Research finds bounds for knots in hexagonal lattice and classifies 11-stick knots.
problem Determining the stick number and edge length of knots in a hexagonal lattice.
method Introducing a linear transformation between lattices to prove strict inequalities and classifying knots.
result Only trefoil and figure-eight knots are 11-stick knots in the hexagonal lattice.
New lattice stick knot condition identified.
problem Determining if 2D lattice knots project to 3D lattice sticks.
method Provided a necessary and sufficient condition.
result Identified a condition for 2D lattice knots to project to 3D lattice sticks.
The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p,p+1)-torus knots, and show how composing and taking satellites can be used to obtain t…
Upper bound for lattice stick number of spatial graphs.
problem Finding the minimum number of sticks in a cubic lattice to represent spatial graphs.
method Defined lattice stick number for spatial graphs and presented an upper bound in terms of crossing number.
result An upper bound for the lattice stick number of spatial graphs.
The lattice stick number of a knot type is defined to be the minimal number of straight line segments required to construct a polygon presentation of the knot type in the cubic lattice. In this paper, we mathematically prove that the trefoil knot 31 and the figure-8 knot 41 are the only knot types of lattice stic…
The lattice stick number sL(K) of a knot K is defined to be the minimal number of straight line segments required to construct a stick presentation of K in the cubic lattice. In this paper, we find an upper bound on the lattice stick number of a nontrivial knot K, except trefoil knot, in terms of the minimal c…
Knots and links have been considered to be useful models for structural analysis of molecular chains such as DNA and proteins. One quantity that we are interested on molecular links is the minimum number of monomers necessary to realize them. In this paper we consider every link in the cubic lattice. Lattice stick numb…
Vertex distortion measures how far lattice knots deviate from straight lines.
problem Measuring how much lattice knots deviate from straight paths.
method Analogous to smooth knots, study vertex distortion in lattice knots.
result Vertex distortion is 1 only for the unknot and can be arbitrarily high.
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.
This paper finds upper bounds for lattice stick numbers of rational links with specific stick configurations.
problem Finding upper bounds for the lattice stick number of rational links with exactly 4 z-sticks.
method Using 2-circuit presentations, the paper constructs lattice stick numbers with exactly 4 z-sticks and derives upper bounds.
result Upper bounds for the lattice stick number of rational links with exactly 4 z-sticks are derived.
Improved bounds on stick numbers of knots up to 13 crossings.
problem Finding better bounds on the stick number of knots.
method Simulated annealing with knot-type preserving moves.
result Comprehensive table of stick number bounds on all knots through 13 crossings.
Exact stick number of two knots with 10 crossings found.
problem Determining the stick number of specific knots.
method Analyzing specific knots 13n592 and 15n41,127 to find their stick number. result First non-torus prime knots with more than 9 crossings have stick number 10.
The stick index of a knot is the least number of line segments required to build the knot in space. We define two analogous 2-dimensional invariants, the planar stick index, which is the least number of line segments in the plane to build a projection, and the spherical stick index, which is the least number of great c…
New bounds on stick number of knots found using random polygon generation.
problem Understanding the minimum number of segments needed to build a polygonal knot.
method Monte Carlo approach to generating and analyzing large ensembles of random polygons.
result Improved bounds on stick number for over 40% of knots with 10 or fewer crossings.
An equilateral stick number s=(K) of a knot K is defined to be the minimal number of sticks required to construct a polygonal knot of K which consists of equal length sticks. Rawdon and Scharein [12] found upper bounds for the equilateral stick numbers of all prime knots through 10 crossings by using algorithm…
Negami found an upper bound on the stick number s(K) of a nontrivial knot K in terms of the minimal crossing number c(K) of the knot which is s(K)≤2c(K). Furthermore McCabe proved s(K)≤c(K)+3 for a 2-bridge knot or link, except in the case of the unlink and the Hopf link. In this paper we const…
The study proves all prime knots up to 10 crossings have superbridge index ≤ 5.
problem Determining the maximum superbridge index for prime knots up to 10 crossings.
method New upper bounds on stick numbers and equilateral stick numbers for specific knots, leading to conclusions about superbridge index.
result All prime knots through 10 crossings have a superbridge index ≤ 5.
