphs, Enseign. Math. 44 (1998) 325

[26] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58 | DOI

[27] P Ozsváth, Z Szabó, A cube of resolutions for knot Floer homology, J. Topol. 2 (2009) 865 | DOI

[28] I Petkova, V Vértesi, Combinatorial tangle Floer homology, Geom. Topol. 20 (2016) 3219 | DOI

[29] J H Przytycki, P Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1988) 115

[30] H Queffelec, D E V Rose, The sln foam 2–category : a combinatorial formulation of Khovanov–Rozansky homology via categorical skew Howe duality, Adv. Math. 302 (2016) 1251 | DOI

[31] H Queffelec, D E V Rose, Sutured annular Khovanov–Rozansky homology, Trans. Amer. Math. Soc. 370 (2018) 1285 | DOI

[32] H Queffelec, D E V Rose, A Sartori, Annular evaluation and link homology, preprint (2018)

[33] J A Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University (2003)

[34] J Rasmussen, Some differentials on Khovanov–Rozansky homology, Geom. Topol. 19 (2015) 3031 | DOI

[35] L H Robert, E Wagner, A closed formula for the evaluation of foams, Quantum Topol. 11 (2020) 411 | DOI

[36] L H Robert, E Wagner, Symmetric Khovanov–Rozansky link homologies, J. Éc. Polytech. Math. 7 (2020) 573 | DOI

[37] R Rouquier, Khovanov–Rozansky homology and 2–braid groups, from: "Categorification in geometry, topology, and physics" (editors A Beliakova, A D Lauda), Contemp. Math. 684, Amer. Math. Soc. (2017) 147 | DOI

[38] L Rozansky, H Saleur, Quantum field theory for the multi-variable Alexander–Conway polynomial, Nuclear Phys. B 376 (1992) 461 | DOI

[39] G Torres, On the Alexander polynomial, Ann. of Math. 57 (1953) 57 | DOI

[40] V G Turaev, The Conway and Kauffman modules of a solid torus, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 167 (1988) 79

[41] P Vaz, A categorification of the quantum sl(N)–link polynomials using foams, PhD thesis, Universidade do Algarve (2008)

[42] O Y Viro, Quantum relatives of the Alexander polynomial, Algebra i Analiz 18 (2006) 63

[43] P Wedrich, Exponential growth of colored HOMFLY-PT homology, Adv. Math. 353 (2019) 471 | DOI

[44] H Wu, A colored sl(N) homology for links in S3, Dissertationes Math. 499 (2014) 1 | DOI