Geometry & Topology Volume 26, issue 8 (2022)

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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Legendrian weaves: $N$–graph calculus, flag moduli and applications

Roger Casals and Eric Zaslow

Geometry & Topology 26 (2022) 3589–3745

DOI: 10.2140/gt.2022.26.3589

Bibliography

1	M Aganagic, C Vafa, Mirror symmetry, D–branes and counting holomorphic discs, preprint (2000) arXiv:hep-th/0012041
2	M Aganagic, C Vafa, Large N duality, mirror symmetry, and a Q–deformed A–polynomial for knots, preprint (2012) arXiv:1204.4709
3	J W Alexander, G B Briggs, On types of knotted curves, Ann. of Math. 28 (1926/27) 562 MR1502807
4	V I Arnold, Critical points of smooth functions, from: "Proceedings of the International Congress of Mathematicians" (editor R D James), Canad. Math. Congress (1975) 19 MR0431217
5	V I Arnold, Wave front evolution and equivariant Morse lemma, Comm. Pure Appl. Math. 29 (1976) 557 MR436200
6	V I Arnold, Indexes of singular points of 1–forms on manifolds with boundary, convolutions of invariants of groups generated by reflections, and singular projections of smooth surfaces, Uspekhi Mat. Nauk 34 (1979) 3 MR535708
7	V I Arnold, Singularities of caustics and wave fronts, 62, Kluwer (1990) MR1151185
8	V I Arnold, A B Givental, Symplectic geometry, from: "Dynamical systems, IV" (editor S P Novikov), Encyclopaedia Math. Sci. 4, Springer (2001) 1 MR1866631
9	D Auroux, Mirror symmetry and T–duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. 1 (2007) 51 MR2386535
10	D Auroux, Special Lagrangian fibrations, wall-crossing, and mirror symmetry, from: "Surveys in differential geometry, XIII : Geometry, analysis, and algebraic geometry" (editors H D Cao, S T Yau), Surv. Differ. Geom. 13, International (2009) 1 MR2537081
11	D Bennequin, Entrelacements et équations de Pfaff, from: "IIIe rencontre de géométrie du Schnepfenried" (editors D Bernard, T Hangan, R Lutz), Astérisque 107, Soc. Math. France (1983) 87 MR753131
12	D Bennequin, Caustique mystique (d’après Arnold et al), from: "Séminaire Bourbaki, 1984/85", Astérisque 133–134, Soc. Math. France (1986) 19 MR837213
13	J S Birman, Braids, links, and mapping class groups, 82, Princeton Univ. Press (1974) MR0375281
14	J A Bondy, U S R Murty, Graph theory, 244, Springer (2008) MR2368647
15	F Bourgeois, J M Sabloff, L Traynor, Lagrangian cobordisms via generating families: construction and geography, Algebr. Geom. Topol. 15 (2015) 2439 MR3402346
16	T Bridgeland, I Smith, Quadratic differentials as stability conditions, Publ. Math. Inst. Hautes Études Sci. 121 (2015) 155 MR3349833
17	R Casals, Lagrangian skeleta and plane curve singularities, J. Fixed Point Theory Appl. 24 (2022) MR4405603
18	R Casals, H Gao, Infinitely many Lagrangian fillings, Ann. of Math. 195 (2022) 207 MR4358415
19	R Casals, E Gorsky, M Gorsky, I Le, L Shen, J Simental, Cluster structures on braid varieties, preprint (2022) arXiv:2207.11607
20	R Casals, E Gorsky, M Gorsky, J Simental, Algebraic weaves and braid varieties, preprint (2020) arXiv:2012.06931
21	R Casals, E Murphy, Differential algebra of cubic planar graphs, Adv. Math. 338 (2018) 401 MR3861709
22	R Casals, E Murphy, Legendrian fronts for affine varieties, Duke Math. J. 168 (2019) 225 MR3909897
23	R Casals, E Murphy, F Presas, Geometric criteria for overtwistedness, J. Amer. Math. Soc. 32 (2019) 563 MR3904160
24	R Casals, L Ng, Braid Loops with infinite monodromy on the Legendrian contact DGA, J. Topol. 15 (2022) 1927
25	Y Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) 441 MR1946550
26	K Cieliebak, Y Eliashberg, From Stein to Weinstein and back : symplectic geometry of affine complex manifolds, 59, Amer. Math. Soc. (2012) MR3012475
27	K Cieliebak, Y Eliashberg, Stein structures: existence and flexibility, from: "Contact and symplectic topology" (editors F Bourgeois, V Colin, A Stipsicz), Bolyai Soc. Math. Stud. 26, János Bolyai Math. Soc. (2014) 357 MR3220946
