arXiv math.GT digest — 29 May 2026 (5 papers)

All 5 new submissions from the arXiv math.GT listing of 29 May 2026. Each entry reproduces the original title, authors, and abstract verbatim, followed by structured commentary on the key result, techniques, and proof idea.

Authors: Lia Buchbinder, Yunjia Kou, Bala Krishnamoorthy, Kevin R. Vixie

arXiv: 1

Abstract: We study the existence of area-minimizing homotopies between homotopic curves in the plane. While the classical Plateau problem establishes the existence of least-area surfaces spanning a single Jordan curve, the corresponding existence theory for homotopies between curves is more subtle and is not directly covered by the same framework. Existing results in the plane are mainly based on combinatorial and algebraic methods, such as decomposing curves into self-overlapping subcurves. These methods are highly effective in the planar setting, but they are often tied to special classes of curves and rely strongly on the local structure of the self-intersections, frequently assuming transverse crossings. In contrast, our approach is geometric and variational, and does not depend on the local structure of the self-intersections. In this paper, we develop a variational existence theory for minimum-area homotopies of immersed planar curves. Our approach adapts classical minimal surface methods by lifting an immersed planar curve with self-intersections into higher co-dimension, where it becomes embedded. For such a lifted curve, we apply Douglas's solution of the Plateau problem to obtain an area-minimizing disk. For closed curves of class , we prove uniform convergence of the Douglas minimizers and show that the limiting map minimizes area among all spanning maps of the original planar curve. We then extend the construction to closed Lipschitz curves using approximation and Sobolev compactness arguments. Since the limiting minimizing disk lies in the original plane, it directly produces a null homotopy whose swept area is minimal among all admissible homotopies of the original curve. In this way, the construction connects Plateau theory with minimal homotopy area minimization.

Key result. The paper proves existence of area-minimizing null homotopies for immersed closed planar curves of class and, by extension, for closed Lipschitz curves. This fills a gap in the existence theory that classical Plateau methods did not directly cover.

Techniques and tools. The central device is a co-dimension lift: an immersed curve in the plane is lifted to an embedded curve in higher-dimensional space, where the classical Douglas solution of the Plateau problem applies. Uniform convergence of Douglas minimizers is established for curves; Sobolev compactness and approximation arguments handle the Lipschitz extension.

Core proof idea. Project the area-minimizing disk produced by Douglas's theorem back to the plane. Because the disk is minimizing in the higher-dimensional ambient space, the projected homotopy is area-minimizing among admissible homotopies of the original planar curve. The key step is showing the projection does not increase swept area, which exploits the embedding property of the lift.

Authors: Stavros Garoufalidis, Rinat Kashaev, Sakie Suzuki

arXiv: 2

Abstract: It is well known that every compact oriented 3-manifold admits an ideal triangulation, and that any two such triangulations with at least two ideal tetrahedra are related by a sequence of Pachner $2$-$3$ moves. Motivated by constructions in quantum topology, we give a combinatorial description of closed $3$-manifolds in terms of ordered ideal triangulations and ordered Pachner $2$-$3$ and $0$-$2$ moves.

Key result. A combinatorial calculus for closed oriented 3-manifolds using ordered ideal triangulations: any two such triangulations of the same manifold are related by a finite sequence of ordered Pachner $2$-$3$ moves and ordered $0$-$2$ moves.

Techniques and tools. The framework extends the classical Matveev–Piergallini calculus on ideal triangulations. The ordering data (a total order on the vertices of each tetrahedron) is the new ingredient; it is essential for the quantum-topological applications motivating the paper (e.g. constructing partition functions that depend on the ordering).

Core proof idea. The unordered version of the calculus is known. The authors show that any ordering on one triangulation can be transported across moves by introducing the $0$-$2$ move (which inserts/removes a pair of tetrahedra) as a means of reordering. The combination of $2$-$3$ and $0$-$2$ moves in the ordered setting is then shown to be sufficient to connect any two ordered triangulations of the same closed manifold.

Authors: Natsuya Takahashi

arXiv: 3

Abstract: We classify knot traces with trisection genus at most 2. We give infinitely many knots whose traces have trisection genus 3, and infinitely many knots whose traces have trisection genus 4. We also show that there exist infinite families of knots whose traces have arbitrarily large trisection genus. In addition, we determine or give sharp bounds for the trisection genus of the traces of several well-known knots, such as the figure-eight knot, the $(p, pq+1)$-torus knots, and the $(-2, 3, 2n-1)$-pretzel knots.

