math.GT daily digest: 6 new submissions for 3 August 2026

New submissions

Hard unknots are often easy from a different perspective

• Authors: Jason Cantarella; Henrik Schumacher; Clayton Shonkwiler
• arXiv: 2607.28772
• Subjects: Geometric Topology (math.GT)
• Comments: 23 pages, 3 figures

Recent attempts to train AI models to recognize knots have produced millions of "hard" unknot diagrams resistant to simplification by Reidemeister moves, pass moves, or random walks on the Reidemeister graph. Most are easy for non-diagrammatic methods such as simplifying triangulations of the knot complement (Regina) or presentations of the knot group (SnapPy). We present ReAPR (Re-embedding And Pass Rerouting), which alternates pass-move reduction with a geometric re-embedding step. The re-embedding minimizes the total variation of a height function on the diagram subject to crossing constraints. We show that for an $n$-crossing diagram, the minimum total variation is , where $k$ is the least number of crossings one must virtualize to make the diagram virtually alternating; this is a combinatorial invariant of the diagram. Reprojecting the resulting embedding from a new viewpoint reveals previously hidden simplifications. ReAPR successfully simplifies every published hard-unknot example we are aware of, as well as several new collections (approximately 2.6 million examples in total) in under 30 seconds of total CPU time. This includes a set of Kauffman's "challenge" unknots, presented as rational tangles, which appear to be surprisingly difficult for both non-diagrammatic methods.

Normalizers of lattices and isometry groups of arithmetic hyperbolic manifolds

• Authors: Mikhail Belolipetsky; Tam Cheetham-West
• arXiv: 2607.28949
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)
• Comments: 12 pages. Comments welcome

We prove that every arithmetic lattice in PSL and every arithmetic lattice of the simplest type in PO$(n,1)$, , is the normalizer of arbitrarily many of its sublattices. Combined with previous work, this result implies that every lattice in PSL has this property. In this way, we prove that the set of profinitely flexible lattices in PSL is either empty or countably infinite. Another result is that every finite group is realized as the full isometry group of an arithmetic hyperbolic $n$-manifold. The proof of this theorem is based on study of normalizers of lattices and subgroup growth theory.

LA-MOKA: A Combinatorial Discretization Algorithm for the Braid Monodromy of Line Arrangements

• Authors: Benoît Guerville-Ballé
• arXiv: 2607.29333
• Subjects: Geometric Topology (math.GT)
• Comments: 13 pages, 3 Figures and 1 Tables

Computing braid monodromy is a key tool for studying the topology of algebraic curves in the complex projective plane . We present an algorithm, named LA-MOKA, to compute this invariant in the specific case of complex line arrangements. To safely manage the problem of floating-point approximations when working over number fields, we translate the continuous geometry into a discrete combinatorial structure. We establish explicit conditions on this discretization that guarantee the topological correctness of the computed braid monodromy.

Cross submissions

On totally symplectically aspherical manifolds

• Cross-list: math.SG
• Authors: Luca F. Di Cerbo; Alexander Dranishnikov; Ekansh Jauhari
• arXiv: 2607.28763
• Subjects: Symplectic Geometry (math.SG); Algebraic Geometry (math.AG); Algebraic Topology (math.AT); Differential Geometry (math.DG); Geometric Topology (math.GT)
• Comments: 21 pages, 1 figure

We construct examples of totally symplectically aspherical Kaehler manifolds with non-trivial second homotopy group in every even dimension greater than or equal to six. In dimension four, we construct examples of totally c-symplectically aspherical near-symplectic manifolds with non-trivial second homotopy group.

Piecewise isometry groups of Euclidean tessellations

• Cross-list: math.GR
• Authors: Robert Bieri; Alex Feingold; Daniel Studenmund
• arXiv: 2607.28893
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

Given a tessellation of Euclidean or hyperbolic space, the piecewise isometry group is the group whose elements are given by cutting space into finitely many tessellated convex subsets and gluing them back together. Groups of piecewise isometries of tessellations generalize Houghton's groups and Thompson's group $V$, and for cubical tessellations were studied by Bieri and Sach. We prove structure results about groups of piecewise isometries of sufficiently nice tessellations of Euclidean space, such as tessellations associated to crystallographic root systems, in particular proving that they are elementary amenable. Future work in progress will prove finite generation and higher finiteness properties.

On the construction of geographical maps: Lagrange, Chebyshev, Darboux and Milnor

• Cross-list: math.DG
• Authors: Hideki Miyachi (MPIM); Ken'Ichi Ohshika (MPIM); Athanase Papadopoulos (IRMA, MPI)
• arXiv: 2607.29263
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT); Metric Geometry (math.MG)

Lagrange, Chebyshev, and Darboux, in 1779, 1856, and 1911, respectively, wrote articles all bearing the same title, \emph{On the Construction of Geographical Maps}. In 1969, Milnor wrote a paper in which he refers to Chebyshev's paper, of which he provides a new formulation and proof. In the present article, we review the results of all these papers, explaining the main ideas they contain and pointing out connections between them. We give complete proofs of the statements by Darboux and Milnor, both of which aim to make explicit and provide a proof of Chebyshev's result, but whose contents are different. Although Chebyshev did not state explicitly what the word ``best'' means, his conclusion, like that of Darboux and of Milnor, is that a best geographical map is characterised by the fact that its conformal factor is constant on the boundary of the region represented. Our statement and proof of Milnor's theorem work in a more general setting than the one he gives. The final version of this paper will appear in the Handbook of Mathematics in the Arts and Sciences (second edition), ed. Bharath Sriraman, Springer, 2027.