Self-affine arcs without inner weak separation are parabolic segments.
problem Characterizing self-affine Jordan arcs without parabolic segments.
method Analyzing the weak separation property and proving implications for arc types.
result Self-affine Jordan arcs without parabolic segments are attractors of multizippers.
Unified description of aesthetic curves through self-affinities.
problem Characterizing log-aesthetic curves and their properties.
method Reformulating and proving self-affinities of planar curves, integrating equiaffine geometry.
result Unified characterization of constant curvature curves in similarity and equiaffine geometries.
We test for departures from normal and independent and identically distributed (NIID) returns, when returns under the alternative hypothesis are self-affine. Self-affine returns are either fractionally integrated and long-range dependent, or drawn randomly from an L-stable distribution with infinite higher-order moment…
This paper studies closed 3-manifolds which are the attractors of a system of finitely many affine contractions that tile R3. Such attractors are called self-affine tiles. Effective characterization and recognition theorems for these 3-manifolds as well as theoretical generalizations of these results to hig…
We develop tools to study the topology and geometry of self-affine fractals in dimension three and higher. We use the self-affine structure and obtain rather detailed information about the connectedness of interior and boundary sets, and on the dimensions and intersections of boundary sets. As an application, we descri…
The paper examines properties of self-affine Sierpiński sponges using metric invariants.
problem Investigating properties of self-affine Sierpiński sponges using metric invariants.
method Examined through maximal power law property and perfectly disconnectedness.
result Characterized self-affine Sierpiński sponges by their metric properties.
Let A be an expanding d×d matrix with integer entries and D⊂Zd be a finite digit set. Then the pair (A,D) defines a unique integral self-affine set K=A−1(K+D). In this paper, by replacing the Euclidean norm with a pseudo-norm w in terms of A, we…
New tiles in higher dimensions are shown to be homeomorphic to balls.
problem Characterizing self-affine tiles in higher dimensions as balls.
method Using Brouwer's invariance of domain theorem and a horizontal distance tool.
result Necessary and sufficient conditions for tiles to be d-dimensional tame balls. Study finds a measure for sponge components of Lalley-Gatzouras type.
problem Understanding the distribution of δ-connected components in self-affine sponges.
method Generalized existing results to self-affine sponges of Lalley-Gatzouras type, proving a measure relationship.
result Existence of a Bernoulli measure for cylinder components with a specific asymptotic relation.
Self-affine tiles homeomorphic to a ball proven for a specific digit set.
problem Topology of self-affine tiles with collinear digit sets.
method Proving homeomorphism to a ball using integral self-affine tiles with collinear digit sets.
result A large class of integral self-affine tiles with collinear digit sets is homeomorphic to a closed 3-dimensional ball.
New aesthetic curves in equiaffine geometry include the quadratic and logarithmic spiral.
problem Designing aesthetic shapes in equiaffine geometry.
method Introducing a new symmetry (ESA) to characterize planar curves.
result The new class of curves includes the quadratic curve and logarithmic spiral.
In this paper, we consider the connectedness of planar self-affine set T(A,D) arising from an integral expanding matrix A with characteristic polynomial f(x)=x2+bx+c and a digit set D={0,1,…,m}v. The necessary and sufficient conditions only depending on b,c,m are given for the $T(A…
Let M be a 3×3 integer matrix each of whose eigenvalues is greater than 1 in modulus and let D⊂Z3 be a set with ∣D∣=∣detM∣, called digit set. The set equation MT=T+D uniquely defines a nonempty compact set T⊂R3. If T has positive L…
We study the connectedness of the planar self-affine sets T(A,D) generated by an integer expanding matrix A with ∣det(A)∣=3 and a non-collinear digit set D={0,v,kAv} where k∈Z∖{0} and v∈Z2 such that {v,Av} is linearly independent. By chec…
In the paper, we focus on the connectedness of planar self-affine sets T(A,D) generated by an integer expanding matrix A with ∣det(A)∣=3 and a collinear digit set D={0,1,b}v, where b>1 and v∈R2 such that {v,Av} is linearly independent. We discuss the domain of…
Study on Hausdorff dimension of Anosov subgroup limit sets under specific affine complexity.
problem Investigating the Hausdorff dimension of Anosov subgroup limit sets with self-affine complexity.
method Analyzing the Hausdorff dimension of projective limit sets Λ1(Γ) of Anosov subgroups Γ under specific assumptions about their affine complexity. result The Hausdorff dimension of Λ1(Γ) is determined by the critical exponent of the first simple root under partial quasi-self-similarity. We propose a construction which transforms a self-similar zipper in Rn to a self-affine zipper Rn+1 whose attractor is a smooth curve.
