math.GT daily digest: 18 new submissions for 28 July 2026

New submissions

-complexes, Clasp Number, and Triple Linking Number

• Authors: Christopher William Davis; David Lawrence; Jack Paulsen; Nathan Phillips
• arXiv: 2607.22848
• Subjects: Geometric Topology (math.GT)
• Comments: 10 pages, 6 figures

A C-complex is a union of Seifert surfaces for the components of a link which intersect each other in clasps. The clasp number of a link is the minimal number of clasps amongst all C-complexes it bounds. It gives a measure of how far a link is form being a boundary link. This paper provides a new lower bound for the number of clasps of all C-complexes bounded by a given 3-component link improving results of Amundsen-Anderson-D.-Guyer. Furthermore, we construct links that achieve these bounds. In order to do so, we express the triple linking numbers as the area bounded by three curves in the plane, called word curves, and then perform the geometry and discrete optimization needed to minimize the length of these curves.

Incompressible surfaces, hierarchies and unknot recognition

• Authors: Marc Lackenby
• arXiv: 2607.23350
• Subjects: Geometric Topology (math.GT)
• Comments: 54 pages, 14 figures

We present a new algorithm to determine whether a compact orientable surface properly embedded in a compact orientable 3-manifold is incompressible. As a special case, this provides a new algorithm to detect the unknot. The central technique is the use of hierarchies. Unlike previous algorithms, no challenging invariants need to be computed and no lengthy search procedures are required. This algorithm leads to a new proof that determining whether a compact orientable 3-manifold has incompressible boundary is in NP.

A TQFT-based Platform for Efficient Computation of Knot Invariants

• Authors: Amena Al Rawi; Hisham Sati; Vivek Kumar Singh
• arXiv: 2607.23551
• Subjects: Geometric Topology (math.GT); Mathematical Software (cs.MS); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
• Comments: 20 pages, 8 figures

We present an interactive web platform that unifies the construction Feynman ribbon diagrams (FRDs), the evaluation of higher-rank Chern--Simons knot invariants, and the identification of FRD-like knots at higher crossing numbers. These tree-structured diagrams naturally represent arborescent knots, which we refer to throughout as FRD-like knots. Within a single visual environment, users can construct an FRD as a tensor network, evaluate its associated Chern--Simons invariants, and use the resulting invariant data to distinguish and identify the corresponding knot. To our knowledge, this is the first platform to combine diagrammatic construction, tensor-network evaluation, invariant computation, and knot identification within a unified workflow.

Topological line arrangements and their topological invariants

• Authors: Sakumi Sugawara
• arXiv: 2607.23570
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Combinatorics (math.CO)
• Comments: 19 pages, 3 figures

A topological line arrangement is an arrangement of embedded spheres in the complex projective plane that topologically generalizes a complex line arrangement. In this paper, we establish foundational results on the topology of the complement of topological line arrangements. First, we prove that the cohomology ring of the complement is isomorphic to the Orlik-Solomon algebra, as for classical complex line arrangements. Since classical methods are unavailable in this setting, we use a homological method to compute the cohomology ring. We then study the homotopy type of the complement. We prove that the complement of a symplectic line arrangement has the homotopy type of a minimal CW complex. In contrast, every combinatorial type realizable by a topological line arrangement admits a realization with a non-minimal complement. Moreover, every such combinatorial type admits infinitely many realizations whose complements are pairwise non-homotopy equivalent.

An Algorithmic Search for Knots Bounding Möbius Bands in

• Authors: Daniel Lee; Joshua M. Sabloff
• arXiv: 2607.23582
• Subjects: Geometric Topology (math.GT)
• Comments: 29 pages, 8 figures

This study examines the smooth non-orientable -genus of knots using a large-scale computational search. Knot invariant obstructions were computed, and existing computational algorithms were extended to identify prime knots in of crossing number at most that bound Möbius bands in . The findings significantly expand the known examples of knots of non-orientable -genus equal to and contribute new data relevant to the study of the non-orientable -genus.

Abelianization of Symmetric Mapping Class Groups

• Authors: Xiyan Zhong
• arXiv: 2607.24271
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)
• Comments: 28 pages, 3 figures. Comments welcome!