Key result. A complete classification of knot traces by trisection genus , together with infinite families realising every genus and families with unbounded trisection genus. Explicit sharp bounds are computed for figure-eight, $(p,pq+1)$-torus, and $(-2,3,2n-1)$-pretzel knot traces.

Techniques and tools. Trisections of 4-manifolds (Gay–Kirby theory), Kirby calculus for knot traces, and the interplay between the classical knot invariants (genus, bridge number, unknotting number) and trisection genus. Stabilisation arguments control lower and upper bounds.

Core proof idea. The trisection genus of a knot trace is bounded below by topological invariants of $K$ and bounded above by explicit handle-decomposition constructions. For the classification in genus the author shows by exhaustion (using Kirby diagrams) that only specific knot families can achieve these small values, and then constructs matching trisections to confirm the upper bounds.

Authors: Makoto Ozawa

arXiv: 4

Abstract: We develop a framework for discrete p-density and compression-radius profiles of lattice knots. For lattice polygons representing a fixed knot type, we define scale-free density quantities by dividing lattice length by chord-length spread functionals, and we define corresponding compression radii using a raw lattice thickness convention. These profiles are studied both on length-filtered lattice representative sets and on seed-generated finite search spaces arising from local move systems such as BFACF moves. The framework is intended as a computable counterpart of continuous scale-free geometric invariants of knot types. It separates density, compression, and ropelength-like quantities, and records how these quantities behave under length filtration and finite move-graph exploration. We emphasize throughout the distinction between exhaustive lattice-knot invariants and seed-generated computational profiles. As a case study, we compute raw p-density and compression-radius values along an explicit BFACF merge certificate connecting a 30-edge simple cubic figure-eight seed to its reflected mirror seed at length bound $N=32$. The extracted path has 21 states and 20 BFACF moves and passes through a 32-edge connecting state. Along this path, density and compression-radius values are not monotone: the connecting state has larger raw density and slightly larger -compression radius than the seeds, while an intermediate 30-edge state has smaller compression radius but larger density. This illustrates that density minimisation, compression minimisation, and ropelength-like minimisation define distinct finite-state optimisation problems.

Key result. A new discrete framework of p-density and compression-radius profiles for lattice knots, with a concrete case study on the figure-eight knot showing that density, compression, and ropelength minimisation are three genuinely distinct optimisation problems on the BFACF move graph.

Techniques and tools. Lattice knot theory on the simple cubic lattice, BFACF local moves, chord-length spread functionals for scale-free density, and lattice thickness conventions for compression radius. Computations are supported by supplementary Python code archived at Zenodo.

Core proof idea. The framework is largely definitional and computational rather than proof-based. The non-monotonicity result follows directly from the explicit 21-state BFACF path: the intermediate and connecting states are exhibited with numerically computed density and compression values that violate monotonicity, demonstrating by example that the three optimisation criteria pull in different directions.

Authors: Ulysse Remfort-Aurat

arXiv: 5

Abstract: Let be the fundamental group of a closed orientable surface of genus . The outer automorphism group naturally acts on the character variety for any Lie group $G$. We consider the set of simple-stable representations which are modelled on Minsky's primitive-stable representations. We prove that the set of conjugacy classes of simple-stable representations of in is a domain of discontinuity for this action, strictly larger than the set of conjugacy classes of convex cocompact representations.

Key result. The set of conjugacy classes of simple-stable representations forms a domain of discontinuity for the -action on the -character variety, and this domain strictly contains the convex cocompact locus.

Techniques and tools. The paper adapts Minsky's theory of primitive-stable representations (originally for ) to the complex hyperbolic setting . Key ingredients include the geometry of the complex hyperbolic plane , the curve complex, and coarse-geometric estimates on the action of via the representation.

Core proof idea. Simple-stability is defined by requiring that simple closed curves on the surface act with uniform quasi-geodesic behaviour in . The domain-of-discontinuity property is then established by showing that orbits of simple-stable representations cannot accumulate: any sequence leaving every compact subset of the simple-stable locus eventually fails the simple-stability criterion. Strict containment of the convex cocompact locus follows by exhibiting simple-stable representations outside it.