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…
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48 results for “Folding”
Study on folded ribbon knots and their minimum length.
problem Finding the minimum length of folded ribbon knots.
method Using Kauffman's model of folded ribbon knots and analyzing their properties.
result Proved bounds on the minimum folded ribbonlength for various types of knots.
This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed norm…
This paper confirms a bound for folded ribbonlength of 2-bridge knots.
problem Bounding the folded ribbonlength of 2-bridge knots.
method Investigated the folded ribbonlength of 2-bridge knots and proved a linear upper bound.
result The folded ribbonlength of a 2-bridge knot is bounded above by .
The paper proves the existence of a folded annulus with multiple creases.
problem Existence of a folded annulus with multiple creases.
method Analyze developable surfaces, use normal curvature and relative torsion, compute geometric descriptors, prove propagation of folds.
result Proves the existence of a folded annulus with multiple creases.
First steps towards a classification of irreducible symplectic 4-folds whose integral 2-cohomology with 4-tuple cup product is isomorphic to that of Hilb^2(K3). We prove that any such 4-fold deforms to an irreducible symplectic 4-fold of Type A or Type B. A 4-fold of Type A is a double cover of a (singular) sextic hype…
Upper bounds on ribbonlength of various knots, showing linear and sub-linear behavior.
problem Estimating the ribbonlength of different types of knots.
method Using Kauffman's model of folded ribbon knots, we derive upper bounds on ribbonlength for specific knot types.
result Upper bounds on ribbonlength are linear in crossing number for some knots and sub-linear for others.
Enhanced coloring invariant distinguishes folded molecular chain topologies.
problem Apparent indistinguishability of folded chain topologies using current coloring invariants.
method Introduced Boltzmann weights to improve the resolving power of quandle colorings.
result Improved resolution in distinguishing folded chain topologies.
Origami can create complex knots, with minimum creases defining a new knot invariant.
problem Creating complex knots using origami folds.
method Developed a new knot invariant called the fold number, defined as the minimum number of creases required to obtain an equivalent knot.
result No proper foldings can produce nontrivial knots, but improper foldings can.
New contact structures on folded sums of contact mapping tori are tight under certain conditions.
problem Understanding tight contact structures on folded sums of contact mapping tori.
method Alternative bundle-theoretical construction and gluing process near the fold.
result Folded contact structures on folded sums of contact mapping tori are tight under specific conditions.
Every oriented 4-manifold admits a folded symplectic structure, which in turn determines a homotopy class of compatible almost complex structures that are discontinuous across the folding hypersurface ("fold") in a controlled fashion. We define folded holomorphic maps, i.e. pseudo-holomorphic maps that are discontinuou…
This paper explores non-periodic folding of Spidron units, revealing nonlinear dynamics.
problem Understanding the kinematics and nonlinear phenomena of Spidron units.
method Analysis of single unit cell kinematics and recursive construction of multiple cells.
result Non-periodic folding restricts isotropic folding as the number of unit cells increases.
A smooth map having only fold singularities is called a fold-map. We will give effective conditions for a continuous map to be homotopic to a fold-map from the viewpoint of the homotopy principle.
A generic smooth map of a closed -manifold into -space has a finite number of cusps (-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 (-singularities). Two fold maps are fold bordant if the…
New formulas for mean curvature of submanifolds in geometries with torsion.
problem Characterizing submanifolds in geometries with intrinsic torsion.
method Deriving formulas for mean curvature of associative, coassociative, and Cayley submanifolds.
result New obstructions to the local existence of coassociative 4-folds in G2-structures with torsion.
Study on the ribbonlength of knots and links, improving upper bounds.
problem Finding the infimal folded ribbonlength of knots and links.
method Creating new folded ribbon knots and applying them to improve upper bounds.
result Improved upper bounds for -torus links, twist knots, and pretzel links.
The study examines K-polystability on Fano 4-folds with specific Lefschetz defects.
problem Investigating K-polystability on Fano 4-folds with Lefschetz defect at least 2.
method Examining 19 families of Fano 4-folds with Lefschetz defect 3 and 175 families with Lefschetz defect 2, proving K-polystability and instability.
result Exactly 5 out of 19 families of Fano 4-folds with Lefschetz defect 3 are K-polystable, and 5 out of 175 Casagrande-Druel Fano 4-folds with Lefschetz defect 2 are K-polystable.
