arXiv math.GT digest — 28 May 2026 (13 papers)

Today's arXiv math.GT listing for 28 May 2026 brings 13 entries: 5 new submissions, 3 cross-list submissions, and 5 replacements. Topics span plane-curve combinatorics, Nielsen realization for 4-manifolds, braid complexity of satellite links, octahedral decompositions of virtual links, intrinsic linking of simplicial complexes, Masur–Veech volumes via DR-cycles, twisted Alexander polynomials and Σ-invariants, twisted string nets for pivotal categories, Seiberg–Witten constraints on embedded surfaces, Morse fundamental groups, stable commutator length in 2-orbifolds, discrete knot theory via lattice-filtered move graphs, and pillowcase-tiled surfaces.

Authors: Boris Shapiro

arXiv: 1

Abstract: We study the minimum number of inflection points among generic immersed closed plane curves with a fixed embedded shadow. The word immersed is essential: a genuinely embedded Jordan curve has inflection minimum zero. For tree-like shadows, inflection criterion converts inflection-free realizability into a finite coorientation problem on the building polygons of the shadow. We sharpen this viewpoint into an exact finite formula for the minimum number of normalized inflections and record a dynamic-programming computation on the block tree. We then push the method beyond the tree-like case. For every embedded shadow the same coorientation model gives a universal lower bound. For a natural larger class, called tree–necklace shadows, in which the non-tree-like blocks are separated annular cycles, the lower bound is exact after imposing an explicit holonomy condition around each necklace. We also record the algorithmic status of the exact minimization problem and formulate a likely NP-hardness problem for unrestricted shadows. Finally, we introduce a related invariant: the minimum possible least multiplicity of the Gauss map, equivalently the smallest guaranteed number of oriented parallel tangencies. This "parallel-tangent load" is controlled by the same inflection folds but is not determined by their number alone.

Key result: An exact combinatorial formula for the minimum number of normalized inflection points of an immersed closed plane curve realizing a given tree-like embedded shadow, computed via dynamic programming on the shadow's block tree. For the broader class of tree–necklace shadows, exactness is recovered by adding a holonomy condition around each necklace cycle. A new invariant — the "parallel-tangent load" measuring the minimum multiplicity of the Gauss map — is introduced and shown to be related to but not determined by the inflection count alone.

Techniques and tools: Shadow combinatorics and the coorientation model on building polygons of planar shadows; block-tree decomposition; dynamic programming; holonomy around necklace cycles; Gauss map analysis; algorithmic complexity arguments (NP-hardness formulation for unrestricted shadows).

Core proof idea: A coorientation on each building polygon of the shadow encodes whether an inflection point is forced at that polygon's boundary arcs. For tree-like shadows the absence of cycles means the coorientation constraints can be satisfied greedily, yielding an explicit minimization formula. For tree–necklace shadows, the single non-trivial topology (annular cycles between tree-like parts) introduces a obstruction; once that holonomy is checked, the tree-like argument extends. The parallel-tangent load is then analyzed by tracking which inflection folds in the Gauss diagram contribute oriented parallel tangencies, showing it depends on a finer structure than the total inflection count.

Authors: Ethan Pesikoff

arXiv: 2

Abstract: Given a smooth, oriented, simply-connected 4-manifold $M$, the homological Nielsen realization problem asks: when does a finite group of isometries preserving the intersection form lift isomorphically to a finite group of orientation-preserving diffeomorphisms? We study this question for the smooth, positive-definite 4-manifolds . Even though every isometry of is induced by some orientation-preserving diffeomorphism, not necessarily of finite order, we show that Nielsen realization is sparse: as , a random subgroup of is asymptotically almost never realizable in ; the same is true for random odd order elements of . We present both positive realization results in certain cases and a range of obstructions to realization in other cases. The proofs combine equivariant connected-sum constructions, fixed-point theory for group actions on 4-manifolds, finite group actions on surfaces, analytic combinatorics, and previous work of Hambleton–Tanase.

Key result: For (the connected sum of $n$ copies of ), homological Nielsen realization is sparse: as , a random subgroup of is asymptotically almost surely not realizable by finite-order diffeomorphisms, despite every individual isometry being induced by some diffeomorphism (not necessarily of finite order). An analogous sparsity result holds for random odd-order elements.

Techniques and tools: Equivariant connected-sum constructions; fixed-point theory for finite group actions on 4-manifolds; finite group actions on surfaces (equivariant topology); analytic combinatorics to count realizable vs. total subgroups asymptotically; Hambleton–Tanase's foundational results on group actions on .