Let T:=T(A,D) be a disk-like self-affine tile generated by an integral expanding matrix A and a consecutive collinear digit set D, and let f(x)=x2+px+q be the characteristic polynomial of A. In the paper, we identify the boundary ∂T with a sofic system by constructing a ne…
In this paper, we formulate a new local move on virtual knot diagram, called arc shift move. Further, we extend it to another local move called region arc shift defined on a region of a virtual knot diagram. We establish that these arc shift and region arc shift moves are unknotting operations by showing that any virtu…
Study on unknotting twisted knots using arc shift and region arc shift moves.
problem Unknotting twisted knots and finding bounds for region arc shift number.
method Introduced arc shift move and region arc shift move for twisted knots.
result Found families of twisted knots with specific arc shift and region arc shift numbers.
It is shown that the projection image of an oriented spatial arc to any oriented plane is approximated by a unique arc diagram (up to isomorphic arc diagrams) determined from the spatial arc and the projection. In a separated paper, the knotting probability of an arc diagram is defined as an invariant under isomorphic …
NT probability measures knotting in 3D arc systems.
problem Measuring knotting in 3D arc systems.
method Transforming polygonal arcs into unique diagrams, generalizing NT probability.
result Properties of NT probability for 3D arc systems are shown.
Non-trivialization probability of arc system in 3D space
problem Defining and generalizing the knotting probability of an arc diagram in 3D space
method Transforming polygonal arcs in 3D space into unique arc diagrams
result Introducing and generalizing the Non-Trivialization probability (NT probability) for arc systems in 3D space
The study of smoothing arcs and curves on surfaces, proving tautness and arc length spectrum properties.
problem Analyzing the geometric and combinatorial effects of smoothing intersections in arcs or curves.
method Geometric and combinatorial analysis, proving tautness and arc length spectrum properties.
result Shortest arcs with self-intersections have exactly or at most one more self-intersection than the self-intersection number.
Minimal grid diagrams for 15,735 knots with 14 crossings and arc index 14.
problem Representing prime knots with 14 crossings and specific arc indices using grid diagrams.
method Enumerated all prime knots with 14 crossings, categorized by arc index, and found minimal grid diagrams for those with arc index 14.
result 8,027 knots with arc index 13 and 15,735 knots with arc index 14 were represented by minimal grid diagrams.
This paper calculates stick numbers for rail arcs and knot classes.
problem Calculating the minimum number of sticks needed for rail arcs and knot classes.
method Rail isotopies, ambient isotopies, winding number invariant, and lattice stick number.
result Calculates stick numbers for rail arcs and knot classes with crossing number at most 9.
Counts arcs in surfaces, proving convergence of geodesic currents.
problem Counting arcs of the same type in compact surfaces and related geometries.
method Derives convergence of geodesic currents to prove arc counts.
result Proves convergence of geodesic currents, leading to arc counting results.
Study arcs on surfaces, focusing on topological aspects and group actions.
problem Understanding arcs and their complements on surfaces.
method Characterize infinite-type surfaces via homeomorphic subsurfaces, construct actions on arc graphs.
result New characterisation of infinite-type surfaces and actions on arc graphs.
Listed 19,513 prime knots with arc index 12-16.
problem Tabulating prime knots with specific arc indices.
method Provided list of prime knots with minimal grid diagrams.
result 19,513 prime knots with arc index 12-16.
As a supplement to the authors' article "Prime knots with arc index up to 11 and an upper bound of arc index for non-alternating knots", to appear in the Journal of Knot Theory and its Ramifications, we present minimal arc presentations of the prime knots up to arc index 11.
The grand arc graph's asymptotic dimension is shown to be infinite.
problem Determining the asymptotic dimension of the grand arc graph.
method Using Gromov-hyperbolic and cocompact arc and curve models, the asymptotic dimension is shown to be infinite for a broad class of surfaces.
result The asymptotic dimension of the grand arc graph is infinite.