28	M D E Conder, Some results on quotients of triangle groups, Bull. Austral. Math. Soc. 30 (1984) 73 MR753563
29	R Diestel, Graph theory, 173, Springer (2017) MR3644391
30	G Dimitroglou Rizell, Knotted Legendrian surfaces with few Reeb chords, Algebr. Geom. Topol. 11 (2011) 2903 MR2846915
31	G Dimitroglou Rizell, Legendrian ambient surgery and Legendrian contact homology, J. Symplectic Geom. 14 (2016) 811 MR3548486
32	T Dimofte, M Gabella, A B Goncharov, K–decompositions and 3d gauge theories, J. High Energy Phys. (2016) MR3594814
33	T Ekholm, J Etnyre, L Ng, M Sullivan, Filtrations on the knot contact homology of transverse knots, Math. Ann. 355 (2013) 1561 MR3037024
34	T Ekholm, J B Etnyre, L Ng, M G Sullivan, Knot contact homology, Geom. Topol. 17 (2013) 975 MR3070519
35	T Ekholm, J Etnyre, M Sullivan, The contact homology of Legendrian submanifolds in ℝ²ⁿ⁺¹, J. Differential Geom. 71 (2005) 177 MR2197142
36	T Ekholm, J Etnyre, M Sullivan, Non-isotopic Legendrian submanifolds in ℝ²ⁿ⁺¹, J. Differential Geom. 71 (2005) 85 MR2191769
37	T Ekholm, K Honda, T Kálmán, Legendrian knots and exact Lagrangian cobordisms, J. Eur. Math. Soc. 18 (2016) 2627 MR3562353
38	T Ekholm, L Ng, V Shende, A complete knot invariant from contact homology, Invent. Math. 211 (2018) 1149 MR3763406
39	B Elias, M Khovanov, Diagrammatics for Soergel categories, Int. J. Math. Math. Sci. (2010) MR3095655
40	B Elias, G Williamson, Soergel calculus, Represent. Theory 20 (2016) 295 MR3555156
41	Y Eliashberg, Topological characterization of Stein manifolds of dimension > 2, Internat. J. Math. 1 (1990) 29 MR1044658
42	Y Eliashberg, Legendrian and transversal knots in tight contact 3–manifolds, from: "Topological methods in modern mathematics" (editors L R Goldberg, A V Phillips), Publish or Perish (1993) 171 MR1215964
43	Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, from: "Visions in mathematics" (editors N Alon, J Bourgain, A Connes, M Gromov, V Milman), Birkhäuser (= GAFA special volume, II) (2000) 560 MR1826267
44	Y Eliashberg, N Mishachev, Introduction to the h–principle, 48, Amer. Math. Soc. (2002) MR1909245
45	J B Etnyre, Legendrian and transversal knots, from: "Handbook of knot theory" (editors W Menasco, M Thistlethwaite), Elsevier (2005) 105 MR2179261
46	J Etnyre, V Vértesi, Legendrian satellites, Int. Math. Res. Not. 2018 (2018) 7241 MR3883132
47	V V Fock, A B Goncharov, Cluster 𝒳–varieties, amalgamation, and Poisson–Lie groups, from: "Algebraic geometry and number theory" (editor V Ginzburg), Progr. Math. 253, Birkhäuser (2006) 27 MR2263192
48	V Fock, A Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006) 1 MR2233852
49	S Fomin, A Zelevinsky, Cluster algebras, I : Foundations, J. Amer. Math. Soc. 15 (2002) 497 MR1887642
50	K Fukaya, Y G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory : anomaly and obstruction, I, 46, Amer. Math. Soc. (2009) MR2553465
51	M Gabella, P Longhi, C Y Park, M Yamazaki, BPS graphs : from spectral networks to BPS quivers, J. High Energy Phys. (2017) MR3688368
52	D Gaiotto, G W Moore, A Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys. 299 (2010) 163 MR2672801
53	D Gaiotto, G W Moore, A Neitzke, Spectral networks, Ann. Henri Poincaré 14 (2013) 1643 MR3115984
54	D Gaiotto, G W Moore, A Neitzke, Spectral networks and snakes, Ann. Henri Poincaré 15 (2014) 61 MR3147409
55	S Ganatra, J Pardon, V Shende, Microlocal Morse theory of wrapped Fukaya categories, preprint (2018) arXiv:1809.08807
56	S Ganatra, J Pardon, V Shende, Sectorial descent for wrapped Fukaya categories, preprint (2018) arXiv:1809.03427
57	S Ganatra, J Pardon, V Shende, Covariantly functorial wrapped Floer theory on Liouville sectors, Publ. Math. Inst. Hautes Études Sci. 131 (2020) 73 MR4106794
58	H Gao, L Shen, D Weng, Augmentations, fillings, and clusters, preprint (2020) arXiv:2008.10793
59	H Gao, L Shen, D Weng, Positive braid links with infinitely many fillings, preprint (2020) arXiv:2009.00499
