Purely cosmetic surgeries on knots with low-span Jones polynomials

We give a new proof of Laudenbach and Poénaru's theorem, which states that any diffeomorphism of the boundary of a 4-dimensional 1-handlebody extends to the whole handlebody. Our proof is based on the classification of Heegaard splittings of double handlebodies and a result of Cerf on diffeomorphisms of the 3-ball. Further, we extend this theorem to 4-dimensional compression bodies, namely cobordisms between 3-manifolds constructed using only 1-handles: when the negative boundary is a product of a compact surface and an interval, we show that every diffeomorphism of the positive boundary extends to the whole compression body. This involves a strong Haken theorem for spheres and disks in sutured Heegaard splittings and a classification of sutured Heegaard splittings of double compression bodies. Finally, we show how this applies to the study of relative trisection diagrams for compact 4-manifolds.

math.GT↗

arXiv2026

Pseudo-Anosov action on the $SU(2)$-character variety of $S_2$

In this article, we continue the study of the action of subgroups of the mapping class group on the $SU(2)$-character variety. We prove the existence of a mapping class subgroup on the surface of genus $2$, containing infinitely many pseudo-Anosov elements, which admit an invariant rational function on $SU(2)$-character variety of $S_2$.

math.GT↗

arXiv2026

Rigidity and Galois theory of 3-manifolds branched over Chebotarev links

The classical Neukirch--Uchida theorem states that the absolute Galois group determines a number field up to isomorphism. We prove an analogue of this theorem for 3-manifolds in the framework of arithmetic topology. We study infinite links $\mathcal{L}$ in 3-manifolds that satisfy a Chebotarev density property, and hence behave like the set of primes. We define the absolute Galois group of a 3-manifold relative to such a link as the inverse limit of profinite completions of finite sublink complements. Our main result shows that two covers of $S^3$ branched over finite sublinks of $\mathcal{L}$ are isomorphic if and only if their absolute Galois groups relative to $\mathcal{L}$ are isomorphic via a characteristic-preserving isomorphism. The proof translates the key ideas from the number-theoretic argument into 3-manifolds, developing Hilbert ramification theory for infinite covers and local-global principles. Doing so also provides a systematic justification for viewing Chebotarev links as topological analogues of the set of prime numbers. In addition, we propose further conditions on Chebotarev links which would further strengthen their connection to the prime numbers.

math.GT↗