The study counts arcs on hyperbolic surfaces, providing asymptotic growth formulas.
problem Counting arcs on hyperbolic surfaces with boundaries and cusps.
method Asymptotic analysis of pure mapping class group orbits and arc lengths.
result The number of arcs of bounded length is asymptotically proportional to L6g−6+2(n+p). Study self-intersections of arcs on a pair of pants, proving natural number spectrum.
problem Understanding self-intersections of arcs on a pair of pants.
method Algorithm to compute self-intersection number, bounds established in terms of word length.
result Spectrum of self-intersection numbers covers all natural numbers.
Solve arc diagrams on surfaces via branched covers.
problem Computing arc diagrams on surfaces via branched covers.
method Represent branched covers combinatorially and solve membership problem.
result Efficient solution for triangulated arc diagrams.
Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the arc index of the knot. We have identified all prime knots with arc index up to 1…
The paper classifies virtual links using the arc shift operation.
problem Classifying \( n \)-component virtual links up to arc shift equivalence.
method Established the arc shift operation as an unknotting tool for \( n \)-homogeneous proper virtual links, explored its connection to the odd writhe, and identified sequences with specific arc shift bounds.
result Identified sequences of virtual link diagrams \( L_n \) with an upper bound of arc shift number equal to \( n \).
Expanded Legendrian knot atlas for 10-arc index knots.
problem Lack of Legendrian knot data for knots with high arc index.
method Created an atlas of Legendrian knots up to arc index 10.
result Legendrian knots of arc index 10 have been cataloged.
Classifies arcs on a 4-punctured sphere that intersect at most once.
problem Classifying arcs on a 4-punctured sphere with intersection constraints.
method Classification of maximal systems of arcs intersecting at most once.
result Maximal systems of arcs on the 4-punctured sphere identified.
The paper shows how to rearrange arcs to form closed curves.
problem Creating closed curves from planar arcs.
method Splitting a curve into arcs and rearranging them to form a closed curve.
result Closed curves can be formed by rearranging arcs under weak assumptions.
For any compact, connected, orientable, finite-type surface with marked points other than the sphere with three marked points, we construct a finite rigid set of its arc complex: a finite simplicial subcomplex of its arc complex such that any locally injective map of this set into the arc complex of another surface wit…
A new method joins two arcs with a degree of freedom.
problem Joining two arcs with a precise point.
method Geometric approach using tangent vectors and points.
result A novel method to determine the join point.
Graph conditions ensure matching arc complexes are connected and hyperbolic.
problem Conditions for connectedness and hyperbolicity of matching arc complexes.
method Conditions on finite simplicial graphs guaranteeing connectedness and hyperbolicity of matching arc complexes.
result Conditions on finite simplicial graphs ensure connectedness and hyperbolicity of matching arc complexes.
Study links' arc index and Turaev genus, proving conjectures.
problem Understanding the arc index and Turaev genus of links.
method Computed arc index, established bounds, and conjectured inequalities.
result Proved conjectures linking crossing number, arc index, and Turaev genus.
Minimal grid diagrams found for 13-crossing prime knots with 13 arc index.
problem Finding minimal grid diagrams for prime knots with specific crossing and arc indices.
method Used Knotscape to generate spanning trees and obtain minimal arc presentations in grid diagrams.
result 9,988 prime knots with 13 crossings and 13 arc index were identified.
Improves arc separation result for homogeneous spaces.
problem Separating regions in homogeneous spaces by arcs.
method Using homogeneity instead of strong local homogeneity, and considering arcs with one interior point.
result Regions in homogeneous spaces of dimension ≥ 2 are not separated by arcs.
A classical combinatorial fact is that the simplicial complex consisting of disjointly embedded chords in a convex planar polygon is a sphere. For any surface F with non-empty boundary, there is an analogous complex Arc(F) consisting of suitable equivalence classes of arcs in F connecting its boundary components. The m…
The paper explores when specific knot operations simplify diagrams.
problem Understanding when arc crossing changes simplify knot diagrams.
method Examined two types of arc crossing changes on link diagrams and determined when they are unknotting operations.
result Any two crossing points in an alternating knot diagram are arc crossing change admissible.
It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams o…