math.GT daily digest: 10 new submissions for 16 June 2026
A structured digest of the 10 papers in the arXiv math.GT daily listing for Tuesday, 16 June 2026, with each paper's title, authors, abstract-level summary, main result, proof idea, and direct arXiv link.
Coverage
This issue covers the arXiv math.GT daily listing for Tuesday, 16 June 2026, which contained 10 papers tagged math.GT, including cross-lists from math.DS and papers also tagged math.AT, math.DG, math.QA, math-ph, and other adjacent areas 1. The short versions below are meant to help you decide what to open first; the abstracts are kept close to the arXiv text.
| arXiv ID | Paper | Main area | Why it may matter |
|---|---|---|---|
| 2606.16588 | Mann--Taylor, pseudo-Anosov orbit spaces | 3-manifold dynamics | Characterizes which bifoliated-plane actions arise from transitive pseudo-Anosov flows. |
| 2606.16577 | Tian--Wang, skein homology | Skein theory / 3-manifolds | Builds a Heegaard-splitting-dependent homology theory extending the Kauffman bracket skein module. |
| 2606.16529 | Elgindi, contact quantization | Contact topology / math physics | Proposes a quantization framework for closed contact 3-manifolds using complex-tangent links and Reeb fields. |
| 2606.16120 | Guth--Kang, bordered Floer concordance formula | Knot Floer theory | Gives a combinatorial way to transport concordance maps through satellite patterns. |
| 2606.15539 | Ding--Gao--Lei--Li--Vesnin, virtual knotoids | Knot invariants | Defines a two-variable parity polynomial for oriented virtual knotoids. |
| 2606.15402 | Zhao, Weyl problem rigidity | Hyperbolic geometry / convexity | Extends rigidity for non-compact convex subsets of hyperbolic 3-space. |
| 2606.15256 | Saurabh, Turk's head knots | Knot polynomials | Proves Fox's trapezoidal conjecture for four-strand Turk's head knots. |
| 2606.14927 | Ghanem, ribbon cobordism | 3-manifold cobordism | Shows rational homology ribbon cobordism is antisymmetric, hence a partial order. |
| 2606.14809 | Bondar, P-categories | LS category / singularity theory | Publishes a translated 1993 dissertation on functions with degenerate critical submanifolds. |
| 2606.14889 | Calegari--Zung, quasimorphisms and flows | Hyperbolic 3-manifold dynamics | Connects adapted quasimorphisms to pseudo-Anosov flows and orbit growth. |
1. A characterization of pseudo-Anosov orbit spaces via bifoliated planes
Authors: Kathryn Mann, Samuel J. Taylor. Tags: math.GT, math.DS. arXiv: 2606.16588 2.
Abstract: The paper characterizes the actions on bifoliated planes that arise as orbit spaces of transitive pseudo-Anosov flows on orientable closed 3-manifolds. It uses branched covers, veering triangulations, and a new compactness criterion to extend earlier work that covered the case with no odd-prong singularities and leafwise-orientation-preserving actions 2.
Main result: A torsion-free group action on a bifoliated plane is realized as the orbit-space action of a transitive pseudo-Anosov flow on a closed oriented 3-manifold exactly when the action is topologically transitive, orientation preserving, and satisfies the closing property, uniformly hyperbolic fixed points, and a finite rectangle condition. The leafwise-orientable version appears as a separate theorem, then the general case allows odd-prong singularities 3.
Proof idea / technique: The proof first treats the leafwise-orientable case by constructing a 3-manifold with an expansive flow and proving compactness from the finite-rectangle and uniform-hyperbolicity hypotheses. The general case passes to a 2-fold orientation branched cover, uses veering triangulations after removing finitely many closed orbits, then applies Dehn filling to return to a closed manifold. The final step checks that the veering triangulation recovers the original bifoliated plane equivariantly 3.
2. Kauffman bracket skein homology from Heegaard splittings
Authors: Yin Tian, Xiao Wang. Tags: math.GT, math.QA. arXiv: 2606.16577 4.
