math.GT daily digest: 5 new submissions for 6 August 2026

New submissions

Polynomial Link Invariants via Decategorification of Subquandle Quiver Filtrations

• Authors: Jose Ceniceros; Benjamin Chidley; Jackson Vaughn
• arXiv: 2608.05014
• Subjects: Geometric Topology (math.GT)

In this paper, we introduce the notion of subquandle filtrations for finite quandles as a foundation to construct algebraic and combinatorial link invariants. A nested sequence of subquandles naturally induces a filtration of the associated quandle coloring quiver by subquivers. By decategorifying these subquivers, we define two new polynomial-valued invariants of links: the disrespectful polynomial and the graded disrespectful polynomial, which measure how a chosen set of quandle endomorphisms respects and disrespects subquandle structure. Finally, we illustrate the distinguishing power of these new tools by comparing them to both classical quandle counting invariants and the quiver in-degree polynomial.

Cross submissions

Homotopy classification of $4$-manifolds with fundamental group

• Cross-list: math.AT
• Authors: Kamen Pavlov
• arXiv: 2608.03245
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)
• Comments: 24 pages

For an odd prime $p$, we show that the homotopy type of a closed $4$-manifold with fundamental group is determined by its quadratic $2$-type together with an additional -valued intersection form. We also give a generalization to infinite groups of a result of Hambleton--Kreck on the polarized homotopy classification of Poincaré $4$-complexes, and we prove some partial results on the homotopy classification that are applicable more generally.

Modular fusion categories with trivial Torelli group actions

• Cross-list: math.QA
• Authors: Liang Chang; Siu-Hung Ng; Yilong Wang
• arXiv: 2608.04462
• Subjects: Quantum Algebra (math.QA); Geometric Topology (math.GT); Representation Theory (math.RT)

In this paper, we study modular fusion categories whose mapping class group representations are trivial on the Torelli groups, with particular emphasis on the congruence properties of the resulting representations of . We prove that, for any , the Torelli group is contained in the kernel of the genus-$g$ mapping class group representation associated with a modular fusion category if and only if is pointed. This refines the corresponding result in \cite{MW25}. We also characterize the modular fusion categories for which the genus-$2$ Torelli group is contained in ; in particular, categories with isotropic adjoint subcategories belong to this class. Finally, we give a complete classification of modular fusion categories with isotropic adjoint subcategories.

The geometry of triples of antipodal ideal chambers of affine buildings

• Cross-list: math.GR
• Authors: Corina Ciobotaru; Corentin Le Bars
• arXiv: 2608.04520
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT); Metric Geometry (math.MG)
• Comments: 33 pages

In this article, we investigate two notions of genericity for triples of antipodal ideal chambers in a locally finite affine building $X$: one defined at the ideal boundary , which we call ideal-genericity, and the other defined from within the affine building \(X\), which we call affine-genericity. While ideal-genericity implies affine-genericity, the latter is the more suitable notion for constructing a barycenter map associated with affine-generic triples of antipodal ideal chambers. This perspective also allows us to establish that this barycenter map is locally constant, and hence continuous. Finally, we provide sufficient geometric conditions on the affine Weyl group associated with $X$ that guarantee both ideal- and affine-genericity for all triples of antipodal ideal chambers of X^\\infty!. These conditions yield an algorithmic method for constructing geometric configurations in (when they exist) that correspond to non-generic triples of antipodal ideal chambers of . Furthermore, our computations for the irreducible finite Weyl groups of rank at least two show that automatic ideal-genericity holds for all triples of antipodal ideal chambers only in types , $G\_2$ and $B\_3$.

Zariski density of discrete subgroups via critical exponents and unitary representations

• Cross-list: math.GR
• Authors: Aleksander Skenderi
• arXiv: 2608.04966
• Subjects: Group Theory (math.GR); Dynamical Systems (math.DS); Geometric Topology (math.GT); Representation Theory (math.RT)
• Comments: 15 pages, no figures. Comments welcome!

Let $G$ be a connected real semisimple linear algebraic group with finite center and no compact factors, let $X$ denote its associated Riemannian symmetric space, and let be the volume growth entropy of $X$. We show that there exists an so that the following holds: if is a discrete subgroup with critical exponent greater than , then $Gamma$ is Zariski dense in $G$. If $G$ is further assumed to be isomorphic to one of the isometry groups of the real, complex, or quaternionic hyperbolic spaces, we determine the smallest possible value of the critical exponent to guarantee Zariski density of the associated subgroup (in other words, the largest possible value of ). In the setting of discrete subgroups of real semisimple Lie groups with no compact factors, this generalizes Borel's Density Theorem for lattices. The key ingredient is input from the theory of unitary representations, particularly the recent work of Benoist--Liang (which was inspired by earlier work of Benoist--Kobayashi) on general temperedness criteria for the quasi-regular representation of for any closed subgroup $H$ of $G$.