arXiv math.GT digest — 05 Jun 2026 (4 papers)

This digest covers the 4 new primary submissions from the arXiv math.GT listing of Friday, 5 June 2026 (session date 4 Jun 2026, arXiv IDs 2606.05313–2606.06299). Cross-list submissions are excluded per channel policy.

Authors: Irving Dai, Abhishek Mallick, Masaki Taniguchi

Abstract: We show that singular instanton Floer homology with the Chern–Simons filtration can be used to produce exotic pairs of slice disks. We moreover construct a strongly invertible -slice knot for which any symmetric pair of -disks are exotic, and remain exotic after stabilizing by or (or by standard or ) for any $n$. Our methods apply more generally to stabilization by any simply connected definite manifold, or by any number of exotic embedded projective planes of the same sign. We also provide an example of a strongly invertible knot which is -slice and equivariantly slice, but not equivariantly -slice. Along the way, we partially compute various symmetry actions on the singular instanton Floer complexes of two-bridge knots via an explicit analysis of their traceless -character varieties.

Key result. The paper establishes that singular instanton Floer homology, equipped with its Chern–Simons filtration, is a sufficiently fine invariant to distinguish exotic pairs of slice disks — that is, smoothly non-isotopic disks bounding the same knot in the 4-ball. The headline example is a strongly invertible -slice knot whose symmetric slice disks remain pairwise exotic after arbitrarily many stabilizations by , , , or , and more generally after stabilization by any simply connected definite 4-manifold. This gives the first examples of exotic disk pairs that are stable under this class of topological modifications.

Techniques and tools. The main algebraic engine is singular instanton Floer homology (associated to a knot in with a singular bundle data along the knot), filtered by the real-valued Chern–Simons functional. The -slice and equivariant slice conditions involve careful comparison of the instanton invariants under the strong inversion symmetry. A key computational ingredient is an explicit analysis of the traceless -representation varieties of two-bridge knots, used to read off the symmetry actions on the associated Floer complexes. The stabilization invariance argument extends the filtration comparison across connected sums with definite manifolds.

Core proof idea. To show a symmetric pair of disks is exotic, one computes a numerical invariant extracted from the Chern–Simons-filtered singular instanton complex for each disk, and observes that the two values differ. The central difficulty is that stabilization changes the ambient 4-manifold; the proof shows the Chern–Simons filtration-based invariant is insensitive to stabilization by definite manifolds (since the latter contribute no self-dual harmonic forms that could shift the filtration levels), so any difference recorded before stabilization persists after. The analysis of -character varieties of two-bridge knots provides enough explicit data to implement these computations in the key examples.

Author: Hongtaek Jung

Abstract: A cataclysm deformation, that shears and twists a given Anosov representation according to data known as a twisted transverse cocycle, is an intuitive and powerful tool for studying Anosov representations. We show that if a sequence of twisted transverse cocycles converges weakly, the sequence of corresponding cataclysm deformations on the space of Anosov representations converges uniformly on compact sets. This result leads to two applications. First, we obtain an extension of the Goldman product formula. Second, we consider strongly dense representations, introduced by Breuillard–Green–Guralnick–Tao and Long–Reid. Using cataclysm deformations, we show that, for a split real form whose Weyl group contains $-1$, the set of strongly dense -Hitchin representations is not open in the -Hitchin component.

Key result. The paper proves a continuity theorem for cataclysm deformations of Anosov representations: weak convergence of the twisting data (twisted transverse cocycles) implies uniform convergence of the deformed representations on compact sets in the representation space. Two applications follow. First, an extension of the Goldman symplectic product formula to the broader setting of Anosov representations. Second, a non-openness result: for any split real Lie group with $-1$ in its Weyl group, the strongly dense representations inside the -Hitchin component do not form an open subset.

Techniques and tools. Cataclysm deformations originate in Thurston's earthquake theory on Teichmüller space and were extended to Anosov representations by Loftin and others. The paper works with the topology on the space of Hitchin representations as a subspace of the character variety . The key analytic tool is a careful control of the cocycle convergence in the weak topology on the space of twisted transverse cocycles, combined with compactness of the relevant limit sets. The Goldman product formula is recovered via integration of the deformation vector fields.

