Algebraic & Geometric Topology Volume 22, issue 8 (2022)

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ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

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$4$–Manifold invariants from Hopf algebras

Julian Chaidez, Jordan Cotler and Shawn X Cui

Algebraic & Geometric Topology 22 (2022) 3747–3807

DOI: 10.2140/agt.2022.22.3747

Bibliography

1	A Abrams, D T Gay, R Kirby, Group trisections and smooth 4–manifolds, Geom. Topol. 22 (2018) 1537 MR3780440
2	B Bakalov, A Kirillov Jr., Lectures on tensor categories and modular functors, 21, Amer. Math. Soc. (2001) MR1797619
3	M Bärenz, J Barrett, Dichromatic state sum models for four-manifolds from pivotal functors, Comm. Math. Phys. 360 (2018) 663 MR3800795
4	J W Barrett, B W Westbury, Invariants of piecewise-linear 3–manifolds, Trans. Amer. Math. Soc. 348 (1996) 3997 MR1357878
5	S Burciu, On the irreducible representations of generalized quantum doubles, preprint (2012) arXiv:1202.4315
6	N A Castro, D T Gay, J Pinzón-Caicedo, Diagrams for relative trisections, Pacific J. Math. 294 (2018) 275 MR3770114
7	N A Castro, B Ozbagci, Trisections of 4–manifolds via Lefschetz fibrations, Math. Res. Lett. 26 (2019) 383 MR3999550
8	J Chaidez, J Cotler, S Cui, Trisection invariant python script (2019)
9	L Chang, S X Cui, On two invariants of three manifolds from Hopf algebras, Adv. Math. 351 (2019) 621 MR3954041
10	L Crane, L H Kauffman, D N Yetter, State-sum invariants of 4–manifolds, J. Knot Theory Ramifications 6 (1997) 177 MR1452438
11	L Crane, D Yetter, A categorical construction of 4D topological quantum field theories, from: "Quantum topology" (editors L H Kauffman, R A Baadhio), Ser. Knots Everything 3, World Sci. (1993) 120 MR1273569
12	X Cui, Higher categories and topological quantum field theories, PhD thesis, University of California, Santa Barbara (2016)
13	R Dijkgraaf, E Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990) 393 MR1048699
14	C L Douglas, D J Reutter, Fusion 2–categories and a state-sum invariant for 4–manifolds, preprint (2018) arXiv:1812.11933
15	D Gay, R Kirby, Trisecting 4–manifolds, Geom. Topol. 20 (2016) 3097 MR3590351
16	J Hass, P Scott, Intersections of curves on surfaces, Israel J. Math. 51 (1985) 90 MR804478
17	M Hennings, Invariants of links and 3–manifolds obtained from Hopf algebras, J. London Math. Soc. 54 (1996) 594 MR1413901
18	V F R Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985) 103
19	R Kashaev, A simple model of 4d–TQFT, from: "2016 MATRIX annals" (editors D R Wood, J de Gier, C E Praeger, T Tao), MATRIX Book Ser. 1, Springer (2018) 277 MR3792526
20	R Kashaev, A Virelizier, Generalized Kuperberg invariants of 3–manifolds, Algebr. Geom. Topol. 19 (2019) 2575 MR4023323
21	L H Kauffman, D E Radford, Invariants of 3–manifolds derived from finite-dimensional Hopf algebras, J. Knot Theory Ramifications 4 (1995) 131 MR1321293
22	G Kuperberg, Noninvolutory Hopf algebras and 3–manifold invariants, Duke Math. J. 84 (1996) 83 MR1394749
23	P Lambert-Cole, Trisections, intersection forms and the Torelli group, Algebr. Geom. Topol. 20 (2020) 1015 MR4092318
24	P Lambert-Cole, J Meier, Bridge trisections in rational surfaces, J. Topol. Anal. 14 (2022) 655 MR4493476
25	R G Larson, D E Radford, Semisimple cosemisimple Hopf algebras, Amer. J. Math. 110 (1988) 187 MR926744
26	W B R Lickorish, The skein method for three-manifold invariants, J. Knot Theory Ramifications 2 (1993) 171 MR1227009
27	D López Neumann, Kuperberg invariants for balanced sutured 3–manifolds, Canad. J. Math. 73 (2021) 1698 MR4350558
28	S Majid, Foundations of quantum group theory, Cambridge University Press (1995) MR1381692
29	J M Paterson, A combinatorial algorithm for immersed loops in surfaces, Topology Appl. 123 (2002) 205 MR1918498
30	J Petit, The dichromatic invariants of smooth 4–manifolds, Glob. J. Pure Appl. Math. 4 (2008)
31	D E Radford, Hopf algebras, 49, World Sci. (2012) MR2894855
32	J Roberts, Skein theory and Turaev–Viro invariants, Topology 34 (1995) 771 MR1362787
33	V G Turaev, Quantum invariants of knots and 3–manifolds, 18, de Gruyter (1994) MR1292673
34	V Turaev, Introduction to combinatorial torsions, Birkhäuser (2001) MR1809561
35	V G Turaev, O Y Viro, State sum invariants of 3–manifolds and quantum 6j–symbols, Topology 31 (1992) 865 MR1191386
36	K Walker, Z Wang, (3+1)–TQFTs and topological insulators, Frontiers of Physics 7 (2012) 150
37	Z Wang, Topological quantum computation, 112, Amer. Math. Soc. (2010) MR2640343
38	D J Williamson, Z Wang, Hamiltonian models for topological phases of matter in three spatial dimensions, Ann. Physics 377 (2017) 311 MR3606141
39	E Witten, Topological quantum field theory, Comm. Math. Phys. 117 (1988) 353 MR953828
40	E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351 MR990772
41	D N Yetter, TQFTs from homotopy 2–types, J. Knot Theory Ramifications 2 (1993) 113 MR1209321