New method finds exponential growth in knot types from sticks.
problem How many knots can be formed with a fixed number of sticks?
method Polygonal self-intersection to sparse real-algebraic chamber problem, braid construction.
result Factorial-scale upper bound for knot types, optimal growth order.
Upper bounds on stick and equilateral stick numbers of spatial graphs derived.
problem Finding upper bounds on stick numbers of spatial graphs.
method Defined stick and equilateral stick numbers of spatial graphs, derived new upper bounds.
result Presented new upper bounds for stick and equilateral stick numbers of spatial graphs.
In 1991, Negami found an upper bound on the stick number s(K) of a nontrivial knot K in terms of the minimal crossing number c(K) of the knot which is s(K)≤2c(K). In this paper we improve this upper bound to s(K)≤23(c(K)+1). Moreover if K is a non-alternating prime knot, then $s(K) \leq…
Study on knots formed by Coxeter galleries, finding bounds and symmetric trefoils.
problem Understanding knots created by Coxeter galleries.
method Examined knots in affine Coxeter complex of type \widewedge{B3}, constructing galleries and proving properties.
result Found bounds on stick number and smallest length of symmetric trefoils.
Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise to a piecewise linear (PL) approximant. This is extended to isotopic convergenc…
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 ncot(π/n) for the ribbonlength of $…
New upper bounds on superbridge index for 49 knots, increasing known results to 49.
problem Determining superbridge index for knots not covered by stick number.
method New upper bounds not derived from stick number.
result Superbridge index of 4 for 49 knots, increasing known results to 49.
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.
New superbridge index calculations for knots with odd edges.
problem Computing superbridge index of knots.
method Polygonal realizations with odd edges and linear programming.
result Exact superbridge index of many new knots, including 9- and 12-crossing knots.
The study examines knot probabilities in confined lattice polygons.
problem Determining the relative knotting probabilities in confined lattice knots.
method Used Monte Carlo algorithms to enumerate conformations of lattice knots in a confined volume.
result Relative knotting probabilities are small, with the model dominated by unknots.
A new surgery formula for knot lattice homology.
problem Developing a new surgery formula for knot lattice homology.
method Provided an iterable version of the surgery formula using doubly-filtered spaces and involutive data.
result Computed knot lattice spaces for specific knots and three-manifolds.
Classifies knots by lattice size, finding unknot ratios and crossing numbers.
problem Understanding the distribution of knots within different lattice sizes.
method Introduced a new knot classification by lattice size, analyzed ratios of unknots and knots with more than 10 crossings, and compared with theoretical estimates.
result Ratio of unknots decreases exponentially with lattice size, and computational results match theoretical estimates.
Smooth knots with odd Conway polynomial terms have inscribed trefoils.
problem Finding inscribed trefoils for smooth knots with specific polynomial terms.
method Using a perturbation of the double-cover of the orientation class and analyzing planar configurations.
result Smooth knots with odd quadratic terms of the Conway polynomial have inscribed trefoils.
One type of switch simplifies operations on lattice knots.
problem Operations on lattice knots are complex.
method Reduced operations to one type of local switch.
result Simplified set of operations on lattice knots.
Study lattice paths from twist knots and double twist knots.
problem Understanding combinatorics of twist knots and double twist knots.
method Analyzing quiver generating series of HOMFLY-PT polynomial limits.
result Lattice path models for twist knots and double twist knots.
Knot lattice homology invariant of smooth knot type in rational homology spheres.
problem Invariance of knot lattice homology in rational homology spheres.
method Proving knot lattice homology invariant through doubly-filtered homotopy type.
result Knot lattice homology invariant of smooth knot type in rational homology spheres.
