arXiv math.GT digest — 02 Jun 2026 (4 papers)

Today's arXiv math.GT listing (Mon 1 Jun 2026) contains 4 new submissions, ranging from quantum knot invariants and mapping-class groups of knotted surfaces to Dehn surgery and combinatorial graph topology.

Authors: Stavros Garoufalidis, Shana Yunsheng Li, Josephine Yu

Abstract: Motivated by the recent work of Harper–Kohli–Song–Tahar, we formulate a positivity, hole-free, and log-concavity conjecture for the Links–Gould polynomial of alternating links and verify it for all 51.3 million alternating knots with at most 19 crossings. All but 544 of those knots satisfy a stronger type-B log-concavity condition characterized by the slopes of edges in the subdivision of the monomial support induced by the log coefficients. 1

Key result. The paper formulates a precise conjecture — positivity, hole-freeness, and log-concavity — for the Links–Gould polynomial of alternating links, and provides strong computational evidence by verifying it across all 51.3 million alternating knots up to 19 crossings. A refined "type-B" log-concavity condition (defined by edge slopes in the subdivision of the monomial support) holds for all but 544 of those knots.

Techniques and tools. The proof strategy relies on large-scale computation over the full census of alternating knots with at most 19 crossings, combined with the combinatorial geometry of Newton polytopes (in particular the subdivision of monomial supports and the slopes of bounding edges). The conjecture is modelled on log-concavity results for the Jones polynomial and builds on the framework of Harper–Kohli–Song–Tahar for related invariants.

Core proof idea. For each alternating knot in the census, the authors compute the Links–Gould polynomial, verify positivity and absence of "holes" in the monomial support, and check log-concavity of coefficients along each row of the support. The type-B strengthening is detected by examining the slopes of edges in the induced subdivision: a slope condition certifies the stronger form. The 544 exceptions to type-B are recorded explicitly, giving the conjecture a precise boundary.

Authors: Oliver Knill

Abstract: The Möbius–Kantor graph MK = G(8,3) is a Cayley graph of three non-abelian groups, the Pauli group P(1), the semi-dihedral group SD(16), as well as the dihedral group D(16) of order 16. In topological graph theory, it illustrates the Heawood number 7 of the torus and leads to the Tucker group Aut(MK), the unique group of genus 2. We compute the Lefschetz numbers to illustrate the Brouwer–Lefschetz fixed point theorem. MK is also the dual of the 2-skeleton complex of the 3-sphere G. The graph represents one of the flat Clifford tori of a Hopf fibration in the 3-sphere G = K(2,2,2,2) reflecting that Coxeter saw that MK is a subgraph of the tesseract G*. It carries a metric d so that (MK, d) has only one algebraic group structure (P(1), *) that preserves the metric. It makes the Pauli group natural, similarly as the Möbius ladder M(16) makes the dihedral group D(16) natural, forcing the algebraic structure from the metric structure. 2

Key result. The paper establishes that the metric geometry of the Möbius–Kantor graph uniquely forces the Pauli group P(1) as its natural algebraic structure: of the three non-abelian groups for which MK is a Cayley graph, only P(1) preserves the graph metric. This is the graph-theoretic analogue of how the Möbius ladder M(16) singles out the dihedral group D(16). As a secondary result, Lefschetz numbers of automorphisms are computed to illustrate the Brouwer–Lefschetz fixed point theorem on this genus-2 graph.

Techniques and tools. The paper uses topological graph theory (genus, Heawood number, Tucker group = Aut(MK)), Cayley graph theory, and the Hopf fibration structure of S³. The connection to the tesseract and flat Clifford tori is exploited via Coxeter's classical embedding. Lefschetz numbers are computed directly from the fixed-point data of automorphisms acting on cellular chain complexes.

Core proof idea. The central argument is metric rigidity: among the three non-abelian groups acting on MK as a Cayley graph, only P(1) acts by metric-preserving transformations with respect to the natural graph metric d. The proof checks, for each of the three group structures, whether the group multiplication is an isometry of (MK, d), and shows uniqueness for P(1).

