Geometry & Topology Volume 26, issue 8 (2022)

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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Invariants of $4$–manifolds from Khovanov–Rozansky link homology

Scott Morrison, Kevin Walker and Paul Wedrich

Geometry & Topology 26 (2022) 3367–3420

DOI: 10.2140/gt.2022.26.3367

Bibliography

1	M Abouzaid, I Smith, Khovanov homology from Floer cohomology, J. Amer. Math. Soc. 32 (2019) 1 MR3867999
2	M M Asaeda, J H Przytycki, A S Sikora, Categorification of the Kauffman bracket skein module of I–bundles over surfaces, Algebr. Geom. Topol. 4 (2004) 1177 MR2113902
3	M Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. 68 (1988) 175 MR1001453
4	J C Baez, L Langford, Higher-dimensional algebra, IV : 2–tangles, Adv. Math. 180 (2003) 705 MR2020556
5	J C Baez, M Neuchl, Higher-dimensional algebra, I : Braided monoidal 2–categories, Adv. Math. 121 (1996) 196 MR1402727
6	D Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443 MR2174270
7	D Bar-Natan, S Morrison, The Karoubi envelope and Lee’s degeneration of Khovanov homology, Algebr. Geom. Topol. 6 (2006) 1459 MR2253455
8	J W Barrett, C Meusburger, G Schaumann, Gray categories with duals and their diagrams, preprint (2012) arXiv:1211.0529
9	A Beliakova, K K Putyra, S M Wehrli, Quantum link homology via trace functor, I, Invent. Math. 215 (2019) 383 MR3910068
10	D Ben-Zvi, A Brochier, D Jordan, Integrating quantum groups over surfaces, J. Topol. 11 (2018) 874 MR3847209
11	C Blanchet, An oriented model for Khovanov homology, J. Knot Theory Ramifications 19 (2010) 291 MR2647055
12	A Brochier, D Jordan, N Snyder, On dualizability of braided tensor categories, Compos. Math. 157 (2021) 435 MR4228258
13	J S Carter, J H Rieger, M Saito, A combinatorial description of knotted surfaces and their isotopies, Adv. Math. 127 (1997) 1 MR1445361
14	J S Carter, M Saito, Reidemeister moves for surface isotopies and their interpretation as moves to movies, J. Knot Theory Ramifications 2 (1993) 251 MR1238875
15	S Cautis, J Kamnitzer, Knot homology via derived categories of coherent sheaves, I : The 𝔰𝔩(2)–case, Duke Math. J. 142 (2008) 511 MR2411561
16	S Cautis, J Kamnitzer, Knot homology via derived categories of coherent sheaves, II : 𝔰𝔩_m case, Invent. Math. 174 (2008) 165 MR2430980
17	S Cautis, J Kamnitzer, S Morrison, Webs and quantum skew Howe duality, Math. Ann. 360 (2014) 351 MR3263166
18	D Clark, S Morrison, K Walker, Fixing the functoriality of Khovanov homology, Geom. Topol. 13 (2009) 1499 MR2496052
19	L Crane, Clock and category: is quantum gravity algebraic?, J. Math. Phys. 36 (1995) 6180 MR1355904
20	L Crane, I B Frenkel, Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases, J. Math. Phys. 35 (1994) 5136 MR1295461
21	L Crane, L H Kauffman, D N Yetter, State-sum invariants of 4–manifolds, J. Knot Theory Ramifications 6 (1997) 177 MR1452438
22	S E Crans, Generalized centers of braided and sylleptic monoidal 2–categories, Adv. Math. 136 (1998) 183 MR1626844
23	M Ehrig, D Tubbenhauer, P Wedrich, Functoriality of colored link homologies, Proc. Lond. Math. Soc. 117 (2018) 996 MR3877770
24	T Ekholm, V Shende, Skeins on branes, preprint (2019) arXiv:1901.08027
25	B Elias, M Khovanov, Diagrammatics for Soergel categories, Int. J. Math. Math. Sci. (2010) MR3095655
26	B Elias, D Krasner, Rouquier complexes are functorial over braid cobordisms, Homology Homotopy Appl. 12 (2010) 109 MR2721032
27	B Elias, Y Qi, A categorification of quantum 𝔰𝔩(2) at prime roots of unity, Adv. Math. 299 (2016) 863 MR3519482