Let be an unbranched regular -fold cyclic cover of a closed orientable surface of genus . Two natural groups are associated with this cover. The first is the centralizer in of a chosen generator of the deck transformation group, denoted by . The second is the finite-index subgroup of consisting of mapping classes that fix the nonzero class corresponding to the cover, denoted by . For , the abelianizations of these groups were computed by Sato. We compute their abelianizations for every odd prime and show that they exhibit a splitting phenomenon different from the case . In most cases, this difference is reflected in the image of the Prym representation; in the remaining cases, it is detected by the existence of a distinguished element in the Johnson kernel.

A De Rham Perspective on the Symplectic Geometry of Teichmüller Space

• Authors: Antoine Ablondi
• arXiv: 2607.24362
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)
• Comments: 40 pages, 12 figures

We present a new proof of Wolpert's Magic Formula, stating that any Fenchel--Nielsen coordinates on the Teichmüller space of a closed surface are Darboux coordinates for the Weil--Petersson symplectic form. Our approach relies on a de Rham cohomology model for the tangent space to the character variety model of Teichmüller space, in which, by the seminal work of Goldman, the Weil--Petersson form is expressed by a natural symplectic form, called the Goldman form.

We extend the work of Fillastre and Seppi, who have managed, using that approach and Stokes's Theorem, to provide a new proof of Wolpert's sum of cosines formula. Using the unique isometric symmetry on any hyperbolic pair of pants, we also introduce a way, given a pair of pants decomposition of a closed surface, to construct an associated linear involution on every tangent space of its Teichmüller space.

That allows us to derive a new proof of Wolpert's Magic Formula and deduce a self-contained proof of the closedness of the Goldman form on Teichmüller space.

Conic reach and polynomial parallel volume in the plane

• Authors: Alejandro Cholaquidis
• arXiv: 2607.24487
• Subjects: Geometric Topology (math.GT)

For a compact set , the local Steiner formula of Hug, Last and Weil expresses the parallel volume through the proximal normal bundle and the truncated fiber lengths . We introduce conic reach, a geometric condition with two requirements: coincides with a cone near each point of a finite, well-separated singular set, and every proximal normal fiber away from those points has length at least . In the plane, these requirements force every link to be a finite union of circular arcs whose complementary gaps have width bounded below. Computing the feet-localized tube of each cone and combining it with the local Steiner formula, we show that is a polynomial of degree at most two on , with explicit coefficients; in particular . A one-dimensional converse shows that a quadratic wall contribution forces a linear cut function. Compact domains with piecewise- boundary, uniformly wedge-like at their reentrant corners, have positive conic reach, and their volume coefficients are given by a Gauss--Bonnet-type formula. For the L-shaped polygon the three invariants separate: , , . A cuspidal notch, two overlapping discs and a Cantor fan of segments show that tangential contact, curvature at a reentrant corner and degenerating link gaps each destroy polynomiality.

Non-kinetic homotopy coherent actions on four-manifolds

• Authors: Sungkyung Kang; JungHwan Park; Masaki Taniguchi
• arXiv: 2607.24548
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)
• Comments: 39 pages, 2 figures

We give the first example of a non-kinetic smooth homotopy coherent action of order two on a closed simply connected smooth four-manifold. This action is obtained by restricting a nontrivial smooth homotopy coherent action of the discrete circle group on a stabilized surface. We also construct relatively non-kinetic smooth homotopy coherent extensions of boundary involutions over compact smooth -manifolds. In addition, we exhibit boundary involutions that admit locally linear topological extensions but no smooth extensions over the same stabilized fillings. Nevertheless, we prove a Wall-type theorem showing that every free involution on a disjoint union of integral homology spheres extends smoothly over any simply connected smooth filling after sufficiently many stabilizations by .

Cross submissions

Infinite Combinatorial Yamabe Flows in Three Dimensions

• Authors: Bohao Ji
• arXiv: 2607.23584
• Cross-list from: math.DG
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)
• Comments: 26 pages

In this paper, we study three-dimensional combinatorial Yamabe flows on locally finite infinite triangulations in Euclidean and hyperbolic background geometries. Under suitable non-degeneracy and bounded-degree assumptions, we establish the short-time existence and uniqueness of the original flows. We further introduce the extended flows by using the continuous extension of solid angles, and prove long-time existence for both extended flows under suitable initial assumptions.

