arXiv math.GT digest — 02 Jun 2026 (8 papers)

Today's digest covers 8 new submissions from the arXiv math.GT listing for Tuesday, 2 June 2026. Topics range from symmetric ribbon numbers of low-crossing knots and trisection diagrams of elliptic surfaces, to real-analytic lifts of Thurston–Tsuboi foliations, new quandle families from Pin groups, and a systematic study of projections of compact sets via Baire category methods.

arXiv:2606.02390 · Sajid Raihan Akash, Eric Corrado, Bishop Placke, Sam Sanketh, Nick Starns, Anok Timothy, Alexander Zupan · 31 pp., 32 figs., 5 tables 1

Abstract (verbatim). Every knot that admits a symmetric union presentation bounds an immersed ribbon disk in , while the converse is an open problem due to Christoph Lamm. The symmetric ribbon number of $K$ is the minimum number of ribbon singularities in any symmetric ribbon disk bounded by $K$. In this paper, we undertake a systematic investigation of symmetric ribbon numbers of knots with at most 12 crossings. Along the way, we exhibit novel lower bounds for arising from knot determinants, Alexander polynomials, Jones polynomials, and Kauffman polynomials.

Key result. A complete computation (or sharp lower bound) of the symmetric ribbon number for every knot with crossing number at most 12 — a tabulation covering all knots through .

Techniques. Four independent families of lower bounds are introduced: one from the knot determinant, one from the Alexander polynomial, one from the Jones polynomial, and one from the Kauffman polynomial. The interaction of these algebraic invariants with the combinatorial structure of symmetric union presentations is the core technical input.

Proof idea. For an upper bound one constructs explicit symmetric ribbon disks; for the lower bounds, one shows that the parity and divisibility data extracted from each polynomial limits how few ribbon singularities a symmetric ribbon disk can have. Comparing the gap between upper and lower bounds determines precisely in many cases and constrains it tightly in the remainder.

arXiv:2606.02159 · Tsukasa Isoshima · 19 pp., 29 figs. 2

Abstract (verbatim). Lambert-Cole and Meier showed that the elliptic surface $E(n)$ admits a $(12n-2,0)$-trisection, considering the property that $E(n)$ is a certain double branched cover of , which is a minimal genus trisection. In this paper, we clarify a way to construct an explicit $(12n-2,0)$-trisection diagram of $E(n)$ from its handle diagram arising from its Lefschetz fibration.

Key result. An explicit, constructive trisection diagram for $E(n)$ of genus $12n-2$, realising the minimal genus established by Lambert-Cole–Meier and making that abstract existence result concrete and diagrammatically accessible.

Techniques. Lefschetz fibration handle diagrams for $E(n)$ serve as the starting point; one translates handle moves into trisection diagram moves using the double branched cover structure over .

Proof idea. The branched cover picture supplies the trisection abstractly; Isoshima traces through the handle calculus explicitly so that each vanishing cycle in the Lefschetz fibration corresponds to a specified arc or curve in the trisection diagram. The result is a combinatorial recipe that works uniformly for all $n$.

arXiv:2606.02023 · Yuichi Kabaya · 18 pp. 3

Abstract (verbatim). We introduce two families of quandles arising from Coxeter quandles. One, which we call a double covering, is the set of roots with binary operation defined by using the negatives of reflections. The double covering is realized as a conjugation quandle in a Pin group. The other, which we call a rotational quandle, is the set of some right angle rotations in the Coxeter group of type with binary operation given by conjugation. We determine their inner automorphism groups, and observe that they are quite similar.

Key result. Two new quandle families — the double-covering quandle (roots with negated-reflection operation, living in a Pin group) and the rotational quandle (right-angle rotations in type- Coxeter groups) — together with a complete computation of their inner automorphism groups.

Techniques. The Pin group realization allows one to use the rich algebraic structure of Clifford algebras to identify the conjugation quandle structure; for rotations, Coxeter group combinatorics tracks the binary operation.

