Knotted tori in the four-sphere with diffeomorphic exteriors

Nifa, Anass

doi:10.5281/zenodo.22963510

Preprint · v1

Knotted tori in the four-sphere with diffeomorphic exteriors

Anass Nifa6 September 2026

Read free PDF Read on Zenodo Zenodo record

Free PDF · 22 pages · 0.71 MB

Abstract

For every positive integer $R$ , there are at least $R$ smooth tori in the standard four-sphere with orientation-preservingly diffeomorphic compact exteriors, no two of which are carried to one another by an ambient homeomorphism. We construct these exteriors by circle surgery on graphs of powers of the trefoil longitude. A circle action identifies the required fillings with the standard four-sphere, for both normal-framing classes. The quotient of the exterior group by its center is a hyperbolic triangle group; rotation numbers distinguish the meridian classes of the fillings. Thus the number of topological embedding types represented by a smooth torus exterior has no uniform finite bound. The same examples admit exterior diffeomorphisms inducing every integral shear in the product-circle direction; the resulting diffeomorphisms of the filled manifolds restrict to the identity on the inserted torus neighborhoods. For the same exteriors, we compute the commutator length of every nonzero power of the corresponding central element. This gives the exact singular spanning genus of the specified surgery direction. The conclusion concerns arbitrarily large finite families, not an infinite family with one fixed exterior.

MSC 2020

57K45Primary: Higher-dimensional knots and links
57R65Primary: Surgery and handlebodies
20F67Secondary: Hyperbolic groups and nonpositively curved groups
57K31Secondary: Invariants of 3-manifolds (including skein modules, character varieties)

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22963510
All versions: 10.5281/zenodo.22963509
Subjects: knotted torus; smooth exterior; torus surgery; circle action; peripheral structure; commutator length
Language: English

Cite this paper

Anass Nifa. Knotted tori in the four-sphere with diffeomorphic exteriors. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.22963510

Download BibTeX Download RIS

References

As listed in the manuscript bibliography.

[1]
Anar Akhmedov and Azer Akhmedov, Nonabelian torus exteriors in S⁴ and torus replacement in four-manifolds, preprint, arXiv:2609.25080, version 1, 2026.
arXiv
[2]
A. F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics 91, Springer-Verlag, New York, 1983.
[3]
E. A. Bertram, Permutations as products of conjugate infinite cycles, Pacific J. Math. 39 (1971), no. 2, 275–284.
[4]
E. A. Bertram, Even permutations as a product of two conjugate cycles, J. Combinatorial Theory Ser. A 12 (1972), no. 3, 368–380.
[5]
D. Calegari, scl, MSJ Memoirs 20, Mathematical Society of Japan, Tokyo, 2009.
[6]
C. Darolles, Un invariant d'isotopie lisse pour des surfaces dans une variété de dimension 4, C. R. Math. Acad. Sci. Paris 335 (2002), no. 10, 811–815.
[7]
D. Eisenbud, U. Hirsch and W. D. Neumann, Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helv. 56 (1981), no. 4, 638–660.
[8]
R. Fintushel, Classification of circle actions on 4-manifolds, Trans. Amer. Math. Soc. 242 (1978), 377–390.
[9]
M. Fukuda and M. Ishikawa, Distinguishing 2-knots admitting circle actions by fundamental groups, Rev. Mat. Complut. 38 (2025), 307–317.
[10]
C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371–415.
[11]
J. A. Hillman, The groups of branched twist-spun knots, preprint, arXiv:2305.00443v4, revised 9 August 2026.
arXiv v4
[12]
M. W. Hirsch, Differential topology, Graduate Texts in Mathematics 33, Springer-Verlag, New York, 1976.
[13]
Z. Iwase, Dehn-surgery along a torus T²-knot, Pacific J. Math. 133 (1988), no. 2, 289–299.
[14]
I. Kapovich and N. Benakli, Boundaries of hyperbolic groups, in Combinatorial and geometric group theory, Contemp. Math. 296, Amer. Math. Soc., Providence, RI, 2002, 39–93.
[15]
H. Konno, A. Mallick and M. Taniguchi, Exotically knotted closed surfaces from Donaldson's diagonalization for families, preprint, arXiv:2409.07287, 2024.
arXiv
[16]
J. Kuhrman, More exotic RP²-knots and homotopy spheres, preprint, arXiv:2507.03798, 2025.
arXiv
[17]
K. Larson, Surgery on tori in the 4-sphere, Math. Proc. Cambridge Philos. Soc. 164 (2018), no. 1, 109–124.
[18]
Y. Liu, Y. Ni, H. Sun and S. Wang, On slope genera of knotted tori in 4-space, Pacific J. Math. 261 (2013), no. 1, 117–144.
[19]
J. Meier and A. Zupan, Knots bounding nonisotopic ribbon disks, J. Topol. 18 (2025), e70047.
[20]
E. Militon, Commutator length of annulus diffeomorphisms, Ergodic Theory Dynam. Systems 34 (2014), no. 3, 919–937.
[21]
J. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3, American Mathematical Society, Providence, RI, and Clay Mathematics Institute, Cambridge, MA, 2007.
[22]
L. Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), no. 3, 737–745.
[23]
P. S. Pao, Non-linear circle actions on the 4-sphere and twisting spun knots, Topology 17 (1978), no. 3, 291–296.
[24]
I. Rivin, Geodesics with one self-intersection, and other stories, Adv. Math. 231 (2012), no. 5, 2391–2412.
[25]
J. W. Wood, Bundles with totally disconnected structure group, Comment. Math. Helv. 46 (1971), 257–273.