Sequence distortion for metric spaces

• Authors: Ilya Kapovich
• arXiv: 2607.23713
• Cross-list from: math.GR
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT); Metric Geometry (math.MG)
• Comments: 28 pages

We introduce \emph{sequence distortion spectrum}, a quasi-isometry invariant recording the large-scale distance profiles of sequences indexed by or in a metric space. For a rate function , extended by for integers , a sequence in a metric space is \emph{-distorted} if there exists an integer such that for all we have

This definition implies that . For rate functions, realizability depends only on the ambient quasi-isometry type and the growth type of . We classify the possible power rates (where for Euclidean spaces: in only the linear rate occurs, while in , , the realizable exponents are exactly . For a geodesic -hyperbolic space , no power rate with occurs. The hyperbolic plane also realizes the logarithmic rate. An exponential packing bound for rules out every rate, but a proper CAT(-1) surface of unbounded geometry realizes a log--log rate. In an arbitrary simplicial tree, every realizable rate is linear up to constants. Finally, we construct two pairs of proper geodesic spaces: the first has equivalent basepoint packing functions and the second equivalent uniform packing functions; both pairs have equal asymptotic dimensions and filling-function growth classes, and isometric asymptotic cones at the chosen wedge points for every common scaling sequence and ultrafilter. Yet sequence distortion distinguishes each pair, and the second pair has bounded geometry.

Local Weyl law and length-minimising loops on hyperbolic surfaces

• Authors: Daniel Meriaz
• arXiv: 2607.24060
• Cross-list from: math-ph
• Subjects: Mathematical Physics (math-ph); Geometric Topology (math.GT); Spectral Theory (math.SP)
• Comments: 50 pages, 11 figures

We study the variance of a local Weyl law over a fixed smooth energy window, when averaged over large Weil--Petersson hyperbolic surfaces. Our results are consistent with the predictions of Berry's random wave model. Our approach allows to explicitly integrate certain test functions which depend on lengths of based geodesic loops, and relate them to the associated lengths of the closed geodesics in their free-homotopy class. We thus utilise the work of Mirzakhani, with exact stationary phase arguments, to identify correct main and error terms, making explicit the asymptotic behaviour of the variance of the local Weyl law. Furthermore, we introduce the geometric notion of length-minimising geodesic loops and sequences, based at a point. We prove a complete characterisation of the topology of these, namely that they are simple. This forms a key ingredient in our study, and yields a new streamlined argument to bound the contributions of remainder terms which depend on lengths of pairs of different short primitive geodesic loops. To illustrate the generality of our results, we further introduce a family of "exploring" loops based at a point, which might be of independent interest.

Complex Analysis and Existence Problems for plane Graphs

• Authors: Fedor Pakovich
• arXiv: 2607.24061
• Cross-list from: math.CO
• Subjects: Combinatorics (math.CO); Algebraic Geometry (math.AG); Complex Variables (math.CV); Geometric Topology (math.GT)

We show that a variety of known and new results concerning connected plane graphs whose vertex and face degrees satisfy prescribed uniformity conditions with at most two exceptions can be deduced from recent results on the Hurwitz existence problem regarding the realizability of branch patterns of rational functions. Our method also yields a description of the Belyi functions corresponding to such graphs.

Nerve-type and invariance theorems for asymptotic dimension

• Authors: Chun-Hung Liu; Sergey Norin
• arXiv: 2607.24146
• Cross-list from: math.CO
• Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM); Geometric Topology (math.GT); Metric Geometry (math.MG)

Asymptotic dimension of metric spaces is a large-scale analog of covering dimension of topological spaces. An intersection graph of a family of sets is the graph whose vertices are the members of the family and whose edges correspond to pairs of members with non-empty intersection.

