math.GT daily digest: 4 submissions for 21 September 2026

New submissions

The Crossing Number and Arc Index of Satellite Knots

• Authors: Sam Ketchell
• arXiv: 2609.20900
• Comments: 50 pages, 23 figures
• Subjects: Geometric Topology (math.GT)

By considering arc presentations and grid diagrams of satellite knots, we prove that if K is a satellite knot with companion C and wrapping number w, then its crossing number satisfies . We also prove other similar bounds regarding the crossing number and arc index of satellite knots and links. This is partial progress on Problem 1.2 of the K3 Problem List.

Chern-Simons invariants and volumes of representations in Nil, Sol, and Euclidean geometries

• Authors: Taiju Suzuki
• arXiv: 2609.21266
• Comments: 25 pages
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

In this paper, we realize volumes of representations as real-valued Chern-Simons invariants in Nil, Sol, and Euclidean geometries. To this end, we formulate a Chern-Simons invariant of a pair of connections on a principal bundle that need not be trivial. For a connected closed oriented 3-manifold $M$ and a representation into the identity component $G$ of the isometry group of one of these geometries, we construct an auxiliary connection on the associated flat $G$-bundle. We show that, for a suitably normalized invariant polynomial, the Chern-Simons invariant of the auxiliary and flat connections equals the volume of the representation. For the holonomy representation of a geometric structure, this invariant recovers the Riemannian volume. We also compute the Chern-Simons invariant of the Levi-Civita connection for representative closed manifolds in each of these geometries.

Finite covers of a product of surfaces with bounded rank and arbitrarily large systole

• Authors: Koji Fujiwara
• arXiv: 2609.21952
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We construct finite covers of a fixed product of two closed hyperbolic surfaces of genus two whose fundamental groups are generated by at most fifteen elements and whose injectivity radii tend to infinity. The construction uses fibre products over finite groups.

These covers are closed aspherical four-manifolds with universal cover .

They give counterexamples to a conjecture of Avramidi and Delzant for symmetric spaces of higher rank, in the case \(\mathbb H^2\times\mathbb H^2\).

Cross submissions

The disjoint disks property for Busemann $G$-spaces

• Authors: Tadashi Fujioka; Shijie Gu
• arXiv: 2609.21357
• Cross-list: math.MG
• Comments: 17 pages, no figures
• Subjects: Metric Geometry (math.MG); Differential Geometry (math.DG); Geometric Topology (math.GT)

We prove that every finite-dimensional Busemann \(G\)-space of dimension at least five has the disjoint disks property (DDP). For a sufficiently small metric sphere \(L=S(c,r)\), we show that every embedded arc contained in an exact distance level is a homotopical \(Z_2\)-set in \(L\). It follows that \(L\) has the disjoint arc-disk property and the disjoint homotopies property. Daverman's product theorem then gives DDP for \(L\times\mathbb R\), and a local avoidance argument at the center yields DDP for the ambient \(G\)-space. Since finite-dimensional Busemann \(G\)-spaces are generalized manifolds, in dimensions at least five the remaining obstruction to the Busemann conjecture is the resolution problem.