28	M Freedman, R Gompf, S Morrison, K Walker, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, Quantum Topol. 1 (2010) 171 MR2657647
29	B Gornik, Note on Khovanov link cohomology, preprint (2004) arXiv:math/0402266
30	E Gorsky, M Hogancamp, P Wedrich, Derived traces of Soergel categories, Int. Math. Res. Not. 2022 (2022) 11304 MR4458553
31	J E Grigsby, A M Licata, S M Wehrli, Annular Khovanov homology and knotted Schur–Weyl representations, Compos. Math. 154 (2018) 459 MR3731256
32	S Gukov, C Manolescu, A two-variable series for knot complements, Quantum Topol. 12 (2021) 1 MR4233201
33	S Gukov, D Pei, P Putrov, C Vafa, BPS spectra and 3–manifold invariants, J. Knot Theory Ramifications 29 (2020) MR4089709
34	S Gukov, P Putrov, C Vafa, Fivebranes and 3–manifold homology, J. High Energy Phys. (2017) MR3686727
35	S Gukov, A Schwarz, C Vafa, Khovanov–Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005) 53 MR2193547
36	A Henriques, D Penneys, J Tener, Categorified trace for module tensor categories over braided tensor categories, Doc. Math. 21 (2016) 1089 MR3578212
37	M Hogancamp, D E V Rose, P Wedrich, A Kirby color for Khovanov homology, preprint (2022) arXiv:2210.05640
38	V F R Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985) 103 MR766964
39	V F R Jones, Planar algebras, I, preprint (1999) arXiv:math/9909027
40	S Kang, Link homology theories and ribbon concordances, preprint (2019) arXiv:1909.06969
41	M M Kapranov, V A Voevodsky, 2–Categories and Zamolodchikov tetrahedra equations, from: "Algebraic groups and their generalizations: quantum and infinite-dimensional methods" (editors W J Haboush, B J Parshall), Proc. Sympos. Pure Math. 56, Amer. Math. Soc. (1994) 177 MR1278735
42	M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359 MR1740682
43	M Khovanov, Link homology and Frobenius extensions, Fund. Math. 190 (2006) 179 MR2232858
44	M Khovanov, Hopfological algebra and categorification at a root of unity: the first steps, J. Knot Theory Ramifications 25 (2016) MR3475073
45	M Khovanov, A D Lauda, A categorification of quantum sl(n), Quantum Topol. 1 (2010) 1 MR2628852
46	M Khovanov, L Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 1 MR2391017
47	P B Kronheimer, T S Mrowka, Khovanov homology is an unknot-detector, Publ. Math. Inst. Hautes Études Sci. 113 (2011) 97 MR2805599
48	A D Lauda, H Queffelec, D E V Rose, Khovanov homology is a skew Howe 2–representation of categorified quantum 𝔰𝔩_m, Algebr. Geom. Topol. 15 (2015) 2517 MR3426687
49	E S Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005) 554 MR2173845
50	A Lobb, A slice genus lower bound from sl(n) Khovanov–Rozansky homology, Adv. Math. 222 (2009) 1220 MR2554935
51	J Lurie, On the classification of topological field theories, from: "Current developments in mathematics, 2008" (editors D Jerison, B Mazur, T Mrowka, W Schmid, R Stanley, S T Yau), International (2009) 129 MR2555928
52	G Lusztig, Introduction to quantum groups, 110, Birkhäuser (1993) MR1227098
53	M Mackaay, Spherical 2–categories and 4–manifold invariants, Adv. Math. 143 (1999) 288 MR1686421
54	M Mackaay, M Stošić, P Vaz, 𝔰𝔩(N)–link homology (N ≥ 4) using foams and the Kapustin–Li formula, Geom. Topol. 13 (2009) 1075 MR2491657
55	M Mackaay, P Vaz, The diagrammatic Soergel category and sl(N)–foams, for N ≥ 4, Int. J. Math. Math. Sci. (2010) MR2671770
56	C Manolescu, I Neithalath, Skein lasagna modules for 2–handlebodies, J. Reine Angew. Math. 788 (2022) 37 MR4445546
57	C Manolescu, K Walker, P Wedrich, Skein lasagna modules and handle decompositions, preprint (2022) arXiv:2206.04616
