Papers 1–2 at a glance
Metadata for the first two submissions in today's listing
arXiv math.GT digest — 10 Jun 2026 (6 papers)
Six new primary submissions from the arXiv math.GT listing of Wednesday 10 June 2026: biquandle arrow weight quiver invariants (Nelson–Sakurai), Stein fillings via relative trisections of genus ≤ 2 (Asano–Takahashi), coloring-allowed invariants and 4-phases functions of knotoids (Chen–An–Li), τ and ε formulas for braided satellite knots plus progress on Hedden's conjecture (Eldridge), realizability of tetrahedral-type sphere-covering branch data via dessins d'enfants (Adrianov–Kreines), and group-theoretic analysis of magic leatherworking braids (Hobkirk–Hollingsworth–Matsumoto–Reid).
Research Brief
Six papers appeared on the arXiv math.GT listing for Wednesday 10 June 2026, spanning knot invariants from multiple angles — biquandle enhancements, concordance invariants of satellite knots, knotoid coloring theory, Stein fillings via trisections, sphere-covering realizability, and a surprising application to leatherworking braids.
1. Biquandle arrow weight quiver representations
arXiv:2606.11073 · Sam Nelson, Migiwa Sakurai · 11 pp. 1
Original abstract. We define an infinite family of quiver representation-valued invariants of classical and virtual knots associated to a choice of data vector consisting of a biquandle, abelian group, set of biquandle arrows weights with values in the abelian group, coefficient ring and set of biquandle endomorphisms. As an application we extract four new polynomial invariants as decategorifications. We provide examples to show that these invariants are proper enhancements of the biquandle counting invariant and biquandle coloring quiver.
Key result. An infinite family of quiver representation-valued knot invariants is constructed, parametrised by a "data vector" (biquandle, abelian group, arrow weights, coefficient ring, endomorphisms). Four new polynomial invariants are extracted by decategorifying these representations, and all are shown to be proper enhancements of both the biquandle counting invariant and the biquandle coloring quiver.
Techniques and tools. The core machinery is the theory of biquandle arrow weights (a cocycle-style enhancement of biquandle colorings) lifted to the setting of quiver representations over a coefficient ring. Decategorification extracts polynomial invariants by passing to numerical invariants of the representation categories.
Proof idea. For each data vector, the set of biquandle colorings of a knot diagram yields a quiver; the arrow weights then define a functor from this quiver to modules over the coefficient ring. Invariance under Reidemeister moves is verified by checking that the module structure transforms coherently. Proper enhancement is demonstrated by exhibiting knot pairs distinguished by the new invariants but not by the biquandle counting invariant.
2. Stein fillings of planar contact 3-manifolds with relative trisection genus 2
arXiv:2606.10764 · Nobutaka Asano, Natsuya Takahashi · 19 pp., 17 figs. 2
Original abstract. We study Stein fillings of planar contact 3-manifolds under certain constraints on their relative trisections. We partially classify the diffeomorphism types of such fillings admitting relative trisections with genus at most 2.
Key result. A partial classification of diffeomorphism types of Stein fillings of planar contact 3-manifolds is achieved, under the constraint that the filling admits a relative trisection of genus ≤ 2. This extends the trisection-based approach to Stein filling classification into the genus-2 range.
Techniques and tools. Relative trisections of 4-manifolds with boundary (following Gay–Kirby and related work) provide the structural framework. Planarity of the contact structure on the boundary constrains the handle decomposition. The classification uses open book decompositions compatible with the contact structure together with the genus bound on the relative trisection.
Proof idea. The genus bound limits the complexity of the relative trisection diagram. The authors enumerate possible trisection diagrams consistent with a planar open book on the boundary, then determine which give rise to distinct diffeomorphism types of Stein fillings — cross-checking with known filling classifications for specific planar contact manifolds via Legendrian surgery diagrams.
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3. Coloring-allowed invariants and the 4-phases functions of knotoids
arXiv:2606.10362 · Haocong Chen, Jiacheng An, Fengling Li · 25 pp., 22 figs. 3
Original abstract. In recent years, numerous polynomial invariants of knotoids have been constructed, some of which are defined with the signs of the crossings. In this paper, the coloring-allowed invariants of planar knotoids, which is a class of planar knotoid invariants defined with the coloring number are introduced. We demonstrate several basic properties of the coloring number and some examples of coloring-allowed invariants, among which the 4-phases functions are discussed in detail, including their invariance and properties.
Key result. A new class of invariants of planar knotoids, called coloring-allowed invariants, is introduced. These use the coloring number as a gate — an invariant is "coloring-allowed" relative to a given coloring structure. Among these, the 4-phases functions are worked out in detail and proved to be genuine knotoid invariants.
Techniques and tools. The paper adapts Fox p-coloring and quandle-coloring methods to the knotoid setting (open-ended knot diagrams in the plane or on the sphere, as introduced by Turaev). The coloring number — a count of valid colorings — controls when the invariant is defined. Properties of the 4-phases functions are established by checking invariance under knotoid Reidemeister moves.