Core proof idea. The convergence theorem reduces to showing that the deformation map — sending a twisted transverse cocycle to the deformed representation — is continuous in the relevant topologies. The argument proceeds by controlling the holonomy perturbations along geodesic laminations: weak convergence of cocycles implies the shearing/twisting data converges at almost every point of the lamination, and standard dominated convergence arguments then promote this to uniform convergence of holonomies on compact sets. The non-openness of strongly dense Hitchin representations follows by constructing, via cataclysm deformations, sequences of representations converging to a strongly dense one, where each term in the sequence fails strong density; the continuity theorem ensures the limit is genuine.

Authors: Daniel Galvin, Patrick Orson, Mark Powell

Abstract: A stabilisation of a $4$-manifold $X$ is the connected sum of $X$ with some number of copies of . If two smooth surfaces in a $4$-manifold are topologically isotopic, we investigate whether they must moreover be smoothly isotopic in some stabilisation of $X$. We prove this result holds whenever the surfaces are trivial in the -homology of $X$. We also produce a large class of fundamental groups of the ambient $4$-manifold for which the result holds; this class includes free products of classical knot groups and, in particular, free groups.

Key result. The main theorem says that if two smooth surfaces in a 4-manifold $X$ are topologically isotopic and represent the trivial class in , then after sufficiently many stabilizations () the surfaces become smoothly isotopic. A second theorem extends this to surfaces in 4-manifolds whose fundamental group is a free product of classical knot groups — a class that includes free groups.

Techniques and tools. The proof strategy follows the Whitney move / finger move framework central to 4-manifold topology, combined with the Whitney tower technology developed by Freedman–Quinn and elaborated by Conant–Schneiderman–Teichner. The -homology triviality condition is used to kill the relevant surgery obstructions, allowing controlled use of the $s$-cobordism theorem in the smooth category after stabilization. The fundamental group condition is handled via van Kampen-type decompositions that allow the argument to be run in each factor separately.

Core proof idea. Given a topological isotopy between two smooth surfaces, one promotes it to a smooth isotopy after stabilization by the following path: (1) use the topological isotopy to produce an immersed concordance between the surfaces in ; (2) perform finger moves to introduce controlled self-intersections that are algebraically cancellable; (3) use the -homology triviality to show the Whitney disks bounding these intersection pairs can be chosen to be embedded (after adding summands to create room); (4) apply Whitney moves to remove the intersections and obtain a smooth isotopy.

Authors: Hong Chang, Xiao Chen, Wujie Shen

Abstract: In this paper, we determine the exact minimal number of curves in a filling $k$-system on an oriented surface of genus $g$ for any positive integers $k$ and $g$.

Key result. For all integers and , the paper gives an explicit formula for the minimum number of curves $f(g,k)$ required to form a filling $k$-system on the closed oriented surface . A filling $k$-system is a collection of simple closed curves in minimal position such that every complementary region is a polygon with at most $k$ sides (i.e., has negative Euler characteristic at most $-k+1$ in the appropriate normalisation). This resolves a combinatorial extremal problem in surface topology that was open in full generality.

Techniques and tools. The proof combines two directions. The lower bound is obtained via an Euler characteristic count: summing the contributions of all complementary regions and using the Gauss–Bonnet formula for the surface gives a constraint on the minimum number of arcs, hence curves. The upper bound is constructive: the authors exhibit explicit configurations of curves achieving the minimal count, using a careful tessellation of adapted to the parameter $k$. The paper includes 28 figures, suggesting that the combinatorial geometry of these tessellations is a significant part of the argument.

Core proof idea. The Euler characteristic approach starts from and tracks the contribution of each region to this total. A filling $k$-system with $n$ curves and $r$ regions satisfies a formula relating $n$, $r$, and the total number of intersection points, derived from the cellulation of by the curve system. The $k$-condition bounds the number of sides of each face from above, which in turn lower-bounds the number of edges (intersection arcs), and hence the number of curves. The matching upper-bound construction shows the resulting bound is tight for all $g$ and $k$.