Our first main result connects the asymptotic dimension of the intersection graph of a family and the Assouad-Nagata dimension of the ambient metric space containing members of under some mild and necessary assumptions. We prove that if is a family of subsets of a metric space of Assouad-Nagata dimension such that every ball of radius intersects at most pairwise disjoint members of of diameter at least for some function , then the asymptotic dimension of the intersection graph of is at most . This result is optimal both quantitatively and qualitatively in several senses. As a corollary of this result, the asymptotic dimension of the intersection graph of any family of compact convex sets of bounded aspect ratio in , such as a family of balls in , is at most .

Our second main result states that the asymptotic dimension of the intersection graph of a family of connected closed sets of a connected topological space with connected boundary equals the asymptotic dimension of the intersection graph of the family of the boundary of the sets in , under a mild condition. In particular, the asymptotic dimension of the intersection graphs of families of spheres in equals or when $n \geq 2.

-weight system does not extend to a graph 4-invariant

• Authors: Daniil Fomichev; Maksim Karev; Fedor Pavutnitskiy; Sergey Usanov
• arXiv: 2607.24217
• Cross-list from: math.CO
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)
• Comments: 14 pages, 5 figures

A long-standing question by S. Lando asks whether the -weight system extends to a unique 4-invariant of graphs. We show that, in full generality, the answer to this question is negative. However, for certain specializations of the weight system, extensions do exist. Explicit formulae for computing two such specializations of the weight system are already known. We construct recurrence relations for one additional such extension and discuss the last remaining specialization, which conjecturally admits an extension. We also study the polynomial coefficients of the -weight system and resolve the questions concerning their extension.

Calabi surgery for Z/2 harmonic 1-forms

• Authors: Jiahuang Chen; Siqi He; Dashen Yan
• arXiv: 2607.24281
• Cross-list from: math.DG
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)
• Comments: 45 pages, 3 figures

We prove a 2-valued analogue of Calabi's intrinsic harmonicity theorem and use it to introduce the Calabi surgery method, a surgery theory for harmonic -forms. Once the ambient metric is allowed to vary, the construction of new harmonic forms can be reduced to cutting and pasting closed 2-valued 1-forms, matching local harmonic models, and controlling the transitivity of the resulting singular foliation. For these constructions, the Nash--Moser-type analytic deformation problem that arises in singular gluing is replaced by local model matching and a global dynamical condition on the foliation. The resulting procedure gives a flexible way to construct and modify harmonic 1-forms under weak regularity assumptions. As applications, we obtain connected-sum and local replacement theorems, blow up isolated ordinary zeros by prescribed Euclidean models, split smooth -nondegenerate branching components, and desingularize graphic singular sets in dimensions 3 and 4 with suitable resolution models.

On Realisability of Twisted Homology

• Authors: Mark Grant; Michael Jung; Baylee Schutte
• arXiv: 2607.24462
• Cross-list from: math.AT
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)
• Comments: more details appear in the thesis of the second author; comments are welcome

We discuss the question of when a homology or cohomology class with twisted integer coefficients of a manifold is realised by a submanifold. While this question is classical in nature, providing an answer requires relatively modern techniques from parametrised homotopy theory. More specifically, we introduce cobordism classes twisted by a coefficient system and then define a twisted Thom space over , which serves as the classifying object for this cobordism theory under a twisted Pontryagin-Thom construction. As a result, a twisted homology class is realisable if and only if its Poincaré dual is the image of the twisted Thom class in under a parametrised map over . Finally, we construct the parametrised Postnikov tower of over to derive obstructions to realisability and conclude by giving the first known examples of non-realisable integer homology classes in non-orientable manifolds.

The conjecture for Artin groups of spherical type

• Authors: Giovanni Paolini
• arXiv: 2607.24659
• Cross-list from: math.GR
• Subjects: Group Theory (math.GR); Algebraic Topology (math.AT); Combinatorics (math.CO); Geometric Topology (math.GT)

In these notes, we introduce the 50-year-old conjecture alongside Coxeter and Artin groups. Roughly speaking, the conjecture states that the complement in of a "symmetric" configuration of hyperplanes is a space. Our end goal is to present a proof of the conjecture in the so-called spherical case, where only a finite number of hyperplanes are removed, through methods from combinatorial topology. This proof draws inspiration from the original proof of the spherical case, which is a special case of a celebrated 1972 theorem by Pierre Deligne.