arXiv research

New surfaces found in a special space.

Two new triply periodic minimal surfaces of genus 4 discovered.

We use bifurcation theory to determine the existence of infinitely many new examples of triply periodic minimal surfaces in $\mathbb R^3$. These new examples form branches issuing from the H-family, the rPD-family, the tP-family, and the tD-family, that converge to some degenerate embedding of the families. As to nonde…

We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz-Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces.

We construct embedded triply periodic zero mean curvature surfaces of mixed type in the Lorentz-Minkowski 3-space with the same topology as the Schwarz D surface in the Euclidean 3-space.

The paper calculates Morse indices and nullities for triply periodic minimal surfaces.

The study constructs periodic surfaces using graph theory and applies cyclically branched coverings to identify their conformal type.

New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.

Study finds the shortest triply periodic graph spanning a cubic lattice.

New 2-parameter family of triply periodic minimal surfaces, sharing properties with Schwarz' D but distinct.

Given a tiling $\mathcal{T}$ of the plane by straight edge polygons, which is invariant by two independent translations, we construct a family of embedded triply periodic minimal surfaces which desingularizes $\mathcal{T}\times\mathbb{R}$. For this purpose, inspired by the work of Martin Traizet, we open the nodes of s…

Researchers create new triply periodic minimal surfaces by gluing saddle towers.

The study finds area estimates for specific surface types in a flat 3-torus.

Existence proof for two families of triply periodic minimal surfaces.

New method for counting distinct tilings with symmetrical surfaces.

We get a continuous one-parameter new family of embedded minimal surfaces, of which the period problems are two-dimensional. Moreover, one proves that it has Scherk second surface and Hoffman-Wohlgemuth example as limit-members.

We find the first examples of triply periodic minimal surfaces of which the intrinsic symmetries are all of horizontal type.

In this paper, we will construct an example of a closed Riemann surface $X$ that can be realized as a quotient of a triply periodic polyhedral surface $Π\subset \mathbb{R}^3$ where the Weierstrass points of $X$ coincide with the vertices of $Π.$ First we construct $Π$ by attaching Platonic solids in a periodic manner a…

New chiral minimal surfaces derived from quartz network.

Paper constructs and proves existence of chiral triply-periodic minimal surfaces.

Discuss Alan Schoen's I-WP minimal surface with geometric realizations.

This study investigates porosity and topological properties of TPMS using machine learning.

Paper defines linking numbers for periodic tangles.

Weaved helices form mechanically stable 3D structures.

Schwarz H surfaces can be continuously deformed into Meeks surfaces.

New framework for folding polyhedra into two planes.

Transforming cylindrical packings into bicontinuous surfaces.

Minimal twin surfaces are found in material science, resembling crystal twins.

New diagrams classify triply periodic entanglements.

Two surfaces minimize variance of Gaussian curvature.

The asymptotic behavior of open plane sections of triply periodic surfaces is dictated, for an open dense set of plane directions, by an integer second homology class of the three-torus. The dependence of this homology class on the direction can have a rather rich structure, leading in special cases to a fractal. In th…

For each braid $β\in Br_n$ we construct a $2$-periodic complex $\mathbb{S}_β$ of quasi-coherent $\mathbb{C}^*\times \mathbb{C}^*$-equivariant sheaves on the non-commutative nested Hilbert scheme $Hilb_{1,n}^{free}$. We show that the triply graded vector space of the hypecohomology $ \mathbb{H}( \mathbb{S}_β\otimes \wed…

Explains connections between monopoles and modules on elliptic curves.

We modify the definition of the Khovanov complex for oriented links in a thickening of an oriented surface to obtain a triply graded homological link invariant with a new homotopical grading.

We use algebraic Backlund transformations (BTs) to construct explicit solutions of the modified 2+1 chiral model from $T^2\times R$ to SU(n), where $T^2$ is a 2-torus. Algebraic BTs are parameterized by $z\in C$ (poles) and holomorphic maps $π$ from $T^2$ to Gr$(k,C^n)$. We apply Bäcklund transformations with carefully…

We trade matrix factorizations and Koszul complexes for Hochschild homology of Soergel bimodules to modify the construction of triply-graded link homology and relate it to Kazhdan-Lusztig theory.

Computes a specific homology for a type of braid.

Study reveals striking uniformity in triply graded link homology for specific braids.

Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth $C^1$-surface built from surface patches of Dupin cyclides, each patch being bounded by curvature lines of the supporting cyclide. An explicit de…

Soft cells fill space without gaps, derived from minimal surfaces and deformed using edge bending.

We propose a framework for unifying the sl(N) Khovanov-Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory which categorifies the HOMFLY polynomial. Moreover, this theory should have an additional formal structure of a fami…

Nonorientable spanning surfaces of periodic knots can have arbitrarily high first Betti number.

New minimal surfaces show stacking disorder in periodic structures.

New invariant restores symmetry in link homology and matches Hilbert scheme ideals.

Categorifies Jones polynomial for odd primes.

Algorithm finds periodic points on Veech surfaces.

Constructs a family of genus three minimal surfaces with parallel ends.

Study finds how periodic surfaces can bend without stretching.