math.GT daily digest: 11 new submissions for 10 July 2026

On the number of shared Dehn surgeries between two knots

• Authors: Patricia Sorya
• arXiv: 2607.07876 2
• Subjects: Geometric Topology (math.GT)

A folklore theorem states that for any pair of distinct knots in , performing -Dehn surgery on each knot yields orientation-preservingly homeomorphic manifolds for at most finitely many slopes . In this paper, we provide a proof based on the JSJ decomposition of knot exteriors. In particular, for any given pair of distinct knots, it provides an effective bound on the maximal number of shared surgeries between the knots.

Exotic 's, RBG Links, and End Floer Homology

• Authors: Sean Eli; Shunyu Wan
• arXiv: 2607.07936 3
• Subjects: Geometric Topology (math.GT)
• Comments: 6 pages, 4 figures. Comments welcome!

We give the first pair of non-diffeomorphic exotic 's made by attaching diffeomorphic Casson handles onto diffeomorphic disk complements. Our examples are obtained using the RBG link construction to find slice knots with diffeomorphic slice disk complements, but whose Whitehead doubled disk complements are not diffeomorphic. We distinguish the exotic 's using end Floer homology.

Probabilistic pseudo knot theory

• Authors: Ioannis Diamantis; Louis H. Kauffman
• arXiv: 2607.08188 4
• Subjects: Geometric Topology (math.GT)

We develop the theory of \emph{probabilistic pseudo knots}, providing a framework for modeling knot diagrams with unresolved crossing information. Pseudo knot diagrams generalize classical diagrams by allowing certain crossings to remain unspecified; in the probabilistic setting, each such \emph{pre-crossing}, namely a crossing with undetermined over--under information, is assigned a probability describing the likelihood of resolving as a positive crossing, with complementary probability assigned to the negative resolution. This induces a probability distribution on complete classical resolutions and, by aggregation, a distribution on classical knot types, capturing uncertainty arising in physical, biological, and computational contexts. We introduce \emph{probabilistic equivalence}, defined via total variation distance between resolution distributions, and extend classical numerical quantities such as writhe and linking number to this setting. We also develop new probabilistic constructions, including the probabilistic chirality index, minimal resolution genus, probabilistic Seifert surface distributions, and polynomial invariants extending the Kauffman bracket. We further discuss matrix-based constructions, including probabilistic Seifert and Goeritz-type matrices, as well as probabilistic surgery producing distributions over 3-manifolds. Finally, we discuss potential applications in molecular biology, materials science, and computational topology.

Coarse embeddings of products of trees as quasi-isometry invariants

• Authors: Mark Hagen; Alessandro Sisto
• arXiv: 2607.08356 5
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR); Metric Geometry (math.MG)
• Comments: 37 pages, 5 figures

We consider the maximal number of factors of a product of bushy trees that can be quasi-isometrically, or even coarsely embedded into various groups of interest, including mapping class groups, Torelli groups, Johnson kernels, surface braid groups, and Bestvina-Brady groups. We use this to quasi-isometrically distinguish groups from the above classes, and also to rule out coarse embeddings between them. All these are applications of general statements about coarse embeddings of products of bushy trees into hierarchically hyperbolic spaces.

Hyperbolic manifolds without positive spun triangulations

• Authors: David Futer; Jessica S. Purcell; Saul Schleimer
• arXiv: 2607.08473 6
• Subjects: Geometric Topology (math.GT)
• Comments: 16 pages

Using a result of Choi, we provide the first examples of pairs consisting of a closed hyperbolic three-manifold and a simple closed geodesic, such that there is no positive spun ideal triangulation for the manifold, spun about the chosen geodesic. In our first two examples, the closed manifold is the third manifold in the SnapPy census, also known as Vol3, and the geodesics are its systole and second systole. This provides evidence for the conjecture that Vol3 has no positive spun ideal triangulation for any choice of geodesic.