[reference-1] [1]
Anar Akhmedov and Azer Akhmedov, Nonabelian torus exteriors in S⁴ and torus replacement in four-manifolds, preprint, arXiv:2609.25080, version 1, 2026.
arXiv

[reference-2] [2]
A. F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics 91, Springer-Verlag, New York, 1983.

[reference-3] [3]
E. A. Bertram, Permutations as products of conjugate infinite cycles, Pacific J. Math. 39 (1971), no. 2, 275–284.

[reference-4] [4]
E. A. Bertram, Even permutations as a product of two conjugate cycles, J. Combinatorial Theory Ser. A 12 (1972), no. 3, 368–380.

[reference-5] [5]
D. Calegari, scl, MSJ Memoirs 20, Mathematical Society of Japan, Tokyo, 2009.

[reference-6] [6]
C. Darolles, Un invariant d'isotopie lisse pour des surfaces dans une variété de dimension 4, C. R. Math. Acad. Sci. Paris 335 (2002), no. 10, 811–815.

[reference-7] [7]
D. Eisenbud, U. Hirsch and W. D. Neumann, Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helv. 56 (1981), no. 4, 638–660.

[reference-8] [8]
R. Fintushel, Classification of circle actions on 4-manifolds, Trans. Amer. Math. Soc. 242 (1978), 377–390.

[reference-9] [9]
M. Fukuda and M. Ishikawa, Distinguishing 2-knots admitting circle actions by fundamental groups, Rev. Mat. Complut. 38 (2025), 307–317.

[reference-10] [10]
C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371–415.

[reference-11] [11]
J. A. Hillman, The groups of branched twist-spun knots, preprint, arXiv:2305.00443v4, revised 9 August 2026.
arXiv v4

[reference-12] [12]
M. W. Hirsch, Differential topology, Graduate Texts in Mathematics 33, Springer-Verlag, New York, 1976.

[reference-13] [13]
Z. Iwase, Dehn-surgery along a torus T²-knot, Pacific J. Math. 133 (1988), no. 2, 289–299.

[reference-14] [14]
I. Kapovich and N. Benakli, Boundaries of hyperbolic groups, in Combinatorial and geometric group theory, Contemp. Math. 296, Amer. Math. Soc., Providence, RI, 2002, 39–93.

[reference-15] [15]
H. Konno, A. Mallick and M. Taniguchi, Exotically knotted closed surfaces from Donaldson's diagonalization for families, preprint, arXiv:2409.07287, 2024.
arXiv

[reference-16] [16]
J. Kuhrman, More exotic RP²-knots and homotopy spheres, preprint, arXiv:2507.03798, 2025.
arXiv

[reference-17] [17]
K. Larson, Surgery on tori in the 4-sphere, Math. Proc. Cambridge Philos. Soc. 164 (2018), no. 1, 109–124.

[reference-18] [18]
Y. Liu, Y. Ni, H. Sun and S. Wang, On slope genera of knotted tori in 4-space, Pacific J. Math. 261 (2013), no. 1, 117–144.

[reference-19] [19]
J. Meier and A. Zupan, Knots bounding nonisotopic ribbon disks, J. Topol. 18 (2025), e70047.

[reference-20] [20]
E. Militon, Commutator length of annulus diffeomorphisms, Ergodic Theory Dynam. Systems 34 (2014), no. 3, 919–937.

[reference-21] [21]
J. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3, American Mathematical Society, Providence, RI, and Clay Mathematics Institute, Cambridge, MA, 2007.

[reference-22] [22]
L. Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), no. 3, 737–745.

[reference-23] [23]
P. S. Pao, Non-linear circle actions on the 4-sphere and twisting spun knots, Topology 17 (1978), no. 3, 291–296.

[reference-24] [24]
I. Rivin, Geodesics with one self-intersection, and other stories, Adv. Math. 231 (2012), no. 5, 2391–2412.

[reference-25] [25]
J. W. Wood, Bundles with totally disconnected structure group, Comment. Math. Helv. 46 (1971), 257–273.