The (isothermic) compressibility of lattice knots can be examined as a model of the effects of topology and geometry on the compressibility of ring polymers. In this paper, the compressibility of minimal length lattice knots in the simple cubic, face centered cubic and body centered cubic lattices are determined. Our r…
Knot lattice homology invariant is preserved under certain 3-manifold diffeomorphisms.
problem Preserving knot lattice homology invariants under 3-manifold diffeomorphisms.
method Examined filtered lattice chain homotopy types of negative-definite forests with one unframed vertex.
result Filtered lattice chain homotopy type is an invariant of the diffeomorphism type of resulting 3-manifolds.
Discrete knot theory models use lattice-filtered graphs to detect merging knot components.
problem Detecting merging knot components in discrete models.
method Lattice-filtered move graphs to model knot types, identifying connected components and merge scales.
result Merge scale defined by connected components of lattice-filtered move graphs, with specific examples for the figure-eight knot.
New deformations of lattice cohomology help calculate knot invariants.
problem Calculating knot invariants using lattice cohomology.
method Using holomorphic triangles counting and lattice cohomology.
result Combinatorial formulae for the upsilon invariant are derived.
Classifies lattices from knot surgeries, defining a concordance invariant.
problem Classifying lattices from knot surgeries.
method Classifies lattices as the intersection form of a four manifold with boundary.
result Defines a concordance invariant and generalizes a theorem on lens space surgeries.
We construct an infinite commutative lattice of groups whose dual spaces give Kauffman finite-type invariants of long virtual knots. The lattice is based "horizontally" upon the Polyak algebra and extended "vertically" using Manturov's functorial map f. For each n, the n-th vertical line in the lattice contains a…
Study links weaving knots with polynomial coefficients and lattice numbers.
problem Understanding polynomial coefficients of weaving knots and their lattice counterparts.
method Established relationships between Jones and Chebyshev polynomials, and derived explicit formulas for Alexander polynomials.
result Proved coefficients of Jones polynomial are Whitney numbers of Lucas lattices and satisfied Fox's trapezoidal conjecture.
New invariant connects knot homology and BPS series for plumbed knot complements.
problem Understanding invariants of plumbed knot complements.
method Introducing an invariant unifying knot lattice homology and BPS series, proving a surgery formula.
result Proved a surgery formula relating the new invariant to the weighted graded root of the surgered 3-manifold.
Proof of Knot Entropy Conjecture for tube lattice polygons.
problem Proving exponential growth rate of knot polygons equals unknot polygons.
method Upper and lower bounds on polygon counts, braid insertions, and pattern theorems.
result Established the Knot Entropy Conjecture for tube lattice polygons.
We show that the knot lattice homology of a knot in an L-space is equivalent to the knot Floer homology of the same knot (viewed these invariants as filtered chain complexes over the polynomial ring Z/2Z [U]). Suppose that G is a negative definite plumbing tree which contains a vertex w such that G-w is a union of rati…
The paper defines and studies discrete p-density and compression-radius profiles of lattice knots.
problem Understanding geometric properties of lattice knots.
method Develops a framework for discrete p-density and compression-radius profiles of lattice knots, studying them on length-filtered sets and finite move-graph exploration.
result Density and compression-radius values are not monotone, illustrating distinct optimization problems.
Conditions for polynomial to be isometry of lattice, answering Hasse principles.
problem Conditions for integral polynomials to be characteristic polynomials of isometries of lattices.
method Necessary and sufficient conditions derived from lattice isometries and Hasse principles.
result Proved a Hasse principle for signatures of knots.
The study determines lens spaces that can be obtained from surgeries on knots in the Poincaré homology sphere.
problem Identifying lens spaces that can be obtained from surgeries on knots in the Poincaré homology sphere.
method Developed a lattice embedding obstruction to realize L-space surgeries on knots in the Poincaré homology sphere.
result Identified the only two knots in the Poincaré homology sphere that admit half-integer lens space surgeries.
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 proof of Alexander polynomial constraints for lens space surgeries.
problem Constraints on Alexander polynomials for lens space surgeries.
method Using changemaker lattices to prove a theorem.
result Constraints on Alexander polynomials for specific surgeries.