Enumerated all genus two handlebody-knots with seven crossings.
problem Counting genus two handlebody-knots with specific crossings.
method Extending an existing table of genus two handlebody-knots.
result Enumerated all genus two handlebody-knots with seven crossings.
The article enumerates doubly symmetric diagrams for knots up to 18 crossings.
problem Enumerating doubly symmetric diagrams for knots.
method Developed an enumeration strategy for prime knots given by doubly symmetric diagrams.
result Determined all cases of doubly symmetric diagrams up to 18 crossings.
A polynomial counts knot states for a specific type of knot.
problem Counting states of two-bridge knots with Conway notation.
method Constructing an integer polynomial to enumerate Kauffman states.
result The polynomial accurately counts states of knots with C(n,r) notation.
We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numb…
This is the third paper in a series devoted to enumerating the prime alternating knots and links. This paper establishes a method for enumerating the prime alternating links. It is shown that one may choose any prime alternating link diagram of a given minimal crossing size and by applications of just two operators (T …
Enumerates knots up to five crossings and describes moves between them.
problem Counting and classifying knots up to a specific number of crossings.
method Generated tables of minimal diagrams and derived moves between knots.
result Conjecture about a lower bound for the triple-crossing number based on Alexander polynomial.
A process enumerates rack elements from a presentation.
problem Systematically enumerating elements of racks from presentations.
method Generalizes Todd-Coxeter process for cosets, adapted for racks.
result Process terminates if and only if rack is finite, outputting operation tables.
Permutations linked to knots and links, with unknots counted by Schröder numbers.
problem Understanding permutations as knots and links.
method Using grid diagrams and Bennequin's inequality.
result Permutations corresponding to unknots and links are counted by Schröder numbers.
New method classifies knots and links uniquely based on bridge number.
problem Classifying knots and links by crossing number is computationally infeasible.
method Using bridge number and geometric topology techniques, proving uniqueness of diagrams.
result Infinitely many knot and link diagrams have a unique simple m-bridge diagram under certain conditions. This is the first in a series of four papers wherein we enumerate all prime alternating knots and links. In this first paper, we introduce four operators on knots and show that, when used according to very simple rules on the prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossing…
The study enumerates virtual quandles up to isomorphism.
problem Classifying virtual quandles up to isomorphism.
method Computer search and classification based on conjugacy class structures of rack automorphism groups.
result Classifications of virtual racks and quandles up to order 8.
We construct a new type of geometric knot theory, plumbers' knots, and solve the problems of distinguishing and enumerating such knots at a fixed level of complexity. (v2) Minor edits, added theorem 3.18. (v3) Substantial revisions, essentially completely rewritten in places.
This paper calculates the growth constant for quantum knot mosaics.
problem Calculating the growth constant for quantum knot mosaics.
method Introduced knot mosaic system, defined quantum knots, and used enumeration to find the growth constant.
result Existence and bounds of the knot mosaic constant δ. Researchers developed an algorithm to count all graph mosaics.
problem Defining and counting graph mosaics to represent graph diagrams.
method Using a recursion formula of state matrices and sixteen graph mosaic tiles.
result Produced the exact enumeration of all graph mosaics.
The crosscap number of a knot in the 3-sphere is defined as the minimal first Betti number of non-orientable subsurfaces bounded by the knot. In this paper, we determine the crosscap numbers of pretzel knots. The key ingredient to obtain the result is the algorithm of enumerating all essential surfaces for Montesinos k…
In low-dimensional topology, many important decision algorithms are based on normal surface enumeration, which is a form of vertex enumeration over a high-dimensional and highly degenerate polytope. Because this enumeration is subject to extra combinatorial constraints, the only practical algorithms to date have been v…
Proved Jones polynomial distinguishes unknots up to 22 crossings.
problem Determining if the Jones polynomial can distinguish the unknot for knots up to 22 crossings.
method Generated knot diagrams, computed their Jones polynomials, tested for unknottedness, employed novel strategies to reduce computation time.
result Verified Jones polynomial distinguishes the unknot for knots up to 22 crossings.