Abstract: The paper extends the Kauffman bracket skein module of 3-manifolds to a combinatorial homology theory. The resulting homology depends on Heegaard splittings of 3-manifolds 4.
Main result: The Heegaard skein complex is invariant under isotopies and handleslides of the Heegaard data, so its homology is an invariant of Heegaard splittings. The paper also proves that this homology is generally not invariant under stabilization, hence it is not a topological invariant of the underlying closed 3-manifold alone 5.
Proof idea / technique: The authors turn Przytycki-style handle-slide presentations of skein modules into a chain complex whose boundary maps are built from handle-sliding relations. They verify the presimplicial identities, construct explicit chain isomorphisms under choices and 2-handle slides, analyze stabilization by building a retraction, and compute the genus-one lens-space case using Chebyshev polynomials 5.
3. Quantization of contact 3-manifolds and the Reeb gravitational field
Author: Ali M. Elgindi. Tags: math.GT, math-ph. arXiv: 2606.16529 6.
Abstract: The paper proposes a canonical geometric quantization scheme for closed contact 3-manifolds using an explicit embedding into
C^3. The contact structure becomes holomorphic along a complex-tangent link, and Stein extension produces a holomorphic line bundle whose restriction defines a finite-dimensional Hilbert space. Under a Sasakian assumption, the Reeb vector field is interpreted as a geodesic, time-Killing field modeling Einstein gravity over the manifold 6.Main result: The proposed Hilbert space is finite-dimensional, with dimension computed as a sum of degrees over components of the complex-tangent link. The paper also proves that the Reeb vector field is geodesic for the Webster metric, and frames the associated Picard-class construction as an invariant able to distinguish tight contact structures on
T^3 7.Proof idea / technique: The finite-dimensionality statement uses Riemann--Roch on each genus-zero component of the binding after constructing the line bundle by Stein extension. The Reeb-field statement follows from the Webster metric definition and the Reeb equations. The physical interpretation is then attached to the geometric package: line-bundle curvature for the electromagnetic part, and the Sasakian Reeb field for the gravitational part 7.
4. Satellites and telescopes: a concordance formula for bordered Floer homology
Authors: Gary Guth, Sungkyung Kang. Tag: math.GT. arXiv: 2606.16120 8.
Abstract: The paper starts from the fact that a knot's knot Floer complex determines the bordered Floer homology of its complement, and conversely. It proves an analogous formula relating certain knot Floer chain endomorphisms to type-D endomorphisms in bordered Floer homology, up to a one-dimensional ambiguity. As an application, satellite concordance maps can be computed combinatorially from the original concordance map and the bordered Floer homology of the pattern complement 8.
Main result: The authors construct a combinatorial correspondence from locally symmetric chain endomorphisms of the
CFK_R knot Floer complex to type-D endomorphisms of the bordered invariant of the knot complement, modulo an explicitly computable class theta^-_K. For a self-concordance and a satellite pattern, the resulting satellite cobordism map is computable up to conjugation from finite algebraic models 9.Proof idea / technique: The proof makes the Lipshitz--Ozsváth--Thurston formula functorial, restricts to locally symmetric homogeneous endomorphisms to fix a cardinality mismatch, then compares the new map with a previously defined reverse map. Hypercubes and telescopes build the knot Floer model, the bordered formula is extended from objects to morphisms, and the
theta^-_K class measures the remaining ambiguity 9.5. Two-variable parity polynomial for virtual knotoids
Authors: Siqi Ding, Suo Gao, Fengchun Lei, Fengling Li, Andrei Vesnin. Tag: math.GT. arXiv: 2606.15539 10.
Abstract: The paper introduces a two-variable parity polynomial invariant for oriented virtual knotoid diagrams. The invariant uses the parity of classical crossings, distinguishing even and odd crossings in its polynomial definition. The authors prove basic properties, show examples not separated by the odd writhe or affine index polynomial, prove it is a Vassiliev invariant of order one, and relate it to the Petit gluing invariant 10.
