arXiv subjects

Bernd Johannes Wuebben

Publications and source records attributed to Bernd Johannes Wuebben.

8 recordsLinked to original sources

arXiv2026

Instanton and pillowcase homology of the $(-2,3,q)$ pretzel knots

Hedden, Herald and Kirk conjectured that every knot admits a decomposition along a Conway sphere for which the Lagrangian Floer homology of the two associated immersed curves in the pillowcase recovers Kronheimer and Mrowka's reduced singular instanton homology. We prove the conjecture for the hyperbolic pretzel knots $P(-2,3,q)$; it was previously known for two-bridge knots and for some torus knots. The decomposition is obtained from Hedden, Herald and Kirk's decomposition of the torus knot $T(3,5)$ by Dehn twists along the Conway sphere, which act on the pillowcase by a linear shear. Building on our analysis of the torus knots $T(3,n)$, we compute the Floer complex and its $\mathbb{Z}/4$ grading by hand; its differential is nonzero for every member of the family except $P(-2,3,7)$. The same method proves the conjecture for a family of twisted torus knots, under a hypothesis on the chirality of the torus-knot decomposition. On the instanton side, we show that the instanton homology of $P(-2,3,q)$ is free abelian of rank $q+2$ for every odd $q\ge3$, extending Lobb and Zentner's rational computation to the integers by means of Manion's integral Khovanov homology; for the hyperbolic members this also follows from work of Daemi and Scaduto.

math.GT↗

arXiv2026

Traceless $\mathrm{SU}(2)$ characters and $\mathbb{Z}/4$ instanton gradings for two-bridge and $(3,n)$-torus knots

For every torus knot, the rank of the differential of the Daemi-Scaduto S-complex of singular instanton homology is determined by the signature and the Alexander polynomial; this follows from theorems of Li-Ye and Daemi-Scaduto. For the torus knots $T(3,n)$ the differential has rank zero or one according to $n$ modulo $6$, and its rank equals Daemi and Scaduto's Frøyshov-type invariant; they proved this in two of the four residue classes and for $n=4,5$, and Hedden, Herald and Kirk computed it for $n\le38$. Following Lewallen, Klassen and Hedden-Herald-Kirk, we describe the generators of the instanton complexes and their $\mathbb{Z}/4$ gradings for two-bridge knots, whose irreducible traceless characters are all binary dihedral, and for $T(3,n)$, where at most one is. We also give a proof through Khovanov homology, using Schütz's computation for three-braids, of Li and Ye's computation of the rank of the instanton homology of $T(3,n)$, and compare the differential of the knot $8_{19}$ with the single pillowcase bigon of Hedden, Herald and Kirk.

math.GT↗

arXiv2026

An exotic $S^2\times S^2$ and an exotic $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$

We prove that a specified Lidman-Piccirillo piece $V$, a symplectic $4$-manifold with the homology of $S^2\times D^2$ built from a genus-$2$ surface bundle over a once-punctured torus by two Luttinger surgeries, is simply connected, for an explicit permitted choice of the two surgery parametrizations. Three consequences follow. The symplectic double $Z=V\cup_σV$ is homeomorphic but not diffeomorphic to $S^2\times S^2$. The Lidman-Piccirillo manifolds $B$ and $W$ are homeomorphic. Since the figure-eight knot is slice in $B$ and not in $W$, they are the first pair of homeomorphic closed $4$-manifolds distinguished by unconstrained knot slicing, that is, by sliceness with no constraint on the homology class of the slice disk; detecting smooth structure this way goes back to Casson. Finally, the regluing of Lidman and Piccirillo's Theorem~2 applied to $Z$ yields a closed simply connected $4$-manifold homeomorphic but not diffeomorphic to $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$. The consequences follow from the simple-connectivity statement by the classifications of Freedman and of Hambleton-Kreck, together with a rigidity analysis of the surgery parameters. The fundamental group is computed in the style of Baldridge and Kirk, from explicit based representatives of every meridian and Lagrangian push off, and the resulting relation system is decided by coset enumeration, after calibration on two configurations whose answers are known independently. The development calculations and finite-presentation decisions can be reproduced from the ancillary files.

math.GM↗

arXiv2026

Doubly isolated Batyrev mirror pairs and non-smoothable Calabi-Yau threefolds

We classify reflexive four-polytopes for which the generic anticanonical hypersurfaces associated with the face fans of the polytope and its polar both have at most isolated singularities. There are exactly 590 such polytopes up to lattice equivalence in the Kreuzer-Skarke classification. Their two-dimensional faces belong to twelve lattice-isomorphism classes, all triangles, zonotopes, or reflexive polygons. The only nonsmoothable local models are the cyclic quotients $\frac13(1,1,1)$ and $\frac15(1,1,3)$ and the anticanonical cone over the first Hirzebruch surface; no singularity has a positive-dimensional reduced deformation space without a smoothing component. Within this classification, the Hirzebruch cone occurs in a unique mirror pair, associated with polytopes having 22 and 26 vertices. We exhibit this pair explicitly: one hypersurface is nonsmoothable, while all singularities of its mirror are locally smoothable. The crepant resolutions have Hodge numbers $(20,26)$ and $(26,20)$. At the opposite extreme, the self-dual 24-cell is the unique member of the classification for which both hypersurfaces are smooth before crepant resolution.

