Alper Ferudun

Sending Loxodromic Elements of Right-Angled Artin Groups to Pseudo-Anosov Mapping Classes: A Partial Answer to Problem 2.28 of the K3 List

Problem 2.28 of the list K3: A New Problem List in Low-Dimensional Topology (Baykur, Kirby and Ruberman, eds., 2026), recorded by T. Koberda, asks: if Γ is a finite graph that is not a join, is there an injective homomorphism from the right-angled Artin group A(Γ) to the mapping...

Spherical Type Is an Isomorphism Invariant of Artin Groups

We prove that an Artin group abstractly isomorphic to an Artin group of spherical type is itself of spherical type. No condition is imposed on the isomorphism or on the rank of the source presentation. The main observation is that, for an irreducible nonspherical Artin group, the...

Soft Density Profiles Do Not Determine Vacant Percolation

For any packing of unit balls in three-dimensional Euclidean space, we construct two fixed modifications preserving its upper and lower soft-density profiles at every fixed outer radius. One modification has an unbounded vacant component at every finite radius. The other has infi...

On Seven Conjectures of Fuchs on Billiard Trajectories in Regular Polygons and Geodesics on the Regular Dodecahedron

In a problem contribution (Arnold Math. J. 7 (2021)), D. Fuchs stated seven conjectures on the billiard trajectories in a regular n-gon which join two vertices (short trajectories) and on their n − 2 types A_0, ..., A_{n−3}, on closed trajectories, and on geodesics on the regular...

Volume-Minimising Projective Subspaces in Berger Projective Spaces: Three Further Cases of a Question of Gil-Medrano

Gil-Medrano asked whether, in the real projective space RP^(2n+1) with a Berger metric g_μ (the round metric with the Hopf fibres rescaled by √μ), the projective subspaces coming from equatorial k-spheres are minimal, and whether they are the only volume-minimising k-cycles in th...

A Common Positive Point for Braid Varieties

We compare two positive subsets of a braid variety: the positive locus of its weave cluster structure and the geometric positive locus defined by the joint nonnegative closure in a twisted product of flag varieties. For every positive braid word of finite crystallographic simple...

The Quaternionic Toledo Spectrum for Split F4 over Products of Surfaces

Let F be the connected linear split real group of type F4, and let Q_rho be its canonical oriented three-plane bundle, obtained from a maximal compact reduction of a flat F-bundle. We determine the range of the characteristic number over all representations of a product of two cl...

The Coxeter Group [5,3,3,3] Has No Torsion-Free Subgroup of Index 14400: A Computer-Assisted Result Related to Problem 4.124 of the K3 List

Problem 4.124 of the K3 problem list asks whether a hyperbolic integer homology 4-sphere exists, whether an arithmetic one exists, and more generally whether there is a closed hyperbolic 4-manifold with Euler characteristic 2. By a theorem of Belolipetsky, an orientable arithmeti...

Quantum Integer Torsion in the Skein Module of a Connected Sum of Two Solid Tori

We extract an explicit infinite family of torsion elements from the two-solid-torus presentation of Bakshi, Le and Przytycki. Over R=Z[q,q^{-1}], put B=R[x_1,x_2] and [m]=(q^{2m}-q^{-2m})/(q^2-q^{-2}). For every m>=2 we construct an element whose exact B-annihilator is ([m]), and...

Quantum Integer Torsion in the Skein Module of a Connected Sum of Two Solid Tori

We extract an explicit infinite family of torsion elements from the two-solid-torus presentation of Bakshi, Le and Przytycki. Over R=Z[q,q^{-1}], put B=R[x_1,x_2] and [m]=(q^{2m}-q^{-2m})/(q^2-q^{-2}). For every m>=2 we construct an element whose exact B-annihilator is ([m]), and...

Minimum-Volume Central Slabs of the Regular Octahedron

We determine the minimum volume of the intersection of a regular octahedron with a centered slab of every prescribed width, including all equality directions. For the octahedron with vertices +/-e_i and slab half-width t, the minimizing normal is diagonal below a single threshold...

Asymmetric Hyperbolic Knots Detect Mapping Classes of 3-Manifolds: A Written Proof for a Problem of the K3 List

Problem 1.19 of the list K3: A New Problem List in Low-Dimensional Topology (Baykur, Kirby and Ruberman, eds., 2026) asks: if Y = Y₁ # Y₂ is a connected sum of 3-manifolds, neither of them S³, and the twist Φ about the connected-sum sphere is not isotopic to the identity, is ther...

Orientation Dependence and Exact Groups for Ridge-Routing Sandpiles

A fitting orientation of a simplicial rooted forest specifies an unsigned chip-firing rule on its nonroot ridges. We show that the resulting sandpile group can depend on that orientation: in every dimension d >= 2, a six-facet rooted forest with relative boundary determinant one...

Algebraic Zero Curves of Homogeneous Plurisubharmonic Polynomials

Let p be a nonzero homogeneous plurisubharmonic polynomial on C^2. We prove that every irreducible affine algebraic curve avoiding the origin on which p vanishes is exactly a nonzero level of a homogeneous holomorphic polynomial of degree at most degree(p). No smoothness or ratio...

Critical Points and Local Maxima of Sparse Binary Restricted Boltzmann Machines

We classify the finite critical points of exact empirical log likelihood for binary restricted Boltzmann machines in which every hidden unit has at most two visible neighbors. All allowed weights and biases are independent real parameters. A pairwise exponential-family representa...

Exact Word-Map Distributions on Four-by-Four Unitriangular Groups

We give the exact probability distribution of every group word, with no constants and any number of variables, on the full unitriangular group UT4(Fq), for every finite field. A nonzero exponent sum modulo the characteristic makes the word map uniform. Otherwise the distribution...

A Conjecture of Guadagnini and Pilo on Quantum Invariants and the Fundamental Group: Counterexamples and a Classification by Gauge Group

Guadagnini and Pilo conjectured that for a closed oriented 3-manifold M and a compact gauge group G the absolute value of the quantum invariant τ^G_r(M), when it does not vanish, depends only on the fundamental group of M; this is Conjecture 7.5 of Ohtsuki's problem list. They pr...

A Conjecture of Guadagnini and Pilo on Quantum Invariants and the Fundamental Group: Counterexamples and a Classification by Gauge Group

Guadagnini and Pilo conjectured that for a closed oriented 3-manifold M and a compact gauge group G the absolute value of the quantum invariant τ^G_r(M), when it does not vanish, depends only on the fundamental group of M; this is Conjecture 7.5 of Ohtsuki's problem list. They pr...

Sharp Ambiguity Bounds for Surviving Conditional Odds Ratios

Fix the support of a binary d-way probability table, its feasible interior one-way margins, and every conditional pairwise odds ratio whose four cells remain positive. We prove that the largest possible dimension of the resulting family is ceil(2^(d+1)/3)-d-1 for d >= 2, even whe...

A Genus-Two Algebraic Family with Vanishing Real Toledo Numbers and Nonzero Generic Witt Class

We give four explicit matrices defining a genus-two surface-group representation over R[s,t,1/(st(1+t^2))], where R denotes the real numbers. Every real specialization has zero Witt class and zero Toledo number. Over R(s,t), however, the Witt class is the anisotropic Pfister form...