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Shadows of $2$–knots and complexity

Hironobu Naoe

Algebraic & Geometric Topology 24 (2024) 4651–4695
Bibliography
1 E Artin, Zur Isotopie zweidimensionaler Flächen im R4, Abh. Math. Sem. Univ. Hamburg 4 (1925) 174 MR3069446
2 R Blair, M Campisi, S A Taylor, M Tomova, Kirby–Thompson distance for trisections of knotted surfaces, J. Lond. Math. Soc. 105 (2022) 765 MR4400936
3 J S Carter, J H Rieger, M Saito, A combinatorial description of knotted surfaces and their isotopies, Adv. Math. 127 (1997) 1 MR1445361
4 F Costantino, Shadows and branched shadows of 3 and 4–manifolds, PhD thesis, Scuola Normale Superiore (2004)
5 F Costantino, Complexity of 4–manifolds, Exp. Math. 15 (2006) 237 MR2253547
6 F Costantino, Stein domains and branched shadows of 4–manifolds, Geom. Dedicata 121 (2006) 89 MR2276237
7 F Costantino, Branched shadows and complex structures on 4–manifolds, J. Knot Theory Ramifications 17 (2008) 1429 MR2469211
8 F Costantino, D Thurston, 3–Manifolds efficiently bound 4–manifolds, J. Topol. 1 (2008) 703 MR2417451
9 M H Freedman, F Quinn, Topology of 4–manifolds, 39, Princeton Univ. Press (1990) MR1201584
10 D Gay, R Kirby, Trisecting 4–manifolds, Geom. Topol. 20 (2016) 3097 MR3590351
11 R E Gompf, M Scharlemann, A Thompson, Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures, Geom. Topol. 14 (2010) 2305 MR2740649
12 M C Hughes, S Kim, M Miller, Isotopies of surfaces in 4–manifolds via banded unlink diagrams, Geom. Topol. 24 (2020) 1519 MR4157558
13 M Ishikawa, Y Koda, Stable maps and branched shadows of 3–manifolds, Math. Ann. 367 (2017) 1819 MR3623239
14 M Ishikawa, H Naoe, Milnor fibration, A’Campo’s divide and Turaev’s shadow, from: "Singularities" (editors M Ishikawa, S Yokura), World Sci. (2020) 95 MR4450732
15 M Jabłonowski, On a banded link presentation of knotted surfaces, J. Knot Theory Ramifications 25 (2016) 1640004 MR3475071
16 A Kawauchi, Ribbonness of a stable-ribbon surface-link, I: A stably trivial surface-link, Topology Appl. 301 (2021) 107522 MR4312972
17 A Kawauchi, Uniqueness of an orthogonal 2–handle pair on a surface-link, Contemp. Math. 4 (2023) 182
18 A Kawauchi, T Shibuya, S Suzuki, Descriptions on surfaces in four-space, I: Normal forms, Math. Sem. Notes Kobe Univ. 10 (1982) 75 MR672939
19 C Kearton, V Kurlin, All 2–dimensional links in 4–space live inside a universal 3–dimensional polyhedron, Algebr. Geom. Topol. 8 (2008) 1223 MR2443242
20 Y Koda, H Naoe, Shadows of acyclic 4–manifolds with sphere boundary, Algebr. Geom. Topol. 20 (2020) 3707 MR4194291
21 Y Koda, B Martelli, H Naoe, Four-manifolds with shadow-complexity one, Ann. Fac. Sci. Toulouse Math. 31 (2022) 1111 MR4506139
22 F Laudenbach, V Poénaru, A note on 4–dimensional handlebodies, Bull. Soc. Math. France 100 (1972) 337 MR317343
23 S J Lomonaco Jr., The homotopy groups of knots, I : How to compute the algebraic 2–type, Pacific J. Math. 95 (1981) 349 MR632192
24 B Martelli, Links, two-handles, and four-manifolds, Int. Math. Res. Not. 2005 (2005) 3595 MR2200121
25 B Martelli, Four-manifolds with shadow-complexity zero, Int. Math. Res. Not. 2011 (2011) 1268 MR2806506
26 S Matveev, Algorithmic topology and classification of 3–manifolds, 9, Springer (2003) MR1997069
27 J Meier, A Zupan, Bridge trisections of knotted surfaces in S4, Trans. Amer. Math. Soc. 369 (2017) 7343 MR3683111
28 J Meier, A Zupan, Bridge trisections of knotted surfaces in 4–manifolds, Proc. Natl. Acad. Sci. USA 115 (2018) 10880 MR3871791
29 H Naoe, Shadows of 4–manifolds with complexity zero and polyhedral collapsing, Proc. Amer. Math. Soc. 145 (2017) 4561 MR3690638
30 D Roseman, Reidemeister-type moves for surfaces in four-dimensional space, from: "Knot theory", Banach Center Publ. 42, Polish Acad. Sci. Inst. Math. (1998) 347 MR1634466
31 M Saito, S Satoh, The spun trefoil needs four broken sheets, J. Knot Theory Ramifications 14 (2005) 853 MR2187601
32 S Satoh, On non-orientable surfaces in 4–space which are projected with at most one triple point, Proc. Amer. Math. Soc. 128 (2000) 2789 MR1652241
33 S Satoh, Triviality of a 2–knot with one or two sheets, Kyushu J. Math. 63 (2009) 239 MR2568770
34 S Satoh, A Shima, The 2–twist-spun trefoil has the triple point number four, Trans. Amer. Math. Soc. 356 (2004) 1007 MR1984465
35 F J Swenton, On a calculus for 2–knots and surfaces in 4–space, J. Knot Theory Ramifications 10 (2001) 1133 MR1871221
36 V G Turaev, Quantum invariants of knots and 3–manifolds, 18, de Gruyter (1994) MR1292673
37 T Yajima, On simply knotted spheres in R4, Osaka Math. J. 1 (1964) 133 MR172280
38 K Yoshikawa, An enumeration of surfaces in four-space, Osaka J. Math. 31 (1994) 497 MR1309400
39 E C Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965) 471 MR195085