A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Enumerated all genus two handlebody-knots with seven crossings.
The article enumerates doubly symmetric diagrams for knots up to 18 crossings.
A polynomial counts knot states for a specific type of knot.
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.
A process enumerates rack elements from a presentation.
Frequently, knots are enumerated by their crossing number. However, the number of knots with crossing number grows exponentially with , and to date computer-assisted proofs can only classify diagrams up to around twenty crossings. Instead, we consider diagrams enumerated by bridge number, following the lead of S…
Permutations linked to knots and links, with unknots counted by Schröder numbers.
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.
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.
Researchers developed an algorithm to count 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…
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.
The study calculates the average 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 . 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.
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.
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.
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.
The paper improves bounds on knot crossings and tabulates minimal diagrams.
New method finds exponential growth in knot types from sticks.
New types of knot mosaics help count and analyze knots efficiently.
Slipknots found in random diagrams almost always.
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.
This paper tabulates prime knot projections 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.
The paper examines circle graphs of Gauss diagrams and finds counterexamples to previous descriptions.
This paper compiles and calculates triple point numbers for surface-links in Yoshikawa's table.
We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 crossings. Following an approach of Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed their Jones polynomials by a divide-and-conquer method, and tested those with triv…
This paper classifies knots with maximal exceptional surgeries on the minimally twisted 5-chain link.
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.
Study links in knitted textiles using knot theory.
Criterion for stopping conjugacy class enumeration in triangle groups.
Algorithm finds essential surfaces in 3D shapes.
Characterizes unknotted curves on Seifert surfaces of twist knots.