math.GT daily digest: 8 new submissions for 28 August 2026

New submissions

The Characterization of Cheng's Multivariate Writhe Polynomial

• Authors: Juno Park
• arXiv: 2608.26201
• Subjects: Geometric Topology (math.GT)

Cheng introduces the multivariate writhe polynomial, leaving a question regarding the characterization of this invariant. In this paper, we resolve this question by proving that a Laurent polynomial $f$ with integer coefficients can be realized as the multivariate writhe polynomial of some multi-virtual knot if and only if and .

Real ideal points, Conway spheres, and left-orderable Dehn fillings

• Authors: Yi Wang
• arXiv: 2608.26564
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We study real ideal points of -character varieties of knot exteriors containing essential Conway spheres. Under explicit deformation-theoretic hypotheses on the two complementary tangles, we show that essential Conway spheres are detected via Culler-Shalen theory with real ideal points; such ideal points are then deformed into arcs of representations on both sides. We then compute the asymptotic behavior of the representations on these arcs and compute the translation numbers of their lifts to representations. Combining this with a result of Gao, we conclude that for certain knots with essential Conway spheres, all sufficiently large positive and negative rational fillings have left-orderable fundamental group. This verifies the $L$-space conjecture for large-slope Dehn fillings of knots which were previously unknown in the literature. We verify the criteria for three infinite families of knots assembled from torus-trivial and twist-trivial tangles, then count the resulting real ideal points, determine the longitudinal translation numbers of all constructed branches, and identify exactly the zero-translation arcs. Computations for the $28$ distinct verified census exteriors listed in Table \ref{tab:instances} agree with the certified branches.

A note on surfaces with large systoles

• Authors: Yifei Cai
• arXiv: 2608.26660
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO)

We show that for every sufficiently large genus $g$, there exists a closed hyperbolic surface with systole . In particular, $$\liminf_{g\to \infty}\frac{\max{\mathrm{sys}(S):S\in \mathcal{M}_g}}{\log g}\geq 1,$$ improving the previously known bound $2/9$. This note is a continuation of our previous work on the diameter of finite covers arXiv:2608.12887, using the same framework of constant-twist pants decomposition to study systoles. The proof was developed by GPT-5.6 Sol through an extended discussion with the author.

Mapping class groups of simply connected closed spin 5-manifolds with no 2- and 3-torsion in homology

• Authors: Huize Jin
• arXiv: 2608.26698
• Subjects: Geometric Topology (math.GT)

We show that the Torelli group of a simply connected closed 5-manifold $M$, which is spin and has no 2-torsion elements in homology, is isomorphic to the bordism group . For with no 2- and 3-torsion we determine this bordism group, and give explicit constructions for the generators of the Torelli group. Furthermore, for the $g$-fold connected sum we completely determine its mapping class group. We apply our results to compute the stabilization and abelianization of the mapping class group of , determine the group of isotopy classes of diffeomorphisms of $M$ that are homotopic to the identity, and study the embeddings of in .

Around the Andreadakis-Johnson filtration

• Authors: Yusuke Kuno
• arXiv: 2608.26934
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Group Theory (math.GR)

The Andreadakis-Johnson filtration and its associated construction, known as the Johnson homomorphism, are useful tools in group theory. They provide a step-by-step approach to studying the automorphisms of a given group. After explaining the basics we survey both classical and recent results around the Andreadakis-Johnson filtration, with emphasis on the mapping class group of a once-bordered surface.

Mapping class groups have a unique Polish group structure

• Authors: Tyrone Ghaswala, Sumun Iyer, Robert Alonzo Lyman, Nicholas G. Vlamis
• arXiv: 2608.27218
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We prove that mapping class groups of surfaces and of locally finite connected graphs support a unique Polish group structure.

Cross submissions

Closed geodesics in homology classes modulo sublattices

• Authors: Noam Pirani
• arXiv: 2608.26311
• Cross-list: math.NT
• Subjects: Number Theory (math.NT); Geometric Topology (math.GT)

Let $M$ be a Weil-Petersson random hyperbolic surface of genus $g$, and let be a lattice of prime index $q$. We study the distribution of primitive closed geodesics in homology classes mod $\Gamma$ in the large genus limit. Averaging over all lattices of index $q$, with , we compute all the centered moments of the corresponding weighted counting functions, and exhibit a transition between Poisson and Gaussian regimes (depending on whether , the expected number of primitive geodesics in a given homology class mod $\Gamma$, tends to or $\infty$). We also study the unnormalized variance of the counts among homology classes, and show that as , averaged over all lattices of prime index $q$, it is asymptotic to in the large genus limit. These results are analogous to phenomena arising in the distribution of primes in arithmetic progressions.

Geometric $K$-homology and operator $K$-theory for Hilbert manifolds

• Authors: Doman Takata
• arXiv: 2608.26560
• Cross-list: math.KT
• Subjects: K-Theory and Homology (math.KT); Geometric Topology (math.GT); Operator Algebras (math.OA)

Poincaré duality is a classical theorem relating the homology and cohomology of closed oriented manifolds. This theorem has been extended to more general settings and to generalized (co)homology theories, including $K$-theory. In this paper, we construct an infinite-dimensional analogue of the $K$-theoretic Poincaré duality homomorphism. More precisely, for an infinite-dimensional Hilbert manifold $\mathcal{M}$, we construct a homomorphism $$K^{geo}*(\mathcal{M})\to K*(\mathcal{A}(\mathcal{M})),$$ where denotes the geometric $K$-homology of Baum and Douglas, and $\mathcal{A}(\mathcal{M})$ is a -algebra associated to $\mathcal{M}$, based on a construction of Gong, Wu, and Yu. We also prove that the constructed homomorphism is non-trivial in certain cases.