Authors: Shunjiang Jiang, Ran Tao

Abstract: We study Dehn fillings on two-bridge knots via non-abelian representations into PSL(2,ℝ) whose meridian image is hyperbolic. For each fixed nontrivial two-bridge knot, we prove that the set of surgery slopes admitting such representations is bounded. Equivalently, Dehn fillings along slopes with sufficiently large absolute value admit no non-abelian PSL(2,ℝ) representations with hyperbolic meridian image. The proof combines the Riley polynomial with Khoi's surgery-slope formula. On each admissible real algebraic branch, we express the meridian and longitude translation parameters as functions of the branch parameter and derive uniform endpoint estimates for their quotient. The resulting bound is effective in principle but not optimized. We also provide examples illustrating the parameter sets and endpoint behavior. 3

Key result. For every nontrivial two-bridge knot, the set of Dehn surgery slopes at which the filled manifold admits a non-abelian representation into PSL(2,ℝ) with hyperbolic meridian image is bounded. In particular, for slopes of sufficiently large absolute value, no such representation exists. This gives a uniform finiteness theorem for a natural class of PSL(2,ℝ) representations across all two-bridge knots.

Techniques and tools. The proof hinges on two classical tools in combinatorial knot theory and representation varieties: the Riley polynomial (whose real roots parameterize non-abelian SL(2,ℝ) / PSL(2,ℝ) representations of two-bridge knot groups) and Khoi's surgery-slope formula, which expresses the surgery slope in terms of meridian and longitude translation parameters of the representation. The real algebraic geometry of the representation variety is analyzed branch by branch.

Core proof idea. On each real algebraic branch of the Riley polynomial, the meridian-translation parameter m and longitude-translation parameter l are expressed as functions of the branch parameter. Their ratio m/l gives the surgery slope. By deriving uniform endpoint estimates for m/l as the branch parameter ranges over all admissible values, the authors show that |m/l| is bounded on every branch, and hence the set of slopes is bounded. The effective bound is in principle computable from the Riley polynomial data, though not optimized in this paper.

Authors: Weizhe Niu

Abstract: Let Σ_g^0 ⊂ S⁴, g ≥ 3, be the standard unknotted closed oriented surface, and let a ⊂ Σ_g^0 be an oriented nonseparating curve. For a nontrivial knot J ⊂ S³, let Σ_{g,a,J} ⊂ S⁴ be the surface obtained by ordinary untwisted rim surgery along a. Assuming a meridian-longitude rigidity condition on the knot group of J, we compute the extendable mapping-class subgroup exactly: E(Σ_{g,a,J}) = Stab_{Mod(Σ_g)}(q_0) ∩ Stab_{Mod(Σ_g)}([a]), where q_0 is the Rokhlin quadratic form of the standard embedding and [a] ∈ H_1(Σ_g; ℤ) is the oriented homology class. 4

Key result. For a rim surgery surface Σ_{g,a,J} ⊂ S⁴ (g ≥ 3, J nontrivial with the meridian-longitude rigidity property), the extendable mapping-class subgroup E(Σ_{g,a,J}) is computed exactly as the intersection of two stabilisers in the mapping class group of Σ_g: the stabiliser of the Rokhlin quadratic form q_0 of the standard embedding, and the stabiliser of the homology class [a] of the surgery curve. This gives a complete algebraic description of which mapping classes of the surface extend to self-diffeomorphisms of S⁴.

Techniques and tools. The key tools are: (1) the Rokhlin quadratic form, which controls which mapping classes preserve the normal framing from the standard embedding; (2) the meridian-longitude rigidity condition on the knot group of J, which forces any extension of a mapping class to S⁴ to fix the isotopy type of J and hence fix [a] in homology; (3) the structure of the mapping class group Mod(Σ_g) and its action on H_1(Σ_g; ℤ) and on the set of quadratic forms enhancing the intersection form. The rim surgery construction replaces a tubular neighbourhood of a in Σ_g^0 × D² with a copy built from J, which is what ties the extendability to the homological data of a and J.

Core proof idea. The proof has two directions. For the upper bound (E ⊆ intersection of stabilisers): any extension of a mapping class φ to a self-diffeomorphism of S⁴ must preserve the Seifert framing of Σ (hence fix q_0) and, by meridian-longitude rigidity of J, must fix [a] in H_1. For the lower bound (intersection of stabilisers ⊆ E): given φ fixing q_0 and [a], one constructs an explicit extension to S⁴ by combining the standard extension for Stab(q_0) with the fact that fixing [a] allows the rim surgery region to be moved compatibly.