Braided Surfaces and Small Exotic 4-Manifolds
Abstract
Let K be the trefoil or figure-eight knot and let MK denote zero surgery on K. We prove that the fiberwise elliptic involution gives a two-fold branched cover MK× S1⟶ T2× S2 whose branch locus is a section together with a connected braided three-section. The corresponding braid words are computed explicitly for both knots and make visible the relation with Luttinger surgery along product Lagrangian tori. More generally, for a genus-g fibered knot with hyperelliptic monodromy, the same construction gives a braided (2g+2)-section; this includes every genus-two fibered knot. We also state the corresponding surface-bundle version over T2. We then track the induced peripheral data and gluing diffeomorphisms in the constructions of YK and its twisted double XK, as studied by the author in \cite[Section~4]{Akh07}, and through the constructions of the small exotic manifolds homeomorphic to CP2#kCPb2, k=3,5 \cite{AkhmedovSmall,ABP,AP2008}. In the symmetric construction of a k=5 example \cite[Section~3.3]{ABP}, each summand admits a double cover branched along a surface, although the marked complements used in the twisted double need not be invariant. Finally, we prove boundary-action, relative-homology, and local-smoothing obstructions to extending these local involutions across the gluing regions to a branched cover of the closed manifold.
Related
- Exotic 4–manifolds with signature zero
- Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds
- Stably exotic knotted surfaces up to equivalence
- Exotic diffeomorphisms on closed aspherical $4$-manifolds
- Braided Multisections and Symplectic Four-Manifolds with the Rational Cohomology of S2× S2
BibTeX
@misc{akhmedov2026braided,
title = {{Braided Surfaces and Small Exotic 4-Manifolds}},
author = {Anar Akhmedov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00409},
eprint = {2610.00409},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00409},
note = {CC0},
}
RIS
TY - UNPB
TI - Braided Surfaces and Small Exotic 4-Manifolds
AU - Anar Akhmedov
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00409
UR - https://arxiv.org/abs/2610.00409
AB - Let K be the trefoil or figure-eight knot and let M_K denote zero surgery on K. We prove that the fiberwise elliptic involution gives a two-fold branched cover M_K× S^1⟶ T^2× S^2 whose branch locus is a section together with a connected braided three-section. The corresponding braid words are computed explicitly for both knots and make visible the relation with Luttinger surgery along product Lagrangian tori. More generally, for a genus-g fibered knot with hyperelliptic monodromy, the same construction gives a braided (2g+2)-section; this includes every genus-two fibered knot. We also state the corresponding surface-bundle version over T^2. We then track the induced peripheral data and gluing diffeomorphisms in the constructions of Y_K and its twisted double X_K, as studied by the author in \cite[Section~4]{Akh07}, and through the constructions of the small exotic manifolds homeomorphic to CP^2#kCPb^2, k=3,5 \cite{AkhmedovSmall,ABP,AP2008}. In the symmetric construction of a k=5 example \cite[Section~3.3]{ABP}, each summand admits a double cover branched along a surface, although the marked complements used in the twisted double need not be invariant. Finally, we prove boundary-action, relative-homology, and local-smoothing obstructions to extending these local involutions across the gluing regions to a branched cover of the closed manifold.
ER -