Band diagrams of immersed surfaces in 4-manifolds

Hughes, Mark¹ ; Kim, Seungwon² ; Miller, Maggie³

¹Department of Mathematics, Brigham Young University, Provo, UT, United States
²Sungkyunkwan University, Suwon, South Korea
³Department of Mathematics, The University of Texas at Austin, Austin, TX, United States

Algebraic and Geometric Topology, Tome 25 (2025) no. 3, pp. 1731-1791

Résumé

We study immersed surfaces in smooth 4-manifolds via singular banded unlink diagrams. Such a diagram consists of a singular link with bands inside a Kirby diagram of the ambient 4-manifold, representing a level set of the surface with respect to an associated Morse function. We show that every self-transverse immersed surface in a smooth, orientable, closed 4-manifold can be represented by a singular banded unlink diagram, and that such representations are uniquely determined by the ambient isotopy or equivalence class of the surface up to a set of singular band moves which we define explicitly. By introducing additional finger, Whitney and cusp diagrammatic moves, we can use these singular band moves to describe homotopies or regular homotopies as well.

Using these techniques, we introduce bridge trisections of immersed surfaces in arbitrary trisected 4-manifolds and prove that such bridge trisections exist and are unique up to simple perturbation moves. We additionally give some examples of how singular banded unlink diagrams may be used to perform computations or produce explicit homotopies of surfaces.

Détail

DOI : 10.2140/agt.2025.25.1731

Keywords: $4$-manifold, surface, band, diagram, knot, trisection

Bibliographie

[1] V I Arnold, A N Varchenko, S M Guseĭn-Zade, Особенности дифференцируемых отображении, Nauka (1982) 304

[2] R İ Baykur, N Sunukjian, Knotted surfaces in 4-manifolds and stabilizations, J. Topol. 9 (2016) 215 | DOI

[3] J S Carter, M Saito, Reidemeister moves for surface isotopies and their interpretation as moves to movies, J. Knot Theory Ramifications 2 (1993) 251 | DOI

[4] J S Carter, M Saito, Knotted surfaces and their diagrams, 55, Amer. Math. Soc. (1998) | DOI

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

[6] M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357

[7] M H Freedman, F Quinn, Topology of 4-manifolds, 39, Princeton Univ. Press (1990)

[8] D Gay, R Kirby, Trisecting 4-manifolds, Geom. Topol. 20 (2016) 3097 | DOI

[9] A Hatcher, J Wagoner, Pseudo-isotopies of compact manifolds, 6, Soc. Math. France (1973)

[10] C Hayashi, K Shimokawa, Heegaard splittings of trivial arcs in compression bodies, J. Knot Theory Ramifications 10 (2001) 71 | DOI

[11] M W Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959) 242 | DOI

[12] M W Hirsch, Differential topology, 33, Springer (1976) | DOI

[13] F Hosokawa, A Kawauchi, Proposals for unknotted surfaces in four-spaces, Osaka Math. J. 16 (1979) 233

[14] M C Hughes, S Kim, M Miller, Isotopies of surfaces in 4-manifolds via banded unlink diagrams, Geom. Topol. 24 (2020) 1519 | DOI

[15] M Jabłonowski, Minimal generating sets of moves for surfaces immersed in the four-space, J. Knot Theory Ramifications 32 (2023) 2350071 | DOI

[16] J M Joseph, M R Klug, B M Ruppik, H R Schwartz, Unknotting numbers of 2-spheres in the 4-sphere, J. Topol. 14 (2021) 1321 | DOI

[17] S Kamada, 2-dimensional braids and chart descriptions, from: "Topics in knot theory", NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 399, Kluwer (1993) 277 | DOI

[18] S Kamada, Unknotting immersed surface-links and singular 2-dimensional braids by 1-handle surgeries, Osaka J. Math. 36 (1999) 33

[19] S Kamada, Braid and knot theory in dimension four, 95, Amer. Math. Soc. (2002) | DOI

[20] S Kamada, A Kawauchi, J Kim, S Y Lee, Presentation of immersed surface-links by marked graph diagrams, J. Knot Theory Ramifications 27 (2018) 1850052 | DOI

[21] A Kawauchi, T Shibuya, S Suzuki, Descriptions on surfaces in four-space, I : Normal forms, Math. Sem. Notes Kobe Univ. 10 (1982) 75

[22] C Kearton, V Kurlin, All 2-dimensional links in 4-space live inside a universal 3-dimensional polyhedron, Algebr. Geom. Topol. 8 (2008) 1223 | DOI

[23] F Laudenbach, Sur les 2-sphères d’une variété de dimension 3, Ann. of Math. 97 (1973) 57 | DOI

[24] F Laudenbach, V Poénaru, A note on 4-dimensional handlebodies, Bull. Soc. Math. France 100 (1972) 337

[25] J Meier, T Schirmer, A Zupan, Classification of trisections and the generalized property R conjecture, Proc. Amer. Math. Soc. 144 (2016) 4983 | DOI

[26] J Meier, A Zupan, Bridge trisections of knotted surfaces in S4, Trans. Amer. Math. Soc. 369 (2017) 7343 | DOI

[27] J Meier, A Zupan, Bridge trisections of knotted surfaces in 4-manifolds, Proc. Natl. Acad. Sci. USA 115 (2018) 10880 | DOI

[28] M Miller, P Naylor, Trisections of nonorientable 4-manifolds, Michigan Math. J. 74 (2024) 403 | DOI

[29] 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

[30] R Schneiderman, P Teichner, The group of disjoint 2-spheres in 4-space, Ann. of Math. 190 (2019) 669 | DOI

[31] S Smale, A classification of immersions of the two-sphere, Trans. Amer. Math. Soc. 90 (1958) 281 | DOI

[32] F J Swenton, On a calculus for 2-knots and surfaces in 4-space, J. Knot Theory Ramifications 10 (2001) 1133 | DOI