58	S Morrison, K Walker, Blob homology, Geom. Topol. 16 (2012) 1481 MR2978449
59	J H Przytycki, Skein modules of 3–manifolds, Bull. Polish Acad. Sci. Math. 39 (1991) 91 MR1194712
60	Y Qi, Hopfological algebra, Compos. Math. 150 (2014) 1 MR3164358
61	Y Qi, J Sussan, Categorification at prime roots of unity and hopfological finiteness, from: "Categorification and higher representation theory" (editors A Beliakova, A D Lauda), Contemp. Math. 683, Amer. Math. Soc. (2017) 261 MR3611717
62	H Queffelec, D E V Rose, The 𝔰𝔩_n foam 2–category : a combinatorial formulation of Khovanov–Rozansky homology via categorical skew Howe duality, Adv. Math. 302 (2016) 1251 MR3545951
63	H Queffelec, D E V Rose, Sutured annular Khovanov–Rozansky homology, Trans. Amer. Math. Soc. 370 (2018) 1285 MR3729501
64	H Queffelec, A Sartori, Mixed quantum skew Howe duality and link invariants of type A, J. Pure Appl. Algebra 223 (2019) 2733 MR3912946
65	H Queffelec, P Wedrich, Extremal weight projectors, Math. Res. Lett. 25 (2018) 1911 MR3934851
66	H Queffelec, P Wedrich, Khovanov homology and categorification of skein modules, Quantum Topol. 12 (2021) 129 MR4233203
67	J Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010) 419 MR2729272
68	J Rasmussen, Some differentials on Khovanov–Rozansky homology, Geom. Topol. 19 (2015) 3031 MR3447099
69	N Y Reshetikhin, V G Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990) 1 MR1036112
70	N Reshetikhin, V G Turaev, Invariants of 3–manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547 MR1091619
71	L H Robert, E Wagner, A closed formula for the evaluation of foams, Quantum Topol. 11 (2020) 411 MR4164001
72	D E V Rose, P Wedrich, Deformations of colored 𝔰𝔩_N link homologies via foams, Geom. Topol. 20 (2016) 3431 MR3590355
73	R Rouquier, 2–Kac–Moody algebras, preprint (2008) arXiv:0812.5023
74	L Rozansky, A categorification of the stable SU(2) Witten–Reshetikhin–Turaev invariant of links in S² × S¹, preprint (2010) arXiv:1011.1958
75	P Seidel, I Smith, A link invariant from the symplectic geometry of nilpotent slices, Duke Math. J. 134 (2006) 453 MR2254624
76	C Stroppel, Categorification of the Temperley–Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126 (2005) 547 MR2120117
77	C Stroppel, Parabolic category 𝒪, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compos. Math. 145 (2009) 954 MR2521250
78	C H Taubes, Compactness theorems for SL(2; ℂ) generalizations of the 4–dimensional anti-self dual equations, preprint (2013) arXiv:1307.6447
79	C H Taubes, Sequences of Nahm pole solutions to the SU(2) Kapustin–Witten equations, preprint (2018) arXiv:1805.02773
80	D P Thurston, Positive basis for surface skein algebras, Proc. Natl. Acad. Sci. USA 111 (2014) 9725 MR3263305
81	D Tubbenhauer, P Vaz, P Wedrich, Super q–Howe duality and web categories, Algebr. Geom. Topol. 17 (2017) 3703 MR3709658
82	V G Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. École Norm. Sup. 24 (1991) 635 MR1142906
83	K Walker, TQFTs, book project (2006)
84	B Webster, Knot invariants and higher representation theory, 1191, Amer. Math. Soc. (2017) MR3709726
85	M Willis, Khovanov homology for links in #^r(S² × S¹), Michigan Math. J. 70 (2021) 675 MR4332675
86	E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351 MR990772
87	E Witten, Fivebranes and knots, Quantum Topol. 3 (2012) 1 MR2852941
88	H Wu, On the quantum filtration of the Khovanov–Rozansky cohomology, Adv. Math. 221 (2009) 54 MR2509322
89	H Wu, Equivariant colored 𝔰𝔩(N)–homology for links, J. Knot Theory Ramifications 21 (2012) MR2885476