One-cusped Dehn fillings of the sisters of the Whitehead and link complements

• Authors: Priyadip Mondal
• arXiv: 2607.08764 7
• Subjects: Geometric Topology (math.GT)
• Comments: 25 pages, 7 figures

In this article, we investigate the arithmeticity of the one-cusped Dehn fillings of the -pretzel link complement and of the Berge manifold, which respectively are the sisters of the Whitehead and link complements. We show that for each such one-cusped hyperbolic Dehn filling, the cusp field, the trace field and the invariant trace field coincide. Moreover, we establish that no one-cusped hyperbolic Dehn filling of the Berge manifold is arithmetic and that the only arithmetic one-cusped hyperbolic Dehn filling of the -pretzel link complement is the sister of the figure eight knot complement. The techniques used to prove these results further show that each knot complement covering a one-cusped hyperbolic Dehn filling of either of these two sisters manifolds admits no hidden symmetries, effectively generalizing already known results in this regard.

Combining cusped triangle groups with Blaschke products: commensurable matings

• Authors: Yusheng Luo; Mahan Mj; Sabyasachi Mukherjee
• arXiv: 2607.07886 8
• Subjects: Dynamical Systems (math.DS); Complex Variables (math.CV); Geometric Topology (math.GT)
• Comments: 20 pages, 6 figures

In this note, we construct algebraic correspondences as matings of Fuchsian -triangle groups with Blaschke products. Combined with the results of [MM25], this proves mateability of all cusped triangle groups with suitable Blaschke products. The proof of the main result involves associating two piecewise analytic circle maps to the triangle group, mating these maps with appropriate Blaschke products to produce two commensurable conformal matings, and finally constructing the desired algebraic correspondence as a common lift of the two conformal matings.

The -algebras of locally finite undirected graphs: A complete description of their K-theory

• Authors: D. Pask
• arXiv: 2607.08271 9
• Subjects: Operator Algebras (math.OA); Geometric Topology (math.GT)

We study the -algebra of a locally finite undirected (Serre) graph and compute its K-theory. The algebra is defined intrinsically, as the graph-of-groups algebra with all groups trivial, and is shown to be independent of the choice of orientation. Its structure is accessed through a passage to directed graphs: for every locally finite there is a row-finite directed graph with , and for essential this identifies with the Cuntz--Krieger algebra of the Bass--Hashimoto (non-backtracking) matrix . This makes the whole directed-graph K-theory machinery available and, unlike the directed case, produces K-groups that read off the geometry of : for a finite essential graph of genus one has and , and in general is governed by the genus, the number of ends and the number of dead-ends. We record the resulting classification by genus: the algebras are finite-dimensional or AF at genus , AT precisely for finite graphs at genus , and unital Kirchberg algebras for finite essential graphs of genus .

Braiding structures on categorical multi-Interval Jones-Wassermann subfactor

• Authors: Zhengwei Liu; Yuze Ruan
• arXiv: 2607.08296 10
• Subjects: Quantum Algebra (math.QA); Mathematical Physics (math-ph); Category Theory (math.CT); Geometric Topology (math.GT); Operator Algebras (math.OA)
• Comments: 51 pages, many figures; comments welcome!

In this paper, we construct braiding structures on the multi-interval Jones-Wassermann subfactor planar algebra associated with any unitary modular fusion category. Utilizing this construction, we provide a new proof of the self-duality of these subfactors. Furthermore, we demonstrate that these braidings induce a projective unitary representation of the balanced superelliptic mapping class group; consequently, these structures effectively encode the non-trivial higher-genus data of the underlying category. As an application of this correspondence, we derive a generalized Verlinde formula as 2-box Fourier duality of the planar algebra.

High-degree cohomology of congruence subgroups of via cohomology of -arithmetic groups

• Authors: Matthew Scalamandre
• arXiv: 2607.08600 11
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT); Number Theory (math.NT); Representation Theory (math.RT)
• Comments: 23 pages

If is a prime ideal of a number ring , then the top-degree cohomology of the principal congruence subgroup of level is naturally a representation of We prove that the multiplicity of the Steinberg representation in this cohomology space is one. When is Euclidean and is suitably small -- for example a universal side divisor -- then we prove that the multiplicity of the Steinberg representation in the next-highest-degree cohomology space is zero. Our proof relies on a computation of the cohomology of an -arithmetic group ouside of a linear range of degrees, derived from work of Blasius--Franke--Grunewald.

Existence of two embedded minimal spheres in with an arbitrary metric

• Authors: Zhichao Wang; Xin Zhou
• arXiv: 2607.08631 12
• Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Geometric Topology (math.GT)
• Comments: 32 pages, 1 figure; comments are welcome

We prove that endowed with an arbitrary Riemannian metric admits at least two embedded minimal spheres. The proof is based on an iterative scheme of relative min-max constructions.