Proof idea. Both quandles are shown to have isomorphic inner automorphism groups despite arising from different geometric data, suggesting a unifying structural pattern. The key step is relating the inner automorphism action to the Weyl group action, enabling a uniform determination of the automorphism group in both cases.

arXiv:2606.01721 · Yongbin Zhou · 33 pp., 1 fig. 4

Abstract (verbatim). We use de Sitter space to construct a concrete model for the measured wall structure of finite-dimensional hyperbolic space in hyperbolic model , which will induce a medianization . We get an upper bound for the Hausdorff distance between and .

Key result. An explicit upper bound on the Hausdorff distance between $n$-dimensional hyperbolic space and its medianization, using de Sitter space to concretely model the measured wall structure.

Techniques. The key tool is the duality between hyperbolic space and de Sitter space, which provides a natural family of half-spaces (walls) and thus a measured wall structure on . The medianization is then a median space carrying the same coarse geometry, and the Hausdorff distance quantifies how faithfully this passage preserves metric data.

Proof idea. Zhou identifies the walls of as level sets of certain de Sitter functions, computes the wall measure explicitly, and from this derives the median metric on . Comparing geodesic distances in with the -type paths in along these walls yields the bound.

arXiv:2606.01119 · Mohammad Alattar, Lewis Tadman 5

Abstract (verbatim). In this paper we further develop the theory of MCS spaces. Our main result shows that MCS spaces, as defined by Perelman, are CS sets with respect to their MCS stratification, and that in fact, the intrinsic stratification agrees with the MCS stratification. As a consequence, we improve on Perelman's result and answer affirmatively a question by Fujioka.

Key result. Every MCS space (in Perelman's sense) is a CS set with respect to its natural MCS stratification, and the intrinsic stratification of the underlying space coincides with the MCS stratification. This resolves affirmatively a question posed by Fujioka.

Techniques. The argument draws on the local conical structure of MCS spaces (finite iterated cones over compact spaces) and the general theory of CS sets (spaces admitting locally cone-like charts with controlled links). The coincidence of stratifications is shown by induction on the depth of the stratification.

Proof idea. For each stratum in the MCS stratification, one verifies the CS condition — roughly, that each point has a distinguished neighbourhood homeomorphic to a cone on the link — by using Perelman's definition directly. The key insight is that the MCS structure already encodes the link data needed for CS, so no additional structure is required to promote Perelman's theorem.

arXiv:2606.01035 · Louis H. Kauffman · 36 pp., 35 figs. 6

Abstract (verbatim). We utilize multi-virtual knot theory where there are a multiplicity of virtual crossings to study strict virtual linkoids. In strict virtual linkoid theory, local moves define all virtual moves and Reidemeister moves. In the strict equivalence, no moves, classical or virtual, can transfer an arc across a linkoid endpoint. By taking closures of strict virtual linkoids that are multi-virtual knots and links, we obtain new invariants for strict virtual linkoids. Generalized bracket polynomial invariants and generalized loop bracket polynomial invariants (for planar strict virtual linkoids) are studied in this context. The paper defines virtual polar links where there are degree two nodes in virtual link diagrams across which isotopies are forbidden. The paper shows how multi-virtual theory and its concepts can be applied to obtain invariants for polar virtual links.

Key result. New invariants for strict virtual linkoids, obtained by closing multi-virtual linkoids to multi-virtual knots/links and applying generalised bracket and loop bracket polynomials. Additionally, a theory of virtual polar links (degree-two nodes blocking isotopies) is introduced and shown to admit analogous invariants.

Techniques. Multi-virtual knot theory replaces the single-type virtual crossing with a family of labelled virtual crossings, allowing finer control over which moves are permitted under strict equivalence. Kauffman's bracket polynomial is generalised to this setting, and the resulting invariants detect features invisible to classical virtual knot invariants.

