arXiv math.GT digest — 11 Jun 2026 (4 papers)

This digest covers the 4 primary math.GT new submissions from the Thursday 11 June 2026 arXiv session. Cross-list entries are excluded per channel policy.

arXiv:2606.12094 | Weizhe Niu | 19 pages 1

Abstract. Freedman and Krushkal introduced a triple torsion linking form for rational homology -spheres and used it to obstruct locally flat embeddings in . For every odd prime , we identify their triple torsion form, computed with parameter on rational homology -spheres whose first homology has exponent , with the mod- triple cup product under torsion-linking duality. For algebraically split -framed surgery links, this gives a signed formula in terms of Milnor's integral length-three invariants , with the framing-sign factor dictated by torsion-linking duality. We then use Borromean band-sums to realize arbitrary mod- triple cup tensors on rational homology -spheres with and fixed hyperbolic ordinary torsion linking form. Finally, using the classical spinor/Klein model for the split six-dimensional quadratic space, we classify the tensors with no dual null Hantzsche pair. This produces, for every odd prime , a rational homology -sphere with hyperbolic ordinary torsion linking form but with no locally flat embedding in , and indeed no locally flat embedding in any integer homology -sphere.

Key result. For every odd prime , the paper produces an explicit infinite family of rational homology 3-spheres that admit a hyperbolic ordinary torsion linking form (the weakest obstruction-theoretic condition for embeddability) but nonetheless fail to embed locally flatly in , or in any integer homology 4-sphere. This separates the ordinary linking obstruction from the more refined triple torsion obstruction.

Techniques and tools. The central bridge is a new identification, valid when , between Freedman–Krushkal's triple torsion linking form (evaluated at parameter ) and the mod- triple Massey-type cup product, leveraging torsion-linking duality. Surgery link calculus provides a signed formula for the form in terms of Milnor's invariants. Borromean band-sum constructions yield the realization results, and the classification of which tensors obstruct embedding passes through the spinor/Klein model of the split six-dimensional quadratic space over .

Core proof idea. The key move is to interpret torsion-linking duality as a canonical isomorphism between the Freedman–Krushkal parameter- triple form and the mod- triple cup product. Once this dictionary is in place, the surgery formula computes the invariant from Milnor data, Borromean band-sums vary it freely across all realizable tensors, and the spinor/Klein classification pinpoints which tensors possess a dual null Hantzsche pair (necessary for embeddability). Tensors lacking such a pair are obstructions — giving the desired non-embeddable examples.

arXiv:2606.11899 | David Pask | MSC: 20F65, 46L05, 20E08, 20F67, 37B50, 37B10, 20M35 2

Abstract. To a full Mealy automaton we associate a bijection , a one-vertex rank-two graph , and a one-vertex -square complex tiled by Wang tiles. We prove that contains an anti-torus if and only if is bi-reversible and is aperiodic. The two hypotheses are independent and play disjoint roles: bi-reversibility is exactly what makes a complete square complex, so that its universal cover splits as a product of two trees and anti-tori can be discussed at all; and, within that setting, an anti-torus is precisely a period-free configuration in the two-sided path space of , whose existence is the aperiodicity condition. Working at the level of configurations removes any appeal to the geometry of products of trees from the main equivalence; the geometric (loop-spanned) form of Wise is shown to be strictly stronger, the lamplighter being aperiodic with no loop-spanned anti-torus.

Key result. A clean biconditional: the -square complex associated with a full Mealy automaton contains an anti-torus if and only if is bi-reversible and the associated rank-two graph is aperiodic. As a byproduct, the paper separates two notions of anti-torus — the configuration-level notion used in the theorem and Wise's loop-spanned notion — showing the latter is strictly stronger via the lamplighter automaton, which is aperiodic yet has no loop-spanned anti-torus.

Techniques and tools. The construction associates to a bijection encoding the automaton's transition structure, a one-vertex rank-two (higher-rank) graph , and a -complex tiled by Wang tiles. Bi-reversibility is shown to be the exact condition making a complete square complex, whose universal cover splits as a product of two trees — the geometric setting in which anti-tori live. Aperiodicity of is then rephrased as period-freeness of configurations in its two-sided path space, connecting symbolic dynamics to the topology of the complex.