Geometric invariants of fold maps and their signatures.
problem Characterizing cobordism groups of fold maps.
method Combining immersion data and stable partial framings.
result A Poincare-Hopf type formula for the signature of manifolds.
Paper explores folding patterns of curved creases preserving their geometric properties.
problem Investigating rigid-ruling folding motions of curved crease-rule patterns.
method Deriving conditions for rigid-ruling foldability and analyzing combinations of creases.
result Constant fold-angle creases are only compatible with other constant fold-angle creases.
This paper is a follow-up to an earlier paper math.DG/0410260 on desingularizations of Calabi-Yau 3-folds with a conical singularity. In math.DG/0410260 we study Calabi-Yau 3-folds M_0 with a conical singularity at x modelled on some Calabi-Yau cone V, and construct a desingularization of M_0 by gluing in an Asymptotic…
Paper constructs fold maps with useful singular value sets.
problem Creating fold maps with specific singular value sets.
method Surgery operations to construct fold maps with crossings.
result Fold maps with singular value sets containing crossings.
Study submersions with definite folds on manifolds with boundary into Euclidean spaces.
problem Understanding differential-topological properties of manifolds with boundary under submersions with definite folds.
method Analyzing submersions with definite folds on manifolds with boundary into Euclidean spaces, focusing on restrictions to the boundary and using results for m-functions.
result Obtained restrictions on the diffeomorphism types of the source manifolds and studied the diffeomorphism types and Euler characteristics of manifolds admitting such maps.
This is the second in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The first paper was math.DG/0008021, which studied special Lagrangian m-folds with large symmetry groups. The third is math.DG/0010036, which uses ideas from this paper to construct families of special Lag…
New linking numbers link complex cycles to Calabi-Yau 3-folds.
problem Understanding complex analytic cycles and their Massey products.
method Relating ABC Massey products to holomorphic linking numbers.
result Constructed a family of Calabi-Yau 3-folds with non-vanishing Massey products.
The paper details folding of branched covers of the 3-sphere over knots.
problem Understanding folding of branched covers of the 3-sphere over knots.
method Presenting detailed foldings of dihedral covers of the 3-sphere branched over torus knots.
result Detailed foldings of dihedral covers of the 3-sphere branched over torus knots are presented.
Sharp lower bound on fold singularities self-intersections.
problem Finding a lower bound on the number of self-intersections of fold singularities.
method Established a sharp lower bound on the number of self-intersections of the boundary of an immersed surface, then applied this to fold singularities.
result Sharp lower bound on the number of self-intersections of fold singularities.
Study shows protein folding rate linked to topological changes.
problem Understanding protein folding kinetics and topology.
method Gauss linking integral, torsion, and sequence-distant contacts.
result Protein topology shifts from right-handed to left-handed with decreasing folding rate, associated with more sequence-distant contacts.
Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular …
Upper bounds and minimum ribbonlengths for folded ribbon knots are derived.
problem Understanding the ribbonlength of folded ribbon knots.
method Analysis of Kauffman's model of folded ribbon knots, upper bounds calculation, and minimization proofs.
result Minimum ribbonlength for a 3-stick unknot is 3√3, achieved by an equilateral triangle.
The paper defines and analyzes -fold vector bundles and their decompositions.
problem Understanding and decomposing -fold vector bundles.
method Introducing -fold vector bundles as functors and studying their cores and decompositions.
result Any -fold vector bundle admits a non-canonical isomorphism to a decomposed -fold vector bundle.
Folding maps on a cross-capmath.DG
Study symmetry of cross-cap surfaces with folding maps.
problem Reflectional symmetry of cross-cap surfaces.
method Characterization of singularities in folding maps.
result Characterized generic singularities on cross-cap.