Proof idea. The key observation is that closing a strict virtual linkoid (fixing its endpoints) into a multi-virtual closed link creates a compact object to which the standard bracket machinery applies — but the endpoint-fixing ensures the resulting polynomial retains information about the open linkoid structure that would be lost under ordinary closure.

arXiv:2606.01017 · Teruaki Kitano, Yoshihiko Mitsumatsu, Shigeyuki Morita 7

Abstract (verbatim). Thurston constructed codimension one foliations on thereby proved that the homomorphism induced by the Godbillon-Vey invariant is surjective. By another real analytic construction, he proved that the homomorphism is also surjective where is a space by Haefliger. Tsuboi proved that the former surjection splits so that . He further showed that the subgroup of generated by all the Thurston's constructions coincides with his direct summand . In this paper, we prove that Thurston's second surjection splits and also that the subgroup of generated by all the Thurston's cycles is equal to our direct summand which is a lift of Tsuboi's one. To show this, we modify the arguments of Thurston and Tsuboi by replacing Reeb components with a real analytic construction. We prove certain uniqueness of them by showing acyclicity of the affine group in the Haefliger group . We also prove the existence of a new kind of characteristic class of foliations in .

Key result. The Godbillon-Vey surjection (Thurston's real-analytic version) splits, producing in the real-analytic category. Additionally, a new characteristic class in is constructed.

Techniques. Thurston's original codimension-one foliation constructions on are replaced by real analytic analogues that avoid Reeb components. The splitting is proved by showing the acyclicity of the affine group inside Haefliger's classifying groupoid .

Proof idea. Tsuboi's smooth splitting argument exploits the fact that Thurston's constructions generate a direct summand of the third homology. Kitano–Mitsumatsu–Morita carry this out in the real-analytic category: they replace smooth Reeb components with real analytic ones, verify that the key acyclicity property still holds for the affine subgroup, and then track through Tsuboi's inductive argument. The new characteristic class arises as an obstruction in the comparison between smooth and real-analytic classifying spaces.

arXiv:2606.00639 · Olga Frolkina · 22 pp. 8

Abstract (verbatim). We apply ideas of geometric measure theory and Baire category theory to topological problems, namely, to topological embeddings of compact sets into Euclidean spaces. In 1947, Borsuk constructed a Cantor set in , , such that its projection onto any $(N-1)$-plane contains an $(N-1)$-dimensional ball. This can be strengthened: a desired Cantor set can be obtained from an arbitrary Cantor set by an arbitrarily small isotopy of the space . The question arises: how do the dimensions of the projections of a compact set behave under a typical ambient isotopy or under a typical ambient homeomorphism? (Typical in the sense of the Baire category.) We solve this problem. As a consequence, we get new criteria of tameness and wildness of a Cantor set in terms of its projections. Our main result strengthens Väisälä's theorem (1979) connecting Hausdorff dimension and Shtan'ko embedding dimension. In its turn, Väisälä's theorem extends results of Nöbeling (1931) and Szpilrajn (1937) on relationship between Hausdorff dimension and topological dimension.

Key result. A complete description of how the Hausdorff dimensions of projections of a compact set behave under a typical ambient isotopy and under a typical ambient homeomorphism (typical in the Baire category sense). As corollaries: new tameness/wildness criteria for Cantor sets in via projections, and a strengthening of Väisälä's 1979 theorem linking Hausdorff dimension to Shtan'ko embedding dimension.

Techniques. Geometric measure theory (Hausdorff dimension, rectifiability) is combined with Baire category arguments in the group of ambient homeomorphisms. The Borsuk (1947) construction of a Cantor set with full-dimensional projections is upgraded to an isotopy perturbation result.

Proof idea. For a generic ambient homeomorphism $h$ (in the Baire sense), the Hausdorff dimension of for any fixed projection takes its maximal possible value consistent with the dimension of $X$. The proof identifies a dense set of homeomorphisms achieving this and shows that any homeomorphism outside this set can be perturbed by an arbitrarily small isotopy into it. The tameness/wildness criteria follow by examining which projection dimensions are realised by a given Cantor set.