math.GT daily digest: 7 new submissions for 17 July 2026

New submissions

Cyclotomic polynomials and homological criteria for mapping class types

• Authors: Thomas Koberda; Łukasz Patryk Michalak
• arXiv: 2607.14599
• Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)

Let be a closed orientable surface and let be the representation induced by the action on first homology. We investigate the characteristic polynomials of integral symplectic matrices arising from mapping classes of algebraically finite type and give a complete characterization in the cyclotomic case: for , the polynomial is realized by a mapping class of algebraically finite type if and only if has at most two distinct prime divisors. Consequently, if is square-free and has at least three distinct prime divisors, then every mapping class with characteristic polynomial is pseudo-Anosov. This gives a cyclotomic complement to the Casson--Bleiler homological criterion and yields a complete criterion for a symplectic polynomial to be realized only by pseudo-Anosov mapping classes.

Baumslag-Solitar subgroups of the mapping class group

• Authors: Pankaj Kapari; Dev Krishna; Kashyap Rajeevsarathy
• arXiv: 2607.14608
• Subjects: Geometric Topology (math.GT)
• Comments: 7 pages, 4 figures

For and nonzero integers , let be the mapping class group of a closed oriented surface of genus , and let be the Baumslag-Solitar group. We provide necessary and sufficient conditions under which two mapping classes generate a subgroup isomorphic to . In particular, if embeds in , then and are reducible mapping classes of infinite order. We also construct subgroups isomorphic to and for . Finally, we show that every infinite metacyclic subgroup and every Baumslag-Solitar subgroup of lifts to an isomorphic subgroup of , that is, these subgroups satisfy the generalized Nielsen realization.

Equivariant rational sliceness and Klein amphichirality of odd-stranded Turk's head knots

• Authors: Suman Saurabh
• arXiv: 2607.14638
• Subjects: Geometric Topology (math.GT)
• Comments: 21 pages, 2 figures

We establish a sharp parity dichotomy for the equivariant -sliceness and Klein amphichirality of odd-stranded Turk's head knots. When the twisting parameter is odd, we construct a commuting pair of ambient involutions by lifting explicit symmetries of the signed, arrowed Gauss diagram through the associated abstract link diagram and supporting sphere. This proves that the knots are Klein amphichiral and hence equivariantly -slice. When is even, we prove that the knots are not equivariantly -slice with respect to any strong inversion. The obstruction is the equivariant Fox-Milnor square condition of Di Prisa-Şavk: we show that the Alexander polynomial is not a Laurent square. The even- argument combines a Burau square factorization arising from Garside conjugacy with a mod- Seifert-matrix computation.

Several families of incommensurable noncompact hyperbolic Coxeter polytopes

• Authors: Lizi Guo; Jiming Ma; Yourong Zang; Fangting Zheng
• arXiv: 2607.14715
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR); Number Theory (math.NT)

We classify all 141 finite-volume hyperbolic Coxeter five-dimensional polytopes with eight facets, of which 125 are noncompact. Using maximal-cusp density and a noncompact analog of Bogachev-Douba-Raimbault's argument, we construct infinitely many pairwise incommensurable noncompact Coxeter polytopes in dimensions 4, 5, 6, 7, and 9, with the number of commensurability classes growing at least exponentially in volume.

The dark side of translation surfaces: It is not all about dynamics

• Authors: Gianluca Faraco
• arXiv: 2607.14822
• Subjects: Geometric Topology (math.GT)
• Comments: 70 pages, 26 figures. Survey to appear as a chapter in the book "In the Tradition of Thurston IV" (K. Ohshika and A. Papadopoulos ed.)

This survey offers a geometric and topological perspective on translation surfaces, with a focus on aspects that are often overshadowed by the dynamical viewpoint. After presenting several equivalent definitions of translation surfaces, particularly those based on polygonal models and fundamental domains, we explore the structure of their moduli spaces in the spirit of Thurston, including natural stratifications. The final part is devoted to the study of the period map, the realisation of representations as period characters and the fibres of the holonomy map. This approach highlights a foundational, yet sometimes underrated, facet of the theory.

A novel construction of homotopy 4-spheres and real projective 4-spaces à la Cappell-Shaneson

• Authors: Rafael Torres
• arXiv: 2607.15062
• Subjects: Geometric Topology (math.GT)

We produce new collections of homotopy 4-spheres and homotopy real projective 4-spaces by extending a classical construction due to Cappell-Shaneson. We use the extension to devise a mechanism that produces sets of smooth 4-manifolds within a given homeomorphism class that satisfies a mild topological condition.

Knotted surfaces with simply-connected complements

• Authors: Audrick Pyronneau
• arXiv: 2607.15165
• Subjects: Geometric Topology (math.GT)
• Comments: 19 pages

We study locally flat, compact, oriented, genus g, surfaces with boundary a fixed knot K, properly embedded in simply-connected 4-manifolds with boundary the 3-sphere, and whose complements have trivial fundamental group. We show that if two such surfaces are homologous then they are topologically ambiently isotopic rel. boundary.