60	H Geiges, An introduction to contact topology, 109, Cambridge Univ. Press (2008) MR2397738
61	R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148 (1998) 619 MR1668563
62	A B Goncharov, Ideal webs, moduli spaces of local systems, and 3d Calabi–Yau categories, from: "Algebra, geometry, and physics in the 21st century" (editors D Auroux, L Katzarkov, T Pantev, Y Soibelman, Y Tschinkel), Progr. Math. 324, Springer (2017) 31 MR3702383
63	A B Goncharov, R Kenyon, Dimers and cluster integrable systems, Ann. Sci. Éc. Norm. Supér. 46 (2013) 747 MR3185352
64	A Goncharov, L Shen, Donaldson–Thomas transformations of moduli spaces of G–local systems, Adv. Math. 327 (2018) 225 MR3761995
65	M Gromov, Partial differential relations, 9, Springer (1986) MR864505
66	S Guillermou, M Kashiwara, P Schapira, Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems, Duke Math. J. 161 (2012) 201 MR2876930
67	S Guillermou, P Schapira, Microlocal theory of sheaves and Tamarkin’s non displaceability theorem, from: "Homological mirror symmetry and tropical geometry" (editors R Castano-Bernard, F Catanese, M Kontsevich, T Pantev, Y Soibelman, I Zharkov), Lect. Notes Unione Mat. Ital. 15, Springer (2014) 43 MR3330785
68	A Hurwitz, Ueber algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1892) 403 MR1510753
69	T Kálmán, Contact homology and one parameter families of Legendrian knots, Geom. Topol. 9 (2005) 2013 MR2209366
70	M Kashiwara, P Schapira, Microlocal study of sheaves, 128, Soc. Math. France (1985) 235 MR794557
71	M Kashiwara, P Schapira, Sheaves on manifolds, 292, Springer (1990) MR1074006
72	B Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. 27 (1994) 63 MR1258406
73	M Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, Internat. J. Math. 18 (2007) 869 MR2339573
74	M Kontsevich, Y Soibelman, Motivic Donaldson–Thomas invariants : summary of results, from: "Mirror symmetry and tropical geometry" (editors R Castaño-Bernard, Y Soibelman, I Zharkov), Contemp. Math. 527, Amer. Math. Soc. (2010) 55 MR2681792
75	M Kontsevich, Y Soibelman, Wall-crossing structures in Donaldson–Thomas invariants, integrable systems and mirror symmetry, from: "Homological mirror symmetry and tropical geometry" (editors R Castano-Bernard, F Catanese, M Kontsevich, T Pantev, Y Soibelman, I Zharkov), Lect. Notes Unione Mat. Ital. 15, Springer (2014) 197 MR3330788
76	T Kuwagaki, Sheaf quantization from exact WKB analysis, preprint (2020) arXiv:2006.14872
77	Y Laszlo, M Olsson, The six operations for sheaves on Artin stacks, I : Finite coefficients, Publ. Math. Inst. Hautes Études Sci. 107 (2008) 109 MR2434692
78	Y Laszlo, M Olsson, Perverse t–structure on Artin stacks, Math. Z. 261 (2009) 737 MR2480756
79	J W Lawson, Minimal mutation-infinite quivers, Exp. Math. 26 (2017) 308 MR3642109
80	A Lucchini, M C Tamburini, Classical groups of large rank as Hurwitz groups, J. Algebra 219 (1999) 531 MR1706821
81	V A Lunts, D O Orlov, Uniqueness of enhancement for triangulated categories, J. Amer. Math. Soc. 23 (2010) 853 MR2629991
82	A A Markov, Über die freie Äquivalenz der geschlossenen Zöpfe, Rec. Math. Moscou 1 (1936) 73
83	S Morier-Genoud, V Ovsienko, S Tabachnikov, 2–Frieze patterns and the cluster structure of the space of polygons, Ann. Inst. Fourier (Grenoble) 62 (2012) 937 MR3013813
84	E Murphy, Loose Legendrian embeddings in high-dimensional contact manifolds, PhD thesis, Stanford University (2012) MR4172336
85	D Nadler, Microlocal branes are constructible sheaves, Selecta Math. 15 (2009) 563 MR2565051
86	D Nadler, Arboreal singularities, Geom. Topol. 21 (2017) 1231 MR3626601
87	D Nadler, A combinatorial calculation of the Landau–Ginzburg model M = ℂ³, W = z₁z₂z₃, Selecta Math. 23 (2017) 519 MR3595901
88	D Nadler, E Zaslow, Constructible sheaves and the Fukaya category, J. Amer. Math. Soc. 22 (2009) 233 MR2449059
89	A Neitzke, Cluster-like coordinates in supersymmetric quantum field theory, Proc. Natl. Acad. Sci. USA 111 (2014) 9717 MR3263304