Constructs non-Kähler Calabi-Yau 3-folds with large Betti numbers.
problem Finding non-Kähler Calabi-Yau 3-folds with high Betti numbers.
method Smoothing normal crossing varieties with trivial dualizing sheaves.
result Constructs non-Kähler Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers.
Study on manifolds that map to lower dimensions with specific critical points.
problem Characterizing manifolds that map to with round fold maps.
method Analyzing smooth -dimensional closed manifolds with and classifying round fold maps up to --equivalence.
result Determine which manifolds admit round fold maps into and classify these maps.
Generalized Quandle Polynomials and Their Applications to Stuquandles, Stuck Links, and RNA Foldingmath.GT
Generalized quandle polynomial used for stuquandles, stuck links, and RNA folding.
problem Defining polynomial invariants for stuquandles, stuck links, and RNA foldings.
method Introduced a generalized quandle polynomial and proved its invariance for stuquandles. Used this invariant to define polynomials for stuck links and RNA foldings.
result Polynomial invariants for stuquandles, stuck links, and RNA foldings.
Given a complex 4-fold with an (Calabi-Yau 3-fold) anti-canonical divisor , we study relative Donaldson-Thomas invariants for this pair, which are elements in the Donaldson-Thomas cohomologies of . We also discuss gluing formulas which relate relative invariants and invariants for Calabi-Yau 4-folds.
Survey on folding knots in paper to minimize length.
problem Minimizing the ribbonlength of folded ribbon knots.
method Review of Kauffman's model and upper bounds using crossing number.
result Upper bounds of ribbonlength for twist and torus knots.
How to Fold a Manifoldmath.GT
Techniques for constructing codimension 2 embeddings and immersions of the 2 and 3-fold branched covers of the 3 and 4-dimensional spheres are presented. These covers are in braided form, and it is in this sense that they are folded. More precisely the composition of the embedding (or immersion) and the canonical proje…
Whitehead's Lemma is a simple result of Stallings folds.
problem None explicitly stated; Whitehead's Lemma is observed as a consequence.
method Observation of Whitehead's Lemma as a consequence of Stallings folds.
result Whitehead's Lemma is a simple result of Stallings folds.
Paper studies surface knotting and its fold curves.
problem Understanding the knotting of surfaces and their fold curves.
method Examines the relationship between surface knotting and fold curves, defines a new invariant.
result Every surface can be isotoped so that its fold curves form an unlink.
Study characterizes SU(3)-structures with special Lagrangian 3-folds as minimal.
problem Characterizing SU(3)-structures with special Lagrangian 3-folds.
method Derivation of mean curvature formulas and analysis of SU(3)-structures.
result Obtained formulas for mean curvature and characterized SU(3)-structures.
3D manifolds can map to a plane with specific curve patterns.
problem Characterizing 3D manifolds that can map to the plane with certain curve patterns.
method Analyzing fold maps and their critical value sets.
result Closed orientable 3-manifolds admit round fold maps into the plane if and only if they are graph manifolds.
This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students i…
We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable …
In this paper we introduce the area of 2-ruled 4-folds in R^n (n=7 or 8), that is, submanifolds M of R^n that admit a fibration over some 2-fold Sigma such that each fibre is an affine 2-plane in R^n. This is motivated by the paper math.DG/0012060 by Joyce on ruled special Lagrangian 3-folds in C^3 and the work of the …
Study maxfaces and minfaces converging to surfaces with folded singularities.
problem Analyzing surfaces with folded singularities and their convergence properties.
method Constructing families of maxfaces and minfaces with increasing cuspidal crosscaps.
result Maxfaces and minfaces converge to surfaces with folded singularities.
Geometric U-folds explained in 4D supergravity.
problem Understanding geometric U-folds in 4D supergravity.
method Construction of geometric U-folds compatible with non-trivial supergravity extensions.
result Smooth non-trivial U-folds can exist only under specific conditions.
Two Fold Branched Coversmath.GT
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…
Biplanar Foldingsmath.DG
New framework for folding polyhedra into two planes.
problem Creating polyhedra that can be folded into two planes.
method General framework for bifoldable polyhedra construction.
result Examples of infinite triply periodic and fractal bifoldable polyhedra.