math.GT daily digest: 12 new submissions for 3 July 2026

1. Alexander's conjecture for infinite simplicial complexes

• Authors: Martina Iannella, Vadim Weinstein
• arXiv: 2607.01349 2
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO)
• Comments: 10 pages, 5 figures

Alexander's conjecture states that for every two finite triangulations of the same topological space, if they have a common subdivision, then they have a common stellar subdivision. We generalize the recent result of Adiprasito and Pak, who resolved Alexander's conjecture for finite simplicial complexes, to infinite simplicial complexes.

2. Permutation Jones Polynomials

• Authors: Sam Nelson
• arXiv: 2607.01384 3
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)
• Comments: 12 pages

We introduce a generalization of the Jones polynomial for classical and virtual knots and links using colorings by a permutation of a finite set $X$. For and for classical knots, the invariant is equivalent to the usual Jones polynomial; for $X$ with cardinality greater than 1 the invariant expresses distinct information from the Jones polynomial or virtual knots and for classical and virtual links. We establish some properties of the new invariants and compute the polynomials for classical and virtual knots and links of small crossing number for a few small permutations.

3. Three thousand obstructions to knotless embedding

• Authors: Thomas W Mattman, Andrei Pavelescu
• arXiv: 2607.01414 4
• Subjects: Geometric Topology (math.GT)
• Comments: 9 pages, no figures, 2 tables

We present a list of 3028 obstructions to knotless embedding. We survey recent work in this area including: 1) A bibliography of graphs proven to be intrinsically knotted without relying on computers; 2) An updated listing of obstructions in $\nabla \mathrm{Y}$ families including two new large families; 3) Connections with the Colin de Verdière's invariant including a new obstruction with ; and 4) Connectivity of obstructions and their structure near vertices of degree three or four. We address questions raised in earlier work, (re)state several conjectures, and propose new questions.

4. Right-angled Artin groups of large girth and finite volume hyperbolic $3$--manifold groups

• Authors: Thomas Koberda
• arXiv: 2607.01652 5
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)
• Comments: 7 pages

Let $\Gamma$ be a finite simplicial graph of girth at least five. In this short note, we give a proof that if $M$ is a finite volume hyperbolic $3$--manifold, then the right-angled Artin group $A(\Gamma )$ cannot contain $\pi \_1(M)$ as a subgroup; the argument is elementary, modulo the resolution of the Virtual Fibering Conjecture and a splitting theorem due to Belegradek. In particular, if $C\_n$ denotes the $n$--cycle then $A(C\_n)$ cannot contain a finite volume hyperbolic $3$--manifold group for any , thus answering a question of A.~Reid.

5. Constructing depth one laminations transverse to pseudo-Anosov flows

• Authors: Junzhi Huang, Samuel J. Taylor
• arXiv: 2607.01672 6
• Subjects: Geometric Topology (math.GT)
• Comments: 30 pages, 14 figures

Given a pseudo-Anosov flow $\varphi$ on a closed atoroidal $3$--manifold $M$ and a closed surface $S$ almost transverse to $\varphi$, we give a homological characterization of when $S$ can be completed to an almost transverse depth one lamination or foliation whose set of compact leaves is $S$. As a consequence, we show that the cone of classes in that are positive on the closed orbits of $\varphi$, when nonempty, is an entire foliation cone of .

6. Fundamental racks of braid spaces of complex reflection groups

• Authors: Tathagata Basak
• arXiv: 2607.01826 7
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Group Theory (math.GR)
• Comments: 8 pages, 2 figures, submitted for publication

Let $\Gamma$ be a complex reflection group acting on the complex affine or hyperbolic space $X$ with the set of reflecting hyperplanes $\mathcal{H}$. We define an augmented rack associated to the orbifold fundamental group which plays the role of the fundamental rack of a framed link complement as defined by Fenn and Rourke. This yields representations of the orbifold fundamental group $G$ on the cohomology of the associated rack space.