Main result: For an oriented virtual knotoid diagram
D, the polynomial sums even classical crossings into the x variable and odd classical crossings into the y variable, with each term weighted by the crossing sign, an intersection index, and an affine-index-type exponent. The resulting Laurent polynomial is invariant under generalized Reidemeister moves 11.Proof idea / technique: Invariance is checked move by move. A first Reidemeister move contributes a crossing with zero intersection index, the two crossings in a second Reidemeister move cancel because they have opposite sign but matching weight, and the third Reidemeister move preserves sign, weight, parity, and intersection index. Pure virtual and mixed moves leave the relevant classical-crossing data unchanged 11.
6. Rigidity theorems for the Weyl problem of convex surfaces in hyperbolic 3-space
Author: Xinrong Zhao. Tags: math.GT, math.DG. arXiv: 2606.15402 12.
Abstract: The paper studies rigidity of non-compact convex sets in hyperbolic 3-space. It proves that intrinsic isometries between boundaries of certain non-compact closed convex subsets extend to global ambient isometries, under hypotheses on their ideal boundaries. The result generalizes a recent rigidity theorem by allowing disk components in the ideal boundary and gives a uniqueness result for a Weyl problem for convex surfaces in hyperbolic 3-space 12.
Main result: If two closed non-compact convex subsets of hyperbolic 3-space have intrinsically isometric boundaries and ideal boundaries of the specified circle-type form, the boundary isometry extends to an ambient hyperbolic isometry. A second theorem allows a finite circle-type part plus a set of zero one-dimensional Hausdorff measure, and the paper derives a discrete Schwarz lemma as a corollary 13.
Proof idea / technique: The proof uses the Pogorelov map to turn the hyperbolic problem into a Euclidean rigidity problem for locally convex radial surfaces. It fills holes in the resulting surfaces through perturbation and gluing; because infinitely many holes may appear, the construction proceeds transfinitely and takes limits at limit ordinals. Pogorelov's rigidity theorem for compact convex bodies then supplies the final congruence statement 13.
7. Fox's trapezoidal conjecture for the four-strand Turk's head knots
Author: Suman Saurabh. Tag: math.GT. arXiv: 2606.15256 14.
Abstract: For the four-strand Turk's head knot
Th(4,2n+1), earlier work reduced the Alexander polynomial, after the substitution t=-z, to a product involving a core polynomial D_n(z). This paper proves log-concavity of the coefficient sequence of D_n(z), using a four-block smoothing theorem for products of reciprocal quartics and an exact integer-arithmetic positivity certificate. It follows that Fox's trapezoidal conjecture holds for the whole family 14.Main result: For every
n >= 1, the coefficients of D_n(z) are log-concave. Consequently, the absolute values of the Alexander-polynomial coefficients of the four-strand Turk's head knot Th(4,2n+1) form a trapezoidal sequence 15.Proof idea / technique: The paper pairs trigonometric quadratic factors into reciprocal quartics, proves a generalized four-block smoothing theorem by exact positivity certificates, verifies small
n exactly, and groups the large-n product into log-concave blocks. The Hoggar--Keilson--Gerber convolution theorem then passes log-concavity to the full polynomial and hence to the Alexander polynomial 15.8. Rational homology ribbon cobordism is a partial order
Author: William Ghanem. Tag: math.GT. arXiv: 2606.14927 16.
Abstract: The paper proves that rational homology ribbon cobordism defines a partial order on homeomorphism classes of closed, connected, oriented 3-manifolds. More precisely, if
W_0 <= W_1 and W_1 <= W_0, then W_0 and W_1 are orientation-preservingly homeomorphic 16.Main result: The antisymmetry statement resolves the conjectural missing part needed to make rational homology ribbon cobordism a partial order on closed oriented 3-manifolds 17.
Proof idea / technique: The proof first strips off and matches
S^1 x S^2 summands, reducing to manifolds without such summands. It then shows the relevant rational homology ribbon cobordisms are integer homology cobordisms, compares Grushko decompositions of fundamental groups, and matches irreducible summands. Aspherical, spherical non-lens, and lens-space summands are handled with the corresponding rigidity or homology-cobordism inputs 17.9. P-categories and functions with degenerate singular submanifolds
Author: Olha Bondar. Tags: math.GT, math.AT, math.DG, math.DS, math.FA. arXiv: 2606.14809 18.
