ℤ-Torus Exteriors and Small Knot-Surgery Four-Manifolds
Abstract
We study a cut-and-paste operation in which torus neighborhoods in the author's symplectic building blocks YK and XK, associated to a genus-one fibered knot K, are replaced by marked exteriors of tori in S4 with infinite cyclic complement group and two null peripheral slopes. We also recall the exact Luttinger-surgery realization of MK× S1 from Σg× T2, keeping the knot-surgery/fiber-sum and Luttinger-surgery viewpoints in the same framework. For the trefoil block YK, two marked replacements give a simply connected manifold with intersection form H, hence a manifold homeomorphic to S2× S2. A one-exterior gluing gives a smooth homotopy $4$-sphere. For the rank-six construction we use the identity double of two copies of YK∖νΣ2, rather than the involutive gluing defining the original XK. Two marked replacements along the surviving rim tori give a simply connected manifold with e=8 and σ=0. An explicit geometric basis has intersection form 3H, so the resulting manifold is homeomorphic to #3(S2× S2). The Case II small-perturbation Seiberg--Witten invariant vanishes. The Seiberg--Witten invariant of the identity-glued rank-six family also vanishes; in particular, these manifolds are nonsymplectic.
Related
- Nonabelian Torus Exteriors in 𝕊4 and Torus Replacement in Four-Manifolds
- A counterexample to a conjecture of Plotnick on knot exteriors
- Essential-Surface Complexes of Knot Exteriors: Image, Kernel, and Reconstruction
- Singular Loci of SL2(C)-Representation Varieties of Torus Knot Groups
- Brieskorn spheres bound orbit spaces of torus manifolds
BibTeX
@misc{akhmedov2026ℤtorus,
title = {{$\mathbbZ$-Torus Exteriors and Small Knot-Surgery Four-Manifolds}},
author = {Anar Akhmedov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17603},
eprint = {2609.17603},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17603},
note = {CC0},
}
RIS
TY - UNPB
TI - ℤ-Torus Exteriors and Small Knot-Surgery Four-Manifolds
AU - Anar Akhmedov
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.17603
UR - https://arxiv.org/abs/2609.17603
AB - We study a cut-and-paste operation in which torus neighborhoods in the author's symplectic building blocks Y_K and X_K, associated to a genus-one fibered knot K, are replaced by marked exteriors of tori in S^4 with infinite cyclic complement group and two null peripheral slopes. We also recall the exact Luttinger-surgery realization of M_K× S^1 from Σ_g× T^2, keeping the knot-surgery/fiber-sum and Luttinger-surgery viewpoints in the same framework. For the trefoil block Y_K, two marked replacements give a simply connected manifold with intersection form H, hence a manifold homeomorphic to S^2× S^2. A one-exterior gluing gives a smooth homotopy $4$-sphere. For the rank-six construction we use the identity double of two copies of Y_K∖νΣ_2, rather than the involutive gluing defining the original X_K. Two marked replacements along the surviving rim tori give a simply connected manifold with e=8 and σ=0. An explicit geometric basis has intersection form 3H, so the resulting manifold is homeomorphic to #_3(S^2× S^2). The Case II small-perturbation Seiberg--Witten invariant vanishes. The Seiberg--Witten invariant of the identity-glued rank-six family also vanishes; in particular, these manifolds are nonsymplectic.
ER -