math.GT daily digest: 9 new submissions for 24 June 2026

A Combinatorial Characterization of Sol 3-Manifolds

• Authors: Daryl Cooper; Leslie Mavrakis; Priyam Patel 2
• arXiv: 2606.23963 2
• Subjects/tags: Geometric Topology (math.GT) 2

We show that there is a universal compact branched 3-manifold such that a closed 3-manifold immerses into if and only if admits a Sol structure. Equivalently, a closed 3-manifold is Sol if and only if it has a certain type of triangulation. The construction of is based on a regular language that characterizes Sol manifolds.

Morse-Novikov theory for links

• Authors: L. Chen; H. Endo; A. Pajitnov 3
• arXiv: 2606.24009 3
• Subjects/tags: Geometric Topology (math.GT) 3

For a compact 3-manifold W. Thurston introduced a norm on the first cohomology group of the manifold. The unit ball of this norm is a polyhedron and the set of cohomology classes that are representable by fibrations over a circle is a union of cones on some of the open faces of . In the present paper we study the fibred faces of the Thurston polyhedra of exteriors of links in . Our approach is based on the non-abelian Novikov homology associated with the universal covering of the exterior of the link. We prove in particular that for a 2-component 2-bridge link a cohomology class can be represented by a fibration over a circle if and only if its 2-variable Alexander polynomial is -monic. We compute the Morse-Novikov numbers for a majority of 2-component prime links with number of crossings .

Non-asphericity of strata of genus-one differentials and stability spaces

• Authors: Dawei Chen; Jingyin Huang; Yu Qiu; Fei Yu 4
• arXiv: 2606.24135 4
• Subjects/tags: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Algebraic Topology (math.AT); Combinatorics (math.CO); Representation Theory (math.RT) 4

We show that when the number of zeros or poles is at least four, every connected component of the strata of differentials in genus one with prescribed zero and pole orders is not an orbifold . For quadratic differentials, this provides infinitely many counterexamples to a conjecture attributed to Kontsevich, as well as to a folklore conjecture concerning the contractibility of spaces of Bridgeland stability conditions.

Links of Mazur manifolds and exotica

• Authors: Sergey Nersisyan 5
• arXiv: 2606.24220 5
• Subjects/tags: Geometric Topology (math.GT) 5

In this paper, we explore links of Mazur manifolds in simple 4-manifolds. We construct non-split 2-component links in . These are used to produce links in which are split topologically but not smoothly. As a consequence, we obtain exotic pairs of simply connected, definite 4-manifolds with boundary, as well as exotic embeddings of various Mazur manifolds in .

The 4-move kills the Alexander polynomial

• Authors: Nikos Askitas 6
• arXiv: 2606.24268 6
• Subjects/tags: Geometric Topology (math.GT) 6

Whether or not the 4-move is an unknotting operation remains an open problem. In this paper I show that every knot can be reduced to one with a trivial Alexander polynomial via a sequence of -moves and isotopies.

Homotopy Coherent Nielsen Realization Problem for Dehn Twists on K3-Type 4-Manifolds

• Authors: Yujie Lin; Yi Sha 7
• arXiv: 2606.24482 7
• Subjects/tags: Geometric Topology (math.GT) 7

We study the homotopy coherent version of the Nielsen realization problem for smooth -manifolds. Given a finite subgroup , this problem asks whether there is a map such that the induced map on fundamental groups coincides with the inclusion of . Using family Seiberg-Witten theory, we prove that for -type -manifolds, the Dehn twists along -spheres are not homotopy coherently Nielsen realizable. In particular, this gives an alternative proof of the failure of the classical Nielsen realization problem in this setting.

Lengths of simple closed geodesics on hyperbolic surfaces in prescribed homology classes

• Authors: Igor M. Patsankov 8
• arXiv: 2606.24836 8
• Subjects/tags: Geometric Topology (math.GT) 8

A classical question in the theory of hyperbolic surfaces is the study of lengths of closed geodesics under various constraints. A celebrated result in this area is M. Mirzakhani's asymptotic formula for the number of simple closed geodesics of length on a hyperbolic surface of genus with punctures. We investigate the number of simple closed geodesics of length representing a fixed primitive nonzero homology class on a hyperbolic surface . We denote this number by . It follows from Mirzakhani's result that . However, numerical evidence suggests that this bound is apparently not asymptotically sharp. We prove that for a surface of genus with punctures and geodesic boundary components, under the condition that and , there exists a constant such that for sufficiently large the inequality [ h_{S}(L, x) \ge C_1 L^{6(g-1) + 2(n + b-1)} ] holds. In the special case of a torus with two punctures , we obtain the following stronger result: there exists a constant such that for sufficiently large the inequality [ h_{S_{1, 2}}(L, x) \ge C_2 L^{3.011057 \ldots } ] holds.

Central extensions of mapping class groups of surfaces from stated skein algebras

• Authors: Joris Moulai 9
• arXiv: 2606.24378 9
• Subjects/tags: Quantum Algebra (math.QA); Geometric Topology (math.GT) 9

Let be a surface of genus with zero or one boundary component and marked points, and a finite-dimensional factorizable ribbon Hopf algebra. We compute the central extension of the mapping class group of , associated to the projective representation defined from the stated skein algebra of and . Our proof is purely two-dimensional, and makes no use of TQFT arguments.

More on Kashaev limits of the quantum -polynomials

• Authors: A.Morozov 10
• arXiv: 2606.24497 10
• Subjects/tags: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Geometric Topology (math.GT) 10

"Colored" knot polynomials satisfy difference equation w.r.t. the highest weights of the underlying representation -- which in the case of symmetrically colored Jones are named "quantum -polynomials". In the double scaling quasiclassical (Kashaev) limit, when representation size , there are different phases -- in one of them the classical action vanishes and in another one it is a deformation of hyperbolic volume (of a knot complement in ). This corresponds to a splitting of the non-homogeneous version of the quantum -polynomial into two pieces, which we illustrate by more examples than just a figure-eight knot in the original paper. From the point of view of quasiclassics, hyperbolic volume is just an integration constant, which is not fully determined by the -polynomial equation -- and actually remains ambiguous in this formalism. As a byproduct, we expect that classical -polynomial at becomes proportional to Alexander: -- this seems true, but should be consistent with the polynomiality of {\it non-homogeneous quantum} -polynomial, what sometime implies that it is not minimal.