math.GT daily digest: 8 new submissions for 21 August 2026

New submissions

Pearl necklace knots with fewer vertices than an equivalent FCC lattice knot

• Authors: Alexander R. Klotz
• arXiv: 2608.19441
• Subjects: Geometric Topology (math.GT)
• Comments: 4 pages, 2 figures, tables in appendix, four ancillary files

The pearl necklace number, NP, of a knot is the smallest number of unit spheres required to construct a knot if each sphere is tangent to two neighbors and no sphere overlaps with another. It has been speculated that the pearl necklace number is equal to the minimum number of lattice sites required to embed a knot on a face centered cubic (FCC) lattice NL, implying that a trefoil knot cannot be constructed from fewer than 15 spheres. A large language model (LLM) was prompted to find configurations of knot for which NP> The pearl necklace number, NP, of a knot is the smallest number of unit spheres required to construct a knot if each sphere is tangent to two neighbors and no sphere overlaps with another. It has been speculated that the pearl necklace number is equal to the minimum number of lattice sites required to embed a knot on a face centered cubic (FCC) lattice NL, implying that a trefoil knot cannot be constructed from fewer than 15 spheres. A large language model (LLM) was prompted to find configurations of knot for which NP<NL, and this manuscript describes its findings, attempts at validating them, and their implications. No 14-vertex example was found for the trefoil knot, but examples with NP=NL−1 were found for all knots from 5 to 7 crossings as well as 819, and 10124. One example, 81, could be constructed with NP=NL−2.

Deciding if a shadow resolves into a given link: linear-time algorithms

• Authors: Andrea Alba, Gelasio Salazar
• arXiv: 2608.19484
• Subjects: Geometric Topology (math.GT)

A {\em shadow} (or {\em projection}) is obtained from a link diagram by ignoring the over/under information at each crossing. Given a fixed link $L$ we investigate the complexity of deciding whether an input shadow $S$ can be {\em resolved} into $L$, that is, whether we can assign over/under information to its crossings to obtain a diagram of a link isotopic to $L$. We show that if then there exists a linear-time algorithm that decides whether an input shadow $S$ resolves into $L$.

The minimum number of Dehn $\mathbb{Z}$-colors of any nonsplittable $\mathbb{Z}$-colorable link is three

• Authors: Eri Matsudo, Kanako Oshiro
• arXiv: 2608.19559
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO)

Our previous papers [8, 9] are the first and second to discuss minimum numbers of ``region'' colors, while minimum numbers of arc colors such as Fox colors are well-studied. As the third installment, in this paper, we investigate the minimum number of Dehn $\mathbb{Z}$-colors. In particular, we show that the minimum number of Dehn $\mathbb{Z}$-colors of a nonsplittable $\mathbb{Z}$-colorable link is three.

Convex cocompact right-angled Gromov-Thurston polyhedra

• Authors: Sami Douba
• arXiv: 2608.19640
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)
• Comments: 9 pages, 1 figure. Comments welcome

Lee and Marquis exhibited convex cocompact hyperbolic reflection groups in dimension 5 whose limit sets are homeomorphic to the 3-sphere, but none of whose finite-index subgroups can be realized as 4-dimensional real hyperbolic lattices. Using different methods, we furnish right-angled examples.

Decompositions of good involutions on quandles

• Authors: Yasuhito Nakajima, Kentaro Yamaguchi
• arXiv: 2608.19951
• Subjects: Geometric Topology (math.GT)
• Comments: 27 pages

We study the behavior of good involutions under two fundamental constructions of quandles: interaction-free unions and direct products. In particular, we show that the set of good involutions on a quandle decomposes naturally into the set of involutions on its maximal trivial component and the set of good involutions on the complement. Moreover, we construct an example of a connected noninvolutory symmetric quandle with multiple good involutions, which serves as a counterexample to a conjecture by Ta.

The entropy spectrum of hyperbolic surfaces

• Authors: Ara Basmajian, Hugo Parlier
• arXiv: 2608.20143
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Dynamical Systems (math.DS)
• Comments: 62 pages, 14 figures

This article introduces and studies the entropy spectrum of a hyperbolic surface, that is the set of entropies of its subsurfaces. The main results are that the entropy spectrum is a reverse well-ordered multiset, with finite multiplicities, and that there is a quantifiable gap around the value $1$. This gap comes from a counting result on the number of non-filling geodesics which in turn comes from explicit estimates on the number of curves on surfaces with boundary in terms of geometric data. The geometric data includes lengths of boundary geodesics and so-called boundary width, which measures maximal distance to the boundary but can also be interpreted in terms of the topology of the surface and the systole.

Systole Increasing Deformations to Maximal Translation Surfaces

• Authors: Achintya Dey, Bidyut Sanki
• arXiv: 2608.20264
• Subjects: Geometric Topology (math.GT)
• Comments: 23 pages, 15 figures

A unit-area translation surface is called \emph{maximal} if it maximizes the length of the shortest saddle connection among all surfaces in the same stratum. We investigate whether a non-maximal translation surface can be continuously deformed into a maximal one while the systole increases strictly monotonically. Although such a deformation does not exist in general due to the existence of local but not global maxima of the systole function, we prove that it exists for square-tiled surfaces and translation surfaces obtained from regular hexagons and regular octagons. In each case, we explicitly construct a continuous deformation to a maximal translation surface along which the systole is strictly increasing. Finally, the preceding construction yields another family of translation surfaces admitting continuous systole-increasing deformations to maximal surfaces.

Two results on asphericity

• Authors: Roman Mikhailov
• arXiv: 2608.20270
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We construct an explicit finite chain of non-aspherical group presentation complexes in which $K(\mathcal{Y})$ is Cockcroft, both inclusion-induced maps on $\pi \_2$ are zero, and the pair has the identity property. This answers a question of Hitchman. The construction is given directly from a short balanced presentation of the binary icosahedral group. As the second result, we construct an integral Lie ring presentation in which a subpresentation has a nontrivial identity among relations, whereas the presentation itself has none.