Proof idea. The coloring number is shown to be well-defined as a knotoid invariant by verifying it is unchanged under each knotoid move. The 4-phases functions are then defined using the coloring data and shown to remain unchanged under each move. Examples separate knotoids that classical invariants do not distinguish.
4. The Ozsváth–Szabó τ-invariant of braided satellites
arXiv:2606.10351 · Alex Eldridge · 14 pp., 8 figs. 4
Original abstract. We give formulas for the τ and ε concordance invariants of satellite knots whose patterns are braided, meaning they wind around the solid torus without reversing. Our methods lead us to define the class of squeezed patterns, analogous to squeezed knots as defined by Feller–Lewark–Lobb. We show that all braided patterns are squeezed, and we give a τ formula for squeezed patterns as well. Towards a conjecture of Hedden, we show that no squeezed pattern, and thus no braided pattern, with winding number at least 2 induces a homomorphism on the concordance group.
Key result. Closed formulas for both τ and ε of a satellite knot $P(K)$ are obtained when the pattern $P$ is braided (winds monotonically around the solid torus). A new class — squeezed patterns, analogous to Feller–Lewark–Lobb's squeezed knots — is defined, and braided patterns are shown to be squeezed. As a consequence, no braided pattern with winding number ≥ 2 induces a group homomorphism on the smooth concordance group, making progress on a conjecture of Hedden.
Techniques and tools. Heegaard Floer homology — specifically the τ and ε invariants from knot Floer homology — is the main tool. The braid structure of the pattern is exploited via the v-invariant and the satellite formula of Hom–Wu. The squeezed condition is formulated in terms of the knot Floer complex; analogies with Feller–Lewark–Lobb's theory guide the definitions.
Proof idea. For a braided pattern, the monotone winding condition implies a specific form of the knot Floer complex of the satellite, from which τ and ε can be read off by explicit computation. The squeezed property then follows from a comparison of the ε-filtration structure. The non-homomorphism conclusion for winding number ≥ 2 patterns follows from the τ formula showing the τ of the satellite cannot be linear in τ(K).
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5. Almost regular coverings of the sphere: realizability — I. Tetrahedral case
arXiv:2606.10079 · Nikolai M. Adrianov, Elena M. Kreines 5
Original abstract. We prove the realizability of genus-0 branch data of the form and , where $a$, $b$, $c$ are not divisible by 2, 3, 3 respectively. The proof uses an explicit combinatorial description of coverings of the sphere branched over 3 points via dessins d'enfants. As a corollary, we establish realizability for a broader class of branch data with more critical values.
Key result. Two infinite families of genus-0 branch data for coverings of branched over 3 points are proved realizable. The "tetrahedral case" refers to branch data whose local monodromy types correspond to the tetrahedral group's generators (orders 2, 3, 3). The corollary extends realizability to branch data with additional critical values.
Techniques and tools. The key tool is dessins d'enfants: bicolored graphs on surfaces that encode branched coverings via the monodromy of the covering map. The branch data are realized by constructing an explicit dessin with the prescribed ramification profile. Combinatorial bookkeeping of permutation factorizations in underpins the argument.
Proof idea. For each target branch datum, an explicit dessin d'enfant is constructed by a combinatorial assembly procedure — specifying a bicolored graph on whose vertex valences match the prescribed ramification. Verifying the correct degree and branch data reduces to checking permutation identities. The corollary on more critical values follows by gluing smaller realizing dessins.
6. On the group structure of "magic" leatherworking braids
arXiv:2606.10047 · William Hobkirk, Abigail Hollingsworth, Elisabetta A. Matsumoto, Corbin Reid · 8 pp., 13 figs. 6
Original abstract. Magic braids are used in leatherworking to make intricate straps and bands. They appear to be impossible to make. To determine which braids can be made with this leatherworking technique, we explore their relation to several braid groups.
Key result. The paper identifies which braids are achievable by the magic leatherworking technique by mapping the construction to subgroups (or quotients) of standard braid groups. The "apparently impossible" character of magic braids is explained: the leatherworking constraint restricts the reachable elements to a proper, non-obvious subgroup.
Techniques and tools. Artin's braid groups and their presentations are the central structure. The physical constraint of the leatherworking technique (cutting a strap and weaving without detaching ends) is formalized as a restriction on the allowed generators and their compositions. The reachable set is analyzed using group-theoretic and combinatorial methods, with heavy use of diagrams (13 figures).
Proof idea. The leatherworking moves are encoded as generators in an appropriate subgroup of or a related braid-like group. The authors determine which elements of the full braid group lie in the subgroup generated by the allowed moves, thereby characterising exactly which magic braids are realizable. The "impossible" appearance is resolved: the moves generate a restricted but non-trivial subgroup, and an element is achievable if and only if it belongs to this subgroup.
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