This is the second of a part series devoted to enumerating prime alternating knots and links. In Part I, we introduced four operators on knots and showed that if these operators are applied to the set of all prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossings is obtained. In …
The topological underpinnings are presented for a new algorithm which answers the question: `Is a given knot the unknot?' The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot is unknotted if and …
Characterizes knot-theoretic flocks up to 64 elements.
problem Classifying ternary quasigroups for knot theory.
method Group action on flock colorings to improve knot-theoretic invariant.
result Enumerated and characterized knot-theoretic flocks up to 64 elements.
The study calculates the average genus of rational knots and links.
problem Finding the average genus of rational knots and links.
method Enumerating and calculating the number of rational knots and links with a given crossing number.
result A precise formula for the average minimal genus of rational knots and links.
The set consisting of all rotations of the Euclidean plane is equipped with a quandle structure. We show that a knot is colorable by this quandle if and only if its Alexander polynomial has a root on the unit circle in C. Further we enumerate all non-trivial colorings of a torus knot diagram by the quandle u…
A combinatorial framework classifies genus-one knots and links.
problem Classifying knots and links on a torus.
method Maps on surfaces, permutation pairs, bit data, Reidemeister II reductions, state sums.
result Completely enumerated and reproducible genus-one knot and link diagrams.
We prove that if an alternating 3-braid knot has unknotting number one, then there must exist an unknotting crossing in any alternating diagram of it, and we enumerate such knots. The argument combines the obstruction to unknotting number one developed by Ozsváth and Szabó using Heegaard Floer homology, together with o…
Classifies exceptional Legendrian torus knots using contact surgery diagrams.
problem Classifying exceptional Legendrian realisations of non-trivial torus knots.
method Contact surgery diagrams and upper bounds on tight contact structures.
result Classification of exceptional Legendrian realisations of torus knots.
We give constructions to realize an odd number, which is representable as sum of two squares, as determinant of an achiral knot, thus proving that these are exactly the numbers occurring as such determinants. Later we study which numbers occur as determinants of prime alternating achiral knots, and obtain a complete re…
An i-hedrite is a 4-regular plane graph with faces of size 2, 3 and 4. We do a short survey of their known properties and explain some new algorithms that allow their efficient enumeration. Using this we give the symmetry groups of all i-hedrites and the minimal representative for each. We also review the link of 4-hed…
Alternating knots follow a pattern theorem, making them rarer than previously thought.
problem Understanding the scarcity of alternating knots.
method Developed a pattern theorem for alternating knots and used it to prove a conjecture about their rarity.
result Alternating knots are rarer than previously believed.
We compose the table of knots in the thickened torus T x I having diagrams with at most 4 crossings. The knots are constructed by the three-step process. First we list regular graphs of degree 4 with at most 4 vertices, then for each graph we enumerate all corresponding knot projections, and after that we construct the…
Study ribbon concordance and minimal compressions, proving new results about fibered knots.
problem Understanding ribbon concordance and minimal compressions of surface homeomorphisms.
method Proving monotonicity of simplicial volume and dilatation under ribbon concordance, algorithmic enumeration of minimal compressions.
result Every fibered knot has only finitely many predecessors in the ribbon-concordance partial order.
The paper improves bounds on knot crossings and tabulates minimal diagrams.
problem Improving bounds on knot crossings and tabulating minimal diagrams.
method Analyzing triple-crossing and delta-crossing numbers, proving tangle existence, generating tables.
result Improved bounds on knot crossings and tabulated minimal diagrams for prime knots up to delta-crossing number 4.
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.
New types of knot mosaics help count and analyze knots efficiently.
problem Counting and analyzing knots efficiently.
method Introducing period and toroidal knot mosaics and developing algorithms for their enumeration.
result Exact enumeration of period knot mosaics and asymptotics of toroidal knot mosaics.