Core proof idea: Hambleton–Tanase classify which finite group actions on are possible, giving an upper bound on realizable subgroups of . Analytic combinatorics then shows that the number of realizable subgroups grows much more slowly than itself (which is essentially ), so the density of realizable subgroups tends to zero. Obstructions come from fixed-point theory (a finite group action on a 4-manifold constrains the characteristic classes of the fixed-point set) and from the structure of the intersection form.

Authors: Thiago de Paiva, Yi Liu, Paolo Piccione

arXiv: 3

Abstract: We study the relationship between the number of full twists in positive braid representations of satellite links and their companion links. We construct infinitely many satellite links that admit positive braid representations with arbitrarily many full twists, while their companion links do not admit any positive braid representation with more than one full twist. This exhibits an unexpected divergence between the braid-theoretic complexity of a satellite link and that of its companion.

Key result: An infinite family of satellite links whose positive braid representations can have arbitrarily many full twists, yet whose companion knots admit positive braid representations with at most one full twist. This is a sharp and unexpected divergence: the full-twist complexity of a satellite link can far exceed that of its companion.

Techniques and tools: Positive braid theory; satellite link constructions (explicit pattern and companion knots in solid tori); twist analysis in braid word presentations; counting full twists via the writhe or braid index.

Core proof idea: The construction takes companion knots with controlled braid representatives (admitting at most one full twist) and satelliting patterns that inject additional full twists. By choosing the pattern and companion carefully, the satellite link inherits extra full twists from the pattern while the companion's braid structure remains bounded. Infinitely many such examples are produced by varying the companion within a suitable family.

Authors: Lecheng Su

arXiv: 4

Abstract: The octahedral decomposition of classical link complements has been considered and utilised by Weeks, Rubinstein, Sakuma etc. It is even more natural to consider the octahedral decomposition of virtual link complements. In this paper, we explore and generalise the octahedral decomposition used in snappy to links in thickened surface. Using the decomposition, we prove nonpositive curvature of the complement and essential-ness of edges in the decomposition.

Key result: The octahedral decomposition (familiar from SnapPy for classical links) extends to alternating links in thickened surfaces . Two structural theorems are proved: (1) the complement of any such link admits a non-positively curved (NPC) cell decomposition coming from the octahedra, and (2) every edge in the decomposition is essential (cannot be collapsed).

Techniques and tools: Polyhedral decomposition theory (octahedral decompositions of link complements); virtual link/thickened-surface topology; CAT(0) / non-positive curvature criteria for cell complexes; arguments analogous to those of Agol–Thurston for classical alternating links.

Core proof idea: For a classical alternating link, the two-colorable checkerboard surfaces guide an octahedral decomposition of the complement. In the thickened-surface setting one uses the analogous checkerboard surfaces for the link diagram drawn on . The faces of the octahedra are identified with combinatorial data from the crossing regions. Non-positive curvature is checked at each vertex and edge of the complex using angle-sum conditions; essentialness of edges follows from the incompressibility/irreducibility of the resulting cell structure.

Authors: Ryo Nikkuni

arXiv: 5

Abstract: For any positive integer $n$, the author previously constructed several minimal simplicial $n$-complexes which necessarily contain a non-splittable two-component link, consisting of an $(n-1)$-sphere and an $n$-sphere, in any embedding into . In this paper, we present an additional simplicial $n$-complex with the same property.

Key result: A new minimal simplicial $n$-complex (for every ) that is intrinsically linked in : every embedding into contains a non-splittable two-component link consisting of an $(n-1)$-sphere and an $n$-sphere. This adds to the previously known list of minimal intrinsically linked $n$-complexes.

Techniques and tools: Simplicial complex combinatorics; intrinsic linking theory in codimension-$n$ (generalizing the classical $n=1$ theory of intrinsic linking in ); van Kampen-type obstruction theory; linking numbers of spheres in Euclidean space.

Core proof idea: The construction exhibits an explicit simplicial $n$-complex $K$ and proves: (a) minimality — removing any simplex allows an embedding with no non-trivial link; (b) intrinsic linking — for any embedding of $K$ into , a specific pair of subcomplexes (topologically and ) must link non-trivially. Part (b) uses a linking number argument: the algebraic intersection of the $n$-sphere with a Seifert hypersurface for the $(n-1)$-sphere is nonzero for any embedding, as forced by the combinatorics of $K$.