7. Every cusp singularity link admits infinitely many strong symplectic fillings

• Authors: Naohiko Kasuya, Takahiro Oba
• arXiv: 2607.01991 8
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)
• Comments: 15 pages, no figures

In this paper, we show that if the link of an isolated complex surface singularity is either a -manifold or an -manifold with its canonical contact structure, then it admits infinitely many strong symplectic fillings that are pairwise non-diffeomorphic and not related by a sequence of blow-ups or blow-downs. As a consequence, the link of any cusp singularity, exceptional unimodal singularity, or hyperbolic Brieskorn singularity admits infinitely many pairwise non-diffeomorphic minimal strong symplectic fillings.

8. Surgery obstructions for knots in integer homology spheres

• Authors: Yuhui Chen
• arXiv: 2607.02028 9
• Subjects: Geometric Topology (math.GT)

For knot surgery in , Heegaard Floer homology provides an obstruction due to Hom--Karakurt--Lidman. We extend this obstruction to all integer homology spheres $Y$, for both positive and negative 1/m surgeries. This is used to test infinitely many small Seifert fibered examples and hyperbolic examples. Moreover, we deduce a lower bound on the $b\_2(W)$ of smooth cobordism between a pair of integer homology spheres.

9. Torsion in the homology of the Torelli group and the Birman-Craggs-Johnson homomorphism

• Authors: Andrei Vladimirov
• arXiv: 2607.02107 10
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)
• Comments: 21 pages, 3 figures

The Birman-Craggs-Johnson homomorphism is a homomorphism from the Torelli group to a certain $\mathbb{Z}/2\mathbb{Z}$-vector space of Boolean polynomials. In 1983, Johnson computed for and showed, in particular, that the induced homomorphism on is injective when restricted to the subgroup generated by Dehn twists about separating simple closed curves. In this paper, we extend Johnson's result to higher homology groups. Given any collection of pairwise disjoint separating simple closed curves on $\Sigma \_g$, the corresponding Dehn twists pairwise commute and determine a homology class in called an abelian cycle. We prove that the pushforward homomorphism restricted to the subgroup of generated by such abelian cycles is injective for .

10. Relativization of symmetries on quandles

• Authors: Yuki Imamura, Tomoki Yoshida
• arXiv: 2607.02204 11
• Subjects: Geometric Topology (math.GT)
• Comments: 36 pages

This paper introduces relative versions of the inner automorphism group and the transvection group associated with surjective quandle this http URL using the relative inner automorphism group, we define a notion of \emph{connectedness} for surjective homomorphisms. We characterize connected homomorphisms algebraically as quotient maps, and use the relative transvection group to establish a maximal \emph{connected-covering} factorization for arbitrary surjections. Finally, we study surjective homomorphisms for which the relative inner automorphism group acts $2$-transitively on each fiber. Under this assumption, we classify the possible quandle structures of the finite fibers.

11. New constructions relating Real and Complex Contact Structure

• Authors: Ali M. Elgindi
• arXiv: 2607.01264 12
• Subjects: Symplectic Geometry (math.SG); Differential Geometry (math.DG); Geometric Topology (math.GT)

We establish new connections between real and complex contact geometry via embeddings of 3-manifolds into \C^3!. We introduce a new \emph{contact wedge} construction combining two transverse real contact structures to make a new \emph{complex contact} structure, subject to obstructions measured by the Nijenhuis tensor and Dolbeault cohomology. Dually, we form a \emph{hedge} construction which extracts real contact structures from complex ones. Applying these tools, we prove that \C^3! admits uncountably many complex contact structures.

12. Categorification of some Penrose polynomials

• Authors: D. W. Collison, D. Tubbenhauer
• arXiv: 2607.01632 13
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)
• Comments: 34 pages, many figures, comments welcome

We construct doubly- and triply-graded Penrose-type homologies for ribbon graphs. The construction is a TQFT-valued cube of resolutions built from two-dimensional cobordisms, which may be nonorientable. Their Euler characteristics recover specializations of some Penrose polynomials; in particular, the four color case comes with a refinement of the classical Penrose criterion.