Abstract: The paper studies smooth functions on manifolds whose critical-point sets are disjoint unions of submanifolds, all diffeomorphic to a fixed manifold
P; these submanifolds need not be non-degenerate. It gives sufficient conditions for the existence of functions with a minimal number of such submanifolds. The arXiv posting is a translation from Ukrainian of a PhD dissertation defended in 1993 at the Institute of Mathematics of the National Academy of Sciences of Ukraine 18.Main result: The dissertation introduces
P-category and p-length as analogues of Lyusternik--Shnirelman category and cohomological length for functions whose singular sets are P-type submanifolds. It proves lower bounds of the form P-cat(M) >= long^p(M)+1 for manifolds without boundary and P-cat(M) >= long^p(M) for manifolds with boundary, then applies these estimates to products of spheres, tori, and low-dimensional manifolds 19.Proof idea / technique: The argument generalizes the standard category-length obstruction: define categorical sets that contract to a copy of
P, define p-length by restricting nonzero cup products according to codimension, and use cohomological products to force lower bounds on the number of critical submanifolds. Later chapters construct functions realizing or approaching these bounds in explicit manifold families 19.10. Quasimorphisms and pseudo-Anosov flows
Authors: Danny Calegari, Jonathan Zung. Tags: math.DS, math.GR, math.GT. arXiv: 2606.14889 20.
Abstract: The paper describes two links between quasimorphisms and pseudo-Anosov flows without perfect fits on closed hyperbolic 3-manifolds. First, every such flow has adapted quasimorphisms whose coarse restrictions to lifted flowlines are uniform quasi-isometries to
R, and these adapted quasimorphisms form an open convex cone. Second, the authors bound the exponential growth rate of closed orbits, in both hyperbolic and word metrics 20.Main result: For a pseudo-Anosov flow without perfect fits on a closed hyperbolic 3-manifold, adapted uniform quasimorphisms exist and form an open convex cone. The number of closed orbits grows at a rate strictly below the growth rate of the fundamental group, both for word metrics and for the hyperbolic metric 21.
Proof idea / technique: The adapted-quasimorphism construction uses the geometry of flowlines in the universal cover and hyperbolic-group quasimorphism tools. For the word-metric growth estimate, the key contrast is probabilistic: a random group element has quasimorphism value concentrated near zero, while elements represented by flow orbits have adapted quasimorphism value growing linearly. The hyperbolic-metric estimate uses large deviations and a geometric proof involving endpoint sets and Hausdorff dimension bounds from CaTherine wheels 21.
Reading order suggestion
If you only have 30 minutes, start with Mann--Taylor for the broad 3-manifold dynamics classification result, Guth--Kang if your work touches knot Floer or satellites, and Saurabh if Alexander-polynomial coefficient shape is your current concern. The shortest direct hit is Ghanem: the statement is clean, and the proof outline tells you exactly which 3-manifold decomposition inputs do the work.
References
- 1
- 2arXiv:2606.16588
arxiv.org
- 3Mann--Taylor HTML version
arxiv.org
- 4arXiv:2606.16577
arxiv.org
- 5Tian--Wang HTML version
arxiv.org
- 6arXiv:2606.16529
arxiv.org
- 7Elgindi HTML version
arxiv.org
- 8arXiv:2606.16120
arxiv.org
- 9Guth--Kang source introduction
arxiv.org
- 10arXiv:2606.15539
arxiv.org
- 11Ding et al. source file
arxiv.org
- 12arXiv:2606.15402
arxiv.org
- 13Zhao HTML version
arxiv.org
- 14arXiv:2606.15256
arxiv.org
- 15Saurabh HTML version
arxiv.org
- 16arXiv:2606.14927
arxiv.org
- 17Ghanem HTML version
arxiv.org
- 18arXiv:2606.14809
arxiv.org
- 19Bondar HTML version
arxiv.org
- 20arXiv:2606.14889
arxiv.org
- 21Calegari--Zung HTML version
arxiv.org
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