Authors: Martin Möller, Miguel Prado

arXiv: 6

Abstract: We show that the completed volumes introduced by Duriev-Goujard-Yakovlev as an approximation to compute Masur-Veech volumes via Witten-Kontsevich's combinatorial classes agrees with the top intersection of the tautological class on the double ramification cycle, computable as a coefficient of a Chiodo class. For the proof we describe the components of the double ramification cycle and their excess intersection classes to the extent seen by the top tautological intersection. This gives a recursion computing completed volumes in terms of volumes appearing in a certain set of level graphs, not only for quadratic differentials. It also completes the work of Duriev-Goujard-Yakovlev solving the technically most involved case of strata with two singularities.

Key result: The "completed volumes" of Duriev–Goujard–Yakovlev (approximate Masur–Veech volumes via combinatorial classes) are identified with the top tautological intersection on the double ramification (DR) cycle, computable as a coefficient of a Chiodo class. This identification implies a recursion for completed volumes via level graphs, extending beyond quadratic differentials and resolving the hardest case (strata with two singularities).

Techniques and tools: Moduli of differentials; double ramification cycle (Hain–Grushevsky–Zakharov; Janda–Pandharipande–Pixton–Zvonkine); Witten–Kontsevich combinatorial classes; Chiodo classes; intersection theory on moduli of curves; level-graph combinatorics for strata.

Core proof idea: The DR cycle carries a natural filtration by level graphs; the authors carefully identify which components contribute to the top tautological intersection and compute their excess intersection contributions. This identification matches the combinatorial definition of completed volumes, yielding the desired equality and a recursive formula computable from simpler volume data.

Authors: Yongqiang Liu, Alexander I. Suciu

arXiv: 7

Abstract: The twisted Alexander polynomials of a space, associated to a linear representation of the fundamental group, are non-abelian refinements of the classical Alexander polynomial from knot theory. In this paper, we show that they arise naturally from a new family of invariants — the twisted homology jump loci — which extend the rank-one characteristic varieties to higher-rank local systems. Using the tropical geometry of these twisted loci, we obtain sharper upper bounds for the Bieri–Neumann–Strebel–Renz (BNSR) -invariants. For compact orientable 3-manifolds with toroidal or empty boundary, we use a theorem of Friedl–Vidussi to show that the closure of the union of these twisted tropical bounds is sharp: it recovers the fibered faces of the Thurston norm ball exactly, a result that fails without twisting. For compact Kähler manifolds, we prove that the -invariant of is controlled by the orbifold fibrations of $X$ for any representation , and that the twisted Alexander polynomial must equal $0$ or $1$. Both results provide obstructions to geometric realizability that are strictly stronger than their classical untwisted counterparts.

Key result: A new family of invariants, the twisted homology jump loci, is introduced and shown to encode twisted Alexander polynomials naturally. The tropical geometry of these loci gives sharp bounds on BNSR Σ-invariants. For 3-manifolds, the twisted tropical bounds exactly detect fibered faces of the Thurston norm ball (sharper than untwisted bounds). For Kähler manifolds, the Σ¹-invariant is controlled by orbifold fibrations and the twisted Alexander polynomial is 0 or 1.

Techniques and tools: Characteristic varieties and resonance varieties; twisted Alexander polynomials; tropical geometry (tropicalization of algebraic varieties); BNSR Σ-invariants; Thurston norm and fibered 3-manifolds (Friedl–Vidussi theorem); orbifold fibrations of Kähler manifolds; representation theory of fundamental groups.

Core proof idea: The twisted homology jump loci are defined by analogy with classical characteristic varieties but using local systems of higher rank. The tropicalization of these loci captures the directions in which -invariants can be non-trivial. For 3-manifolds, the Friedl–Vidussi result translates fibering conditions into vanishing of twisted Alexander polynomials; using all representations simultaneously, the union of tropical bounds recovers exactly the non-fibered faces, completing the detection of fibered faces. For Kähler manifolds, the result that or $1$ follows from the algebraic geometry of the Albanese variety combined with the orbifold fibration structure.

Authors: Benjamin Haïoun, William Stewart, Filippos Sytilidis

arXiv: 8

Abstract: We develop a graphical calculus for monoidal categories equipped with twisted pivotal structures, which are a generalization of pivotal structures originating from the study of orientation structures in the context of the Cobordism Hypothesis. This graphical calculus depends on a possibly singular foliation, and we use it to construct twisted string net modules for surfaces equipped with a Morse function or a Morse foliation. We prove that, despite the apparent dependence on this Morse function, the twisted string net modules assemble in an oriented categorified 2-TQFT. We study when the twisted string net module of the 2-sphere vanishes, relate it to the distinguished invertible object for finite tensor categories and exhibit examples of non-unimodular finite tensor categories with non-vanishing twisted string net module on the 2-sphere. This vanishing is expected to be the main obstruction for extending our categorified 2-TQFT to a non-compact 3-TQFT.

