Knotted tori in the four-sphere with diffeomorphic exteriors
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.
Related
- Nonabelian Torus Exteriors in 𝕊4 and Torus Replacement in Four-Manifolds
- ℤ-Torus Exteriors and Small Knot-Surgery Four-Manifolds
- Four triangle-free intrinsically knotted graphs with 22 edges
- THE OPTIMAL EIGENVALUE COUNT IN AUBRY'S SPECTRAL SPHERE THEOREM FOR DIMENSIONS AT LEAST FOUR
- Hyperbolic complex tangencies on a four-sphere
BibTeX
@misc{nifa2026knotted,
title = {{Knotted tori in the four-sphere with diffeomorphic exteriors}},
author = {Anass Nifa},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.5281/zenodo.22963510},
url = {https://hal.science/hal-05769989v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Knotted tori in the four-sphere with diffeomorphic exteriors
AU - Anass Nifa
PY - 2026
DA - 2026/09/06
PB - HAL
DO - 10.5281/zenodo.22963510
UR - https://hal.science/hal-05769989v1
AB - 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.
ER -