Slipknots found in random diagrams almost always.
problem The presence of slipknots in random diagrams.
method Developed knotoid diagrams to study slipknots in knot diagrams.
result Almost all knot diagrams are slipknotted.
This paper is a computation of the homotopy type of K, the space of long knots in R^3, the same space of knots studied by Vassiliev via singularity theory. Each component of K corresponds to an isotopy class of long knot, and we `enumerate' the components via the companionship trees associated to the knot. The knots wi…
The paper verifies no cosmetic surgeries on knots and 3-manifolds using hyperbolic geometry.
problem Checking cosmetic surgeries on knots and 3-manifolds.
method Knot invariants and hyperbolic geometry.
result Verification of no cosmetic surgeries on knots and 3-manifolds.
This paper tabulates prime knot projections up to eight double points.
problem Tabulating prime knot projections and their mirror images up to a certain number of double points.
method Systematic flypes and enumeration of tangles with at most four double points, using arrow diagrams.
result Complete table of prime knot projections with their mirror images up to eight double points.
We present a new, practical algorithm to test whether a knot complement contains a closed essential surface. This property has important theoretical and algorithmic consequences; however, systematically testing it has until now been infeasibly slow, and current techniques only apply to specific families of knots. As a …
This study simplifies verification of invariants in oriented virtual knots.
problem Verifying invariants of oriented virtual knots is complex and time-consuming.
method Identifying a minimal generating set of oriented virtual Reidemeister moves.
result A four-element subset serves as a generating set for oriented virtual Reidemeister moves.
The paper examines circle graphs of Gauss diagrams and finds counterexamples to previous descriptions.
problem Problems with previous descriptions of realizable Gauss diagrams.
method Experimental checking and formulation of new descriptions of realizable circle graphs.
result New descriptions of realizable circle graphs and an algorithm for checking realizability.
This paper compiles and calculates triple point numbers for surface-links in Yoshikawa's table.
problem Determining the triple point number of surface-links in Yoshikawa's table.
method Using broken sheet diagrams, the paper compiles known triple point numbers and calculates or bounds the remaining ones.
result Compilation and calculation of triple point numbers for surface-links in Yoshikawa's table.
This paper classifies knots with maximal exceptional surgeries on the minimally twisted 5-chain link.
problem Identifying knots with maximal exceptional surgeries on the minimally twisted 5-chain link.
method Enumerating all hyperbolic knots with maximal distance between exceptional surgeries.
result Examples of knots with maximal distance between exceptional surgeries are not obtained by filling the Berge manifold.
The crosscap number of a knot is an invariant describing the non-orientable surface of smallest genus that the knot bounds. Unlike knot genus (its orientable counterpart), crosscap numbers are difficult to compute and no general algorithm is known. We present three methods for computing crosscap number that offer varyi…
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.
Study links in knitted textiles using knot theory.
problem Identify and classify links in knitted textiles.
method Correspondence between links in thickened torus and 2-periodic weft-knitted textiles, using link invariants and ribbon knots.
result New stitch patterns and links in knitted textiles identified.
Criterion for stopping conjugacy class enumeration in triangle groups.
problem Enumerating all conjugacy classes in cocompact triangle groups.
method Encoding by P. Dehornoy and T. Pinsky; stopping criterion based on geometric length.
result Stopping criterion for the generation of conjugacy classes in cocompact triangle groups.
Algorithm finds essential surfaces in 3D shapes.
problem Detecting closed essential surfaces in 3D shapes.
method Triangulation, ideal triangulation, enumeration, optimisation, normal surface theory.
result Algorithm tests for essential surfaces in 3D shapes.
Characterizes unknotted curves on Seifert surfaces of twist knots.
problem Identifying unknotted curves on Seifert surfaces of twist knots.
method Analyzing homologically essential simple closed curves on Seifert surfaces of genus one knots.
result Characterizes unknotted curves on Seifert surfaces of twist knots, including infinitely many for the figure eight knot and one for Whitehead doubles.