Key result: A fully developed graphical calculus for twisted pivotal categories, and the construction of twisted string net modules for surfaces with Morse foliations. These modules assemble into an oriented categorified 2-TQFT (independent of the Morse function). The vanishing criterion for the string net module of is identified with the distinguished invertible object; examples of non-unimodular finite tensor categories with non-vanishing module are exhibited.

Techniques and tools: String net models (Levin–Wen type); pivotal and twisted pivotal structures on monoidal categories; Cobordism Hypothesis framework; Morse foliations on surfaces; (oriented) 2-TQFTs; distinguished invertible object (Etingof–Gelaki–Nikshych–Ostrik); finite tensor categories; categorified TQFT.

Core proof idea: The key insight is that a twisted pivotal structure provides exactly the data needed to evaluate string net diagrams in the presence of orientation reversals dictated by a foliation on the surface. The Morse function determines a foliation but the string net module is shown to be invariant under Morse modifications (handle slides and cancellations), giving independence. The vanishing criterion follows from the fact that the module is a 1-dimensional module over the center of the category, which is controlled by the distinguished invertible object; unimodularity (triviality of this object) implies vanishing.

Authors: David Baraglia

arXiv: 9

Journal: Math. Z. Volume 313, article number 39 (2026)

Abstract: We calculate the Seiberg-Witten invariants of branched covers of prime degree, where the branch locus consists of embedded spheres. Aside from the formula itself, our calculations give rise to some new constraints on configurations of embedded spheres in 4-manifolds. Using similar methods, we also obtain new constraints on embeddings of real projective planes and spheres with a cusp singularity. Moreover, we show that the existence of certain configurations of surfaces would give rise to 4-manifolds of non-simple type. Our proof makes use of equivariant Seiberg-Witten invariants as well as a gluing formula for the relative Seiberg-Witten invariants of 4-manifolds with positive scalar curvature boundary.

Key result: A formula for the Seiberg–Witten invariants of prime-degree branched covers with branch locus a configuration of embedded spheres. New constraints on: (a) configurations of embedded spheres in 4-manifolds; (b) embeddings of and spheres with cusp singularities. Certain surface configurations are shown to force the ambient 4-manifold to be of non-simple type.

Techniques and tools: Seiberg–Witten invariants; equivariant Seiberg–Witten theory; branched cover formulas; gluing formulas for SW invariants on 4-manifolds with positive scalar curvature boundary; adjunction inequalities; non-simple type obstructions.

Core proof idea: For a prime-degree branched cover branched over embedded spheres , one computes by relating it to equivariant SW invariants of $X$ (relative to the symmetry). The equivariant contribution near the branch locus is controlled by a gluing formula; tracking the resulting constraints on the basic classes of $X$ yields restrictions on the topology of $B$ (genus, self-intersection). Analogous arguments handle embeddings and cusp singularities via appropriate local models.

Authors: Salammbo Connolly

arXiv: 10

Journal: International Journal of Mathematics (2026)

Abstract: We study properties of the continuation map for the Morse fundamental group associated to a Morse-Smale pair $(f,g)$ on a manifold $M$. We get a morphism between and and show that it is functorial. We also define the morphism in the case of Morse data over different manifolds, thanks to the use of grafted trajectories. Finally, given an interpolation function on between two Morse functions (used for example to define the continuation map), we study the Morse fundamental group associated to that function and show that it is isomorphic to a relative fundamental group on .

Key result: A functorial continuation map for the Morse fundamental group , giving morphisms between the Morse fundamental groups associated to different Morse-Smale pairs (even over different manifolds via grafted trajectories). The Morse fundamental group of an interpolation function is shown to be isomorphic to a relative fundamental group of .

Techniques and tools: Morse–Smale theory; Morse fundamental groups (Barraud–Cornea type constructions); continuation maps in Floer/Morse-theoretic settings; grafted trajectories; relative homotopy groups; cobordism-type arguments for Morse data over .

Core proof idea: The continuation map between Morse fundamental groups is defined by counting rigid broken trajectories in a 1-parameter family of Morse-Smale pairs. Functoriality follows from the usual gluing argument for 2-parameter families (the composition of two continuation maps coincides with the map for the concatenated homotopy). For different manifolds, grafted trajectories provide a mechanism to "connect" the two Morse flows. The isomorphism between and a relative fundamental group is established by constructing an explicit identification of the relevant moduli spaces with path spaces.