90	L L Ng, Computable Legendrian invariants, Topology 42 (2003) 55 MR1928645
91	L Ng, Combinatorial knot contact homology and transverse knots, Adv. Math. 227 (2011) 2189 MR2807087
92	L Ng, D Rutherford, Satellites of Legendrian knots and representations of the Chekanov–Eliashberg algebra, Algebr. Geom. Topol. 13 (2013) 3047 MR3116313
93	L Ng, D Rutherford, V Shende, S Sivek, E Zaslow, Augmentations are sheaves, Geom. Topol. 24 (2020) 2149 MR4194293
94	V Ovsienko, R E Schwartz, S Tabachnikov, Liouville–Arnold integrability of the pentagram map on closed polygons, Duke Math. J. 162 (2013) 2149 MR3102478
95	F Palesi, Introduction to positive representations and Fock–Goncharov coordinates, preprint (2015)
96	Y Pan, Augmentations and exact Lagrangian cobordisms, PhD thesis, Duke University (2017) MR3664848
97	Y Pan, Exact Lagrangian fillings of Legendrian (2,n) torus links, Pacific J. Math. 289 (2017) 417 MR3667178
98	L Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991) 198 MR1097259
99	V V Prasolov, A B Sossinsky, Knots, links, braids and 3–manifolds : an introduction to the new invariants in low-dimensional topology, 154, Amer. Math. Soc. (1997) MR1414898
100	K Reidemeister, Elementare Begründung der Knotentheorie, Abh. Math. Sem. Univ. Hamburg 5 (1927) 24 MR3069462
101	D Rolfsen, Knots and links, 7, Publish or Perish (1976) MR0515288
102	D Roseman, Reidemeister-type moves for surfaces in four-dimensional space, from: "Knot theory" (editors V F R Jones, J Kania-Bartoszyńska, J H Przytycki, P Traczyk, V G Turaev), Banach Center Publ. 42, Polish Acad. Sci. Inst. Math. (1998) 347 MR1634466
103	R Rouquier, Categorification of 𝔰𝔩₂ and braid groups, from: "Trends in representation theory of algebras and related topics" (editors J A de la Peña, R Bautista), Contemp. Math. 406, Amer. Math. Soc. (2006) 137 MR2258045
104	H Ruddat, N Sibilla, D Treumann, E Zaslow, Skeleta of affine hypersurfaces, Geom. Topol. 18 (2014) 1343 MR3228454
105	D Rutherford, M Sullivan, Cellular Legendrian contact homology for surfaces, II, Internat. J. Math. 30 (2019) MR3977284
106	D Rutherford, M Sullivan, Cellular Legendrian contact homology for surfaces, III, Internat. J. Math. 30 (2019) MR3977285
107	J M Sabloff, M G Sullivan, Families of Legendrian submanifolds via generating families, Quantum Topol. 7 (2016) 639 MR3593565
108	H Schubert, Knoten und Vollringe, Acta Math. 90 (1953) 131 MR72482
109	V Shende, The conormal torus is a complete knot invariant, Forum Math. Pi 7 (2019) MR4010558
110	V Shende, D Treumann, H Williams, On the combinatorics of exact Lagrangian surfaces, preprint (2016) arXiv:1603.07449
111	V Shende, D Treumann, H Williams, E Zaslow, Cluster varieties from Legendrian knots, Duke Math. J. 168 (2019) 2801 MR4017516
112	V Shende, D Treumann, E Zaslow, Legendrian knots and constructible sheaves, Invent. Math. 207 (2017) 1031 MR3608288
113	S Sivek, A bordered Chekanov–Eliashberg algebra, J. Topol. 4 (2011) 73 MR2783378
114	I Smith, Quiver algebras as Fukaya categories, Geom. Topol. 19 (2015) 2557 MR3416110
115	W Soergel, Kazhdan–Lusztig–Polynome und unzerlegbare Bimoduln über Polynomringen, J. Inst. Math. Jussieu 6 (2007) 501 MR2329762
116	L Starkston, Arboreal singularities in Weinstein skeleta, Selecta Math. 24 (2018) 4105 MR3874691
117	G Tabuada, Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C. R. Math. Acad. Sci. Paris 340 (2005) 15 MR2112034
118	B Toën, The homotopy theory of dg–categories and derived Morita theory, Invent. Math. 167 (2007) 615 MR2276263
119	D Treumann, E Zaslow, Cubic planar graphs and Legendrian surface theory, Adv. Theor. Math. Phys. 22 (2018) 1289 MR3952351
120	M T Trinh, Annular homology of Artin braids, I, preprint (2019)
121	R Vianna, On exotic Lagrangian tori in ℂℙ², Geom. Topol. 18 (2014) 2419 MR3268780
122	A Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math. 6 (1971) 329 MR286137