arXiv research

Study on folded ribbon knots and their minimum length.

Origami can create complex knots, with minimum creases defining a new knot invariant.

New linking numbers link complex cycles to Calabi-Yau 3-folds.

Sharp lower bound on fold singularities self-intersections.

A generic smooth map of a closed $2k$-manifold into $(3k-1)$-space has a finite number of cusps ($Σ^{1,1}$-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities ($Σ^{1,0}$-singularities). Two fold maps are fold bordant if the…

Upper bounds on ribbonlength of various knots, showing linear and sub-linear behavior.

This paper confirms a bound for folded ribbonlength of 2-bridge knots.

We use braids and linking number to explain why automobile shades fold into an odd number of loops.

Study on ribbonlength and crossing number for folded ribbon knots.

Focusing on a small set of proteins that i) fold in a concerted, all-or-none fashion and ii) do not contain knots or slipknots, we show that the Gauss linking integral, the torsion and the number of sequence-distant contacts provide information regarding the folding rate. Our results suggest that the global topology/ge…

Study linking numbers in hyperbolic 3-folds, linking to Siegel modular forms.

Study on the parity of fold map singular points, showing non-invariance for odd-dimensional manifolds.

Machine learning predicts Hodge numbers of Calabi-Yau four-folds.

This note is a report on the observation that some singular varieties admit Calabi--Yau coverings. As an application, we construct 18 new Calabi--Yau 3-folds with Picard number one that have some interesting properties.

Study positive characteristic Fano 4-folds with nef tangent bundles.

Enhanced coloring invariant distinguishes folded molecular chain topologies.

We show by example that the Chern numbers c_1^3 and c_1 c_2 of a complex 3-fold are not determined by the topology of the underlying smooth compact 6-manifold. In fact, we observe that infinitely many different values of a Chern number can be achieved by (integrable) complex structures on a fixed 6-manifold.

The paper proves the existence of a folded annulus with multiple creases.

We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular att…

We construct non-Kähler simply connected Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers by smoothing normal crossing varieties with trivial dualizing sheaves.

The study examines the stretch factors of outer automorphisms and their latent symmetry.

3D manifolds can map to a plane with specific curve patterns.

This paper explores non-periodic folding of Spidron units, revealing nonlinear dynamics.

Proves bounds on ribbonlength for various knot types.

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a ribbon, and it turns out that the way the ribbon is folded influences the ribbonlength. We give an upper bound of $n\cot(π/n)$ for the ribbonlength of $…

This survey reviews Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a ribbon, and the ribbonlength problem asks to minimize the ribbonlength for a given knot type. We give a summary of known results. For the mos…

Many three dimensional manifolds are two-fold branched covers of the three dimensional sphere. However, there are some that are not. This paper includes exposition about two-fold branched covers and many examples. It shows that there are three dimensional homology spheres that do not two-fold branched cover any manifol…

The paper characterizes and contrasts knots with high 4D clasp numbers.

Consider an oriented curve $Γ$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding in the Euclidean space $\boldsymbol R^3$. This can be expressed as the image of an "origami map" $Φ:D\to \boldsymbol R^3$ such that $Γ$ is the singular set of $Φ$, the word "…

Machine learning approximates Calabi-Yau Hodge numbers from weight systems.

Icosahedral virus capsids are composed of symmetrons, organized arrangements of capsomers. There are three types of symmetrons: disymmetrons, trisymmetrons, and pentasymmetrons, which have different shapes and are centered on the icosahedral 2-fold, 3-fold and 5-fold axes of symmetry, respectively. In 2010 [Sinkovits &…

Study finds symplectic fillings' properties for specific contact covers.

In this paper, we study deformations of Brieskorn polynomials of two variables obtained by adding linear terms consisting of the conjugates of complex variables and prove that the deformed polynomial maps have only indefinite fold and cusp singularities in general. We then estimate the number of cusps appearing in such…

This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The previous paper in the series, math.DG/0008155, defined the idea of evolution data, which includes an (m-1)-submanifold P in R^n, and constructed a family of special Lagrangian m-folds N in C^m, which ar…

Paper proves ribbonlength grows linearly with knot complexity.

Study shows invariant curves in tubular origami dynamics, revealing geometric barriers to folding transitions.

This is a survey of the author's series of three papers math.DG/0111324, math.DG/0111326, math.DG/0204343 using analysis to investigate special Lagrangian 3-folds (SL 3-folds) in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, and their sequel math.DG/0011179 on sp…

A new approach uses circuit topology to study complex polymer interactions.

We provide a significant extension of the twisted connected sum construction of G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Some of the main contributions of t…

We speed up Gaussian process cross-validation calculations and improve model diagnostics.

Recent developments in specialized computer hardware have greatly accelerated atomic level Molecular Dynamics (MD) simulations. A single GPU-attached cluster is capable of producing microsecond-length trajectories in reasonable amounts of time. Multiple protein states and a large number of microstates associated with f…

Formula counts rational curves with a specific singular point in projective space.

Model selection is a crucial issue in machine-learning and a wide variety of penalisation methods (with possibly data dependent complexity penalties) have recently been introduced for this purpose. However their empirical performance is generally not well documented in the literature. It is the goal of this paper to in…

Paper proposes a method to compare vector fields across surfaces, useful for analyzing brain folding patterns.

We study quotients $Γ\backslash \mathbb H^n$ of the $n$-fold product of the upper half plane $\mathbb H$ by irreducible and torsion-free lattices $Γ< PSL_2(\mathbb R)^n$ with the same Betti numbers as the $n$-fold product $(\mathbb P^1)^n$ of projective lines. Such varieties are called fake products of projective lines…

Non-autoregressive method speeds up protein folding prediction 23 times.

Classifies Fano varieties with large pseudoindex and non-free rational curves.

We consider a priori generalization bounds developed in terms of cross-validation estimates and the stability of learners. In particular, we first derive an exponential Efron-Stein type tail inequality for the concentration of a general function of n independent random variables. Next, under some reasonable notion of s…