math.GT daily digest: 6 new submissions for 23 July 2026

New submissions

Irreducible proper 2-knots from exotic open 2-handles

• Authors: Sean Eli
• arXiv: 2607.19481
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)
• Comments: 32 pages, 22 figures. Comments welcome!

We construct infinite families of irreducible exotic proper knotted surfaces in , making progress on a question of Gompf. Here irreducible means these surfaces are not end-sums of standard surfaces with exotic planes. To prove the topological equivalence, we give a highly flexible construction of exotic open 2-handles, which generalizes several similar constructions in the literature. We distinguish exotic surfaces through the genus functions and end Floer homology of their double branched covers. By studying these generalized handles further, we construct a new family of topologically slice links.

The support genus does not increase under contact connected sum

• Authors: Miguel Orbegozo Rodriguez; Eric Stenhede
• arXiv: 2607.19892
• Subjects: Geometric Topology (math.GT)
• Comments: 5 pages, 2 figures. Comments welcome!

We show that the support genus of the contact connected sum of two contact -manifolds is at most the maximum of the support genera of the summands. In particular, iterated connected sums of contact manifolds of support genus one still have support genus at most one, and therefore cannot provide candidates for contact manifolds of higher support genus.

Étude on the Delta-unknotting number

• Authors: Sebastian Baader; Lukas Lewark
• arXiv: 2607.20212
• Subjects: Geometric Topology (math.GT)
• Comments: 7 pages, 2 figures, comments welcome

We derive an almost sharp lower bound on the Casson invariant of positive braid knots. As a consequence, we show that the -unknotting number is quasi-additive on the set of positive and negative braids on a fixed number of strands.

Non-determinacy of HOMFLY-PT homology for diagrams

• Authors: Maciej Borodzik; Mikhail Malashchuk
• arXiv: 2607.20256
• Subjects: Geometric Topology (math.GT)
• Comments: 9 pages

We show that for any two knots , in , there exist diagrams and that represent and , respectively, such that the Khovanov--Rozansky triply graded homologies of and are isomorphic. The methods expand on Abel's paper [Abe17].

Morse complexity of homology classes

• Authors: Fedor Manin; Bena Tshishiku; Shmuel Weinberger
• arXiv: 2607.20259
• Subjects: Geometric Topology (math.GT)
• Comments: Formerly an appendix to arXiv:2311.16389, which will be edited to remove the appendix once this article is posted

The Morse complexity of a manifold is the minimal number of handles required to build it. We explore the Morse complexity of manifolds, bordisms, and homology classes, proving nontrivial upper bounds using surgery theory and lower bounds using index theory. Our most involved result shows that for Lie groups which admit discrete series representations, the Morse complexity of their locally symmetric spaces grows linearly with volume. This implies that such locally symmetric spaces do not admit open book decompositions.

A refinement of the asymptotic expansion of Weil-Petersson volumes

• Authors: Marthe Guillermit
• arXiv: 2607.20341
• Subjects: Geometric Topology (math.GT); Spectral Theory (math.SP)

Over the past decade, the study of the asymptotic growth of Weil-Petersson volumes of the moduli space of hyperbolic surfaces has yielded numerous results on the length spectrum and on the spectrum of the Laplacian of typical large genus surfaces. We compute the exact asymptotic value of the volume polynomials for the lengths of the boundary components such that : $$\prod_{j=1}^{n}\frac{x_j}{2}\cdot\frac{V_{g,n}(x_1,\ldots x_n)}{V_{g,n}}!=!\frac{1}{2^n}\exp\left({\frac{|\mathbf{x}|}{2}!-!\frac{1}{8\pi^{2}g}\left(\frac{|\mathbf{x}|}{2}\right)^{2}}\right)!\left(1!+!\mathcal{O}_{n}\left(\frac{1}{\min x_j}\right)\right).$$ This result relies on the analysis of the expansion of Witten-Kontsevitch intersection numbers, for which we obtain an analogous explicit result. We also refine the bound over the coefficients of the expansion in terms of the degree of the expansion. From the expansion of the volumes, we deduce an exact estimate of the average number of non-separating simple geodesics of length of order . Our result therefore explains the behavior of counting functions at the cutoff , at which simple geodesics become negligible with respect to non-simple ones. The existence of this cut-off was conjectured by Lipnowski and Wright and proven by Wu and Xue.