math.AG↗

arXiv2026

Slope stability of tangent bundles of smooth toric Fano varieties

We construct smooth toric Fano $n$-folds of Picard number $n+2$ whose tangent bundles are slope-stable with respect to the anticanonical polarization, for every $n\ge4$. The construction gives toric fibrations with fibre the del Pezzo surface of degree six over products of projective lines, parametrized by multisets of roots of $A_2$. Every nonempty multiset of nonzero roots with vanishing sum yields a stable tangent bundle. Among these multisets, the resulting variety is Kähler-Einstein if and only if the multiset is invariant under negation or under the order-three rotation of the root hexagon. We also construct stable, non-Kähler-Einstein examples with nonzero twist sum in every even dimension at least four. The proofs reduce slope inequalities and barycenter computations to integrals over the moment hexagon. For a smooth toric Fano variety with strictly semistable tangent bundle, we prove that polystability is equivalent to decomposition as a nontrivial product of smooth toric Fano varieties with anticanonically stable tangent bundles. Exact evaluation of Klyachko's criterion extends the stability classification to dimensions five and six, recovering the classifications of Steffens and Reynolds in dimensions three and four. Together with the product criterion, this determines polystability for all $8{,}630$ varieties in dimensions three through six, and hence the existence of Hermitian-Einstein metrics on their tangent bundles with respect to an anticanonical Kähler form.

math.DG↗

arXiv2026

Deformations of toric pairs and the smoothing of Batyrev Calabi-Yau threefolds

Let $Δ$ be a reflexive four-polytope with primitive edges. We construct deformations of the toric fourfold defined by its face fan that smooth the singularities of a generic anticanonical Calabi-Yau hypersurface over a chosen two-dimensional face. The construction applies when the face cone is smoothable and the dual edge has an interior lattice point; its equations are trinomials in Cox coordinates. It gives a smoothing of the hypersurface when this is the only face whose cone is singular. If every singular face satisfies these conditions, simultaneous smoothing follows in the nodal case and, in general, under irreducibility of the semiuniversal deformation space. A seven-vertex example with one ordinary double point and no smoothing shows that the dual-edge condition is sharp. A scan of the Kreuzer--Skarke classification shows that the facewise criterion accounts for 3,774 of the 30,241 smoothings in the Batyrev--Kreuzer census; the other 26,467 require relations among exceptional curve classes from different faces.

math.AG↗

arXiv2026

Non-smoothable Calabi-Yau threefolds from reflexive polytopes

We construct projective Calabi-Yau threefolds with isolated Gorenstein canonical singularities that admit no smoothing, as anticanonical hypersurfaces in Gorenstein Fano toric fourfolds. One example has a unique singularity, the anticanonical cone over the first Hirzebruch surface. Another has a unique singularity whose reduced miniversal base is a smooth curve but whose deformations are all singular. The construction combines Altmann's deformation theory with the smoothing criterion of Corti-Filip-Petracci: a two-dimensional face of the ambient fan polytope with primitive edges obstructs smoothing if it has no Minkowski decomposition into unit segments and unimodular triangles. We also classify the 217 isolated Gorenstein toric threefold singularities whose polygon edge vectors have coordinates in $[-2,2]$ in some lattice basis, recording their deformation types and smoothing components. Subject to completeness and absence of repetitions in the database copy used, a scan of the Kreuzer-Skarke classification finds this obstruction in $39{,}175{,}536$ of its $473{,}800{,}776$ polytopes (approximately $8.27\%$), whose generic anticanonical hypersurfaces therefore admit no smoothing.

math.AG↗

arXiv2026

Beyond De Prado and Cotton: Hierarchical and Iterative Methods for General Mean-Variance Portfolios

Hierarchical Risk Parity (De Pardo) and the Schur-complement generalization of Cotton are among the most widely adopted regularised portfolio construction methods, yet both are signal-blind: they solve only the minimum-variance problem and cannot accommodate an arbitrary expected-return forecast. This paper introduces three methods that incorporate alpha signals into hierarchical and regularised portfolio construction. HRP-$μ$ is a hierarchical allocator that accepts an arbitrary signal $μ$ and nests standard HRP when $γ= 0$ and $μ=\mathbf{1}$. It preserves the tree-based structure of HRP while extending it beyond the minimum-variance setting. HRP-$Σμ$ strengthens this construction by replacing inverse-variance representatives with recursive local mean-variance optima, thereby using richer within-cluster covariance information at the same $O(N^2)$ asymptotic cost. CRISP (Correlation-Regularised Iterative Shrinkage Portfolios) is an iterative solver for $P_γw = μ$ with $P_γ= (1-γ)\operatorname{diag}(Σ) + γΣ$, so that $γ$ interpolates between a diagonal portfolio rule and full Markowitz. At convergence, CRISP is Markowitz applied to a variance-preserving shrunk covariance-diagonal variances unchanged, off-diagonal correlations shrunk-with $γ$ tuned for out-of-sample Sharpe rather than covariance-estimation loss. In Monte Carlo experiments across multiple covariance regimes and estimation ratios, HRP-$μ$ and HRP-$Σμ$ both outperform plain HRP with HRP-$Σμ$ consistently improving on HRP-$μ$. CRISP at intermediate $γ$ is the dominant method in both regimes, outperforming HRP, Cotton, Ledoit-Wolf shrinkage, direct Markowitz, and the signal-aware hierarchical methods.

q-fin.PM↗