Core proof idea. The proof decouples the two conditions: bi-reversibility handles geometry (producing the product-of-trees universal cover), while aperiodicity handles combinatorics (controlling period-free configurations). By working at the level of configurations in the path space of rather than directly in the product-of-trees geometry, the equivalence can be proved without any appeal to the geometry of Cayley graphs or Bass–Serre theory — making the argument more direct and revealing the lamplighter counterexample to loop-spanned anti-tori.

arXiv:2606.11694 | Suhyoung Choi | 14 pages, MSC: 57M50, 83A99, 20C99 3

Abstract. Let be a flat Lorentzian space of signature . A Margulis space-time is a noncompact complete flat Lorentzian -manifold , where the holonomy group is a free group of rank acting freely and properly discontinuously by isometries. We consider the case where contains a parabolic element. We show that sufficiently small deformations of still act properly discontinuously on provided their linear parts are Fuchsian; moreover, the number of conjugacy classes of parabolic elements may increase or decrease under deformation. Our proof combines our previous compactification of relative to parabolic holonomy elements with a partial generalization of the work of Carrière. However, this result depends only on the parts on parabolic actions of our earlier work. We believe that the shortness of the proof of this openness result is of independent interest.

Key result. The deformation space of Margulis space-times with parabolic holonomy is open: sufficiently small deformations of a properly discontinuous Fuchsian-linear free group that contains parabolics remain properly discontinuous, as long as the deformed linear parts stay Fuchsian. The paper also shows the number of conjugacy classes of parabolic elements is not a deformation invariant — it can increase or decrease.

Techniques and tools. The proof draws on Choi's earlier compactification of relative to its parabolic holonomy elements (extracting only the parabolic-action portion of that machinery), combined with a partial generalization of Carrière's openness argument for Anosov representations. The Lorentzian flat geometry of signature is used to control the dynamics of parabolic isometries under perturbation.

Core proof idea. The essential step is showing that proper discontinuity — typically unstable under deformation when parabolics are present — survives small perturbations because the relative compactification provides enough geometric control near the parabolic ends. The paper notes explicitly that only the parabolic-action component of the compactification technology is needed, yielding a short proof whose brevity the author flags as independently interesting.

arXiv:2606.11301 | Suman Saurabh | 13 pages + appendices, MSC: 57K10, 57K14, 33C20 4

Abstract. We study the Alexander polynomials of the 4-strand Turk's head knots , defined as the closures of the braid . Using the reduced Burau representation, we derive an annihilating recurrence of order at most 8 and a rational generating function for the resulting polynomial sequence. By executing a multivariable resultant elimination over the reciprocal constraint, we obtain an exact factorization of the normalized Alexander polynomial in terms of Chebyshev polynomials. This factorization produces a binomial convolution formula for an associated coefficient sequence and a representation by a terminating hypergeometric series. We evaluate the continuous approximation of this representation using the saddle-point method, demonstrating negative curvature in the asymptotic main term. Finally, we describe analytic obstructions to extracting global discrete error bounds via this method, leaving the formal proof of Fox's Trapezoidal Conjecture for this family open.

Key result. The Alexander polynomial of admits an exact factorization into Chebyshev polynomials, which yields a binomial convolution formula for its coefficients and a closed-form expression as a terminating hypergeometric series. As a consequence, the coefficients' asymptotic behavior can be studied via the saddle-point method, which reveals negative curvature in the leading term — making substantial progress toward Fox's Trapezoidal Conjecture for this infinite family, though a complete proof remains open.

Techniques and tools. The reduced Burau representation of the 4-strand braid group is used to write down an explicit matrix whose characteristic polynomial encodes the Alexander polynomial. A recurrence of order and a rational generating function are derived from this matrix, then multivariable resultant elimination over the reciprocal (palindromic) constraint extracts the Chebyshev factorization. The hypergeometric representation follows from the Chebyshev structure via standard hypergeometric identities, and the saddle-point method from complex analysis handles the asymptotics.

Core proof idea. The Burau representation reduces the problem to linear algebra over a polynomial ring. Resultant elimination then enforces the palindromic symmetry of Alexander polynomials, producing Chebyshev factors whose product form is amenable to generating-function techniques. The series emerges as the natural closed form for the resulting coefficient convolution, and saddle-point analysis translates this into asymptotic information — stopping short of Fox's conjecture because extracting global discrete error bounds from the saddle-point approximation encounters analytic obstructions not yet resolved.