arXiv math.GT digest — 04 Jun 2026 (2 papers)

Two new submissions appeared in the arXiv math.GT listing for Thursday 4 June 2026: a picture-valued knot invariant built from the group and plat closures of 3-braids with 4-valent vertices, and a counterexample family to a recent conjecture of Eudave-Muñoz and Ozawa about incompressible spanning planar surfaces in hyperbolic link exteriors.

arXiv:2606.04540 | Seongjeong Kim, Vassily O. Manturov

Abstract. We construct a picture-valued invariant of classical knots by using the group and the plat closure of braids. We introduce a modification of the group and define a map from the braid group to framed 6-valent graphs with leaves. Each 6-valent vertex corresponds to the moment when three points become collinear. By using this construction, we obtain a knot invariant valued in equivalence classes of graphs modulo local moves. Our invariant provides a new graphical approach to the study of classical knots.

Key result. The paper produces a knot invariant taking values in equivalence classes of framed 6-valent graphs with leaves, modulo a prescribed set of local moves. The invariant is defined via the plat closure of modified -braids, and is therefore strictly picture-valued rather than scalar- or polynomial-valued. 1

Techniques and tools. Manturov's groups encode the combinatorics of points in general position and their collinearity events; the version used here, , tracks moments when triples of points become collinear. The paper introduces a modified variant of and constructs a braid-group homomorphism into framed 6-valent graphs with leaves, where each 6-valent vertex represents one such collinearity event. The invariant is extracted via the plat closure, following the classical passage from braids to knots in the plat model. 1

Core proof idea. The central move is to translate braid-group relations in into local graph moves, and to show that these moves are precisely the right equivalences to make the plat-closure construction well-defined on knot isotopy classes. The picture-valued output — a graph up to local moves — is then the invariant; verifying invariance reduces to checking that Reidemeister moves on the knot side correspond to the declared local moves on the graph side. 1

arXiv:2606.04201 | Luis G. Valdez-Sánchez

Abstract. For each integer we construct examples of $N$-component hyperbolic links whose exterior contains an incompressible spanning planar surface with one boundary component on each boundary torus of of nonmeridional and nonintegral slope, thus providing counterexamples to a recent conjecture of M. Eudave-Muñoz and M. Ozawa. The case $N=3$ is the crucial one to consider: all such link pairs $(L,P)$ are classified and found to be generated by the structure of the exterior of hyperbolic Eudave-Muñoz knots. More generally, necessary and sufficient conditions on integers are given for the existence of a 3-component link in whose exterior contains a spanning pants with boundary slopes of the form . A key role in the analysis of 3-component link pairs is played by the properties of the embeddings of three mutually disjoint and nonparallel primitive circles on the boundary of a genus two handlebody. These are classified in general and in the special case when the handlebody is part of a genus two Heegaard decomposition of associated with a 3-component link pair. The hyperbolic links with components whose exterior contains a spanning planar surface with nonmeridional and nonintegral boundary slopes are constructed via an inductive process that starts with any of the classified 3-component hyperbolic link pairs.

Key result. The paper disproves the Eudave-Muñoz–Ozawa conjecture by exhibiting, for every , an $N$-component hyperbolic link whose exterior contains an incompressible spanning planar surface whose boundary slopes on every boundary torus are simultaneously nonmeridional and nonintegral. The $N=3$ case is classified completely. 2

Techniques and tools. The construction for $N=3$ is anchored in the geometry of hyperbolic Eudave-Muñoz knots. A central combinatorial input is a classification of embeddings of three mutually disjoint, nonparallel primitive circles on the boundary of a genus-two handlebody — particularly in the case where the handlebody appears in a genus-two Heegaard splitting of associated to a 3-component link. Necessary and sufficient conditions on denominators for realising spanning-pants boundary slopes of the form are identified. The cases are handled by an inductive construction starting from the classified 3-component pairs. 2

Core proof idea. The Eudave-Muñoz knot exterior provides a template: its structure forces a specific embedding pattern of boundary curves in the Heegaard handlebody, and the classification of primitive-circle triples on the handlebody boundary then controls which boundary slopes are realisable. Once the $N=3$ pairs are in hand, an inductive scheme attaches further hyperbolic components while preserving incompressibility and the nonmeridional–nonintegral slope condition, generating the higher-component counterexamples. 2