Authors: Lvzhou Chen, Nicolaus Heuer

arXiv: 11

Abstract: We show a uniform spectral gap of stable commutator length for all compact hyperbolic 2-orbifolds relative to the peripheral subgroups. Except for the case of a sphere with three cone points, we have an explicit uniform gap $1/36$. These estimates are needed in understanding stable commutator length in 3-manifolds. Our methods use explicit quasimorphisms for the generic case, and use hyperbolic geometry (pleated surfaces) for the exceptional case of a sphere with three cone points.

Key result: A uniform lower bound of on the stable commutator length (scl) spectral gap for all compact hyperbolic 2-orbifolds (relative to peripheral subgroups), with the sole exception of a sphere with three cone points. This uniform gap is a key input for understanding scl in 3-manifolds.

Techniques and tools: Stable commutator length (scl); quasimorphisms (explicit constructions giving scl lower bounds via the Bavard duality); hyperbolic geometry; pleated surfaces (for the exceptional sphere-with-3-cone-points case); orbifold fundamental groups; peripheral subgroups.

Core proof idea: For the generic case, one constructs explicit quasimorphisms on the fundamental group of the orbifold that are bounded on the peripheral subgroups but detect scl; by Bavard duality, the existence of such quasimorphisms with controlled defect gives a lower bound of $1/36$. For the exceptional sphere with three cone points (the most degenerate orbifold), quasimorphisms alone are insufficient; instead pleated surfaces in the hyperbolic orbifold provide the geometric lower bound directly.

Authors: Makoto Ozawa

arXiv: 12

Abstract: We introduce lattice-filtered move graphs as finite-state experimental models for knot types. At level N, vertices are lattice-polygon representatives of a fixed knot type with lattice length at most N, modulo orientation-preserving lattice isometries, and edges are prescribed local moves. Connected components of these graphs are discrete analogues of admissible components in ropelength-filtered knot spaces. The first level at which two initial components become connected defines a discrete merge scale; after subtracting the birth level, the resulting function is an ultrapseudometric whenever the relevant initial components eventually merge. The general framework is move-system independent. We then specialize to the simple cubic lattice and BFACF-type moves, treating BFACF as a chosen local move system rather than as a complete lattice-isotopy calculus. The main seed-generated computation uses a 30-edge simple cubic lattice seed for the figure-eight knot and its reflected mirror seed. With mirror symmetries not identified, the two BFACF components are separated at N=30 and merge at N=32. We also extract an explicit merge certificate: a 21-state, 20-move BFACF path through a 32-edge connecting state. Thus, relative to the supplied seeds and the BFACF move system, the seed-generated merge scale is 32.

Key result: Lattice-filtered move graphs are introduced as a rigorous framework for discrete knot theory, with a well-defined "merge scale" that is an ultrapseudometric. Applied to the figure-eight knot in the simple cubic lattice (BFACF moves), the two mirror components are separated at $N=30$ and merge at $N=32$, with an explicit 21-state merge certificate.

Techniques and tools: Lattice knots (simple cubic lattice); BFACF moves (local lattice isotopy moves); graph-theoretic methods for connected components of move graphs; ultrapseudometric construction; computational enumeration (seed-generated computation); merge certificate extraction.

Core proof idea: The framework filters the space of lattice representatives by length $N$, yielding a sequence of finite graphs whose connected components coarsen as $N$ increases. The merge scale between two components is the first $N$ at which they join; subtracting birth levels gives an ultrapseudometric. For the figure-eight knot, an exhaustive BFACF computation at each lattice length from 30 to 32 shows separation at 30 and connection at 32, with an explicit merge path serving as a constructive proof of the upper bound.

Authors: Malak Abdalla, Gabriela Brown

arXiv: 13

Journal: Geometriae Dedicata 220, 38 (2026)

Abstract: Apisa-Wright conjectured that all branched covers of quadratic differentials in with at most one cylinder in each direction are cyclic covers. We provide infinitely many counterexamples to this conjecture.

Key result: An infinite family of branched covers of quadratic differentials in the stratum that have at most one cylinder in each horizontal and vertical direction, yet are not cyclic covers. This disproves the Apisa–Wright conjecture in this stratum.

Techniques and tools: Translation surfaces and quadratic differentials; pillowcase covers ( is the pillowcase quotient of the torus); cylinder decompositions; Veech groups; branched cover theory for half-translation surfaces; combinatorial classification of cylinder structures.

Core proof idea: The construction starts with the pillowcase surface (the square torus with quotient, having four simple poles). Pillowcase covers are built by choosing a coloring of the squares; the 1-cylinder condition is verified combinatorially (at most one cylinder in each direction). Non-cyclicity is demonstrated by showing that the deck transformation group of the cover is non-cyclic, using the combinatorial data of the coloring. Infinitely many distinct colorings satisfying both conditions are produced explicitly.