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We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a folded ribbon knot. The folded ribbon knot is also a framed knot, and the ribbon linking number is the linking number of the knot and one boundary component of the ribbon. We find the minimum folded ribbonlength for 3 -stick unknots with ribbon linking numbers ± 1 and ± 3 , and we prove that the minimum folded ribbonlength for n -gons with obtuse interior angles is achieved when the n -gon is regular. Among other results, we prove that the minimum folded ribbonlength of any folded ribbon unknot which is a topological annulus with ribbon linking number ± n is bounded from above by 2 n. [ABSTRACT FROM AUTHOR]

We consider the set of connected surfaces in the 4‐ball with boundary a fixed knot in the 3‐sphere. We define the stabilization distance between two surfaces as the minimal g$g$ such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most g$g$. Similarly, we consider a double‐point distance between two surfaces of the same genus that is the minimum over all regular homotopies connecting the two surfaces of the maximal number of double points appearing in the homotopy. To many of the concordance invariants defined using Heegaard Floer homology, we construct an analogous invariant for a pair of surfaces. We show that these give lower bounds on the stabilization distance and the double‐point distance. We compute our invariants for some pairs of deform‐spun slice disks by proving a trace formula on the full infinity knot Floer complex, and by determining the action on knot Floer homology of an automorphism of the connected sum of a knot with itself that swaps the two summands. We use our invariants to find pairs of slice disks with arbitrarily large distance with respect to many of the metrics we consider in this paper. We also answer a slice‐disk analog of Problem 1.105 (B) from Kirby's problem list by showing the existence of non‐0‐cobordant slice disks. [ABSTRACT FROM AUTHOR]

Knot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces L(n,n−1)$L(n,n-1)$ via a quotient by Weiler. While toric constructions can be used to understand the ECH chain complexes of many contact forms adapted to open books with binding the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3‐sphere in terms of their presentation as open books and as Seifert fiber spaces. We provide Morse–Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg–Witten Floer theory developed by Hutchings and Taubes, and use them to compute the T(2,q)$T(2,q)$ knot filtered embedded contact homology, for q$q$ odd and positive. [ABSTRACT FROM AUTHOR]

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface (called a surface-link) embedded in 4-space. In this paper, we investigate surface-links by using charts. In [T. Nagase and A. Shima, The structure of a minimal n -chart with two crossings I: Complementary domains of Γ 1 ∪ Γ n − 1 , J. Knot Theory Ramifactions27(14) (2018) 1850078; T. Nagase and A. Shima, The structure of a minimal n -chart with two crossings II: Neighbourhoods of Γ 1 ∪ Γ n − 1 , Revista de la Real Academia de Ciencias Exactas, Fiskcas y Natrales. Serie A. Math.113 (2019) 1693–1738, arXiv:1709.08827v2 ] we gave an enumeration of the charts with two crossings. In particular, there are two classes for 4-charts with two crossings and eight black vertices. The first class represents surface-links each of which is connected. The second class represents surface-links each of which is exactly two connected components. In this paper, by using quandle colorings, we shall show that the charts in the second class represent different surface-links. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two new polynomial invariants Q D (t) and A D (t) for one-component virtual doodles. We will also show that these polynomial invariants are not invariants of flat virtual knots. [ABSTRACT FROM AUTHOR]

Let K be a tame knot and consider an n beaded necklace T ∘ which is the union of n consecutive disjoint closed round balls (pearls) B j , j = 1 , ... , n. An n pearl chain necklace T is the union of T ∘ and K. We will construct, via the action of a Kleinian group, a sequence of nested pearl chain necklaces T k whose inverse limit is a wild knot of dynamically defined type Λ (K , T 0). In this paper, we will prove some topological properties of this kind of wild knots; in particular, we generalize the construction of cyclic branched coverings for this case, and we show that there exists a wild knot of dynamically defined type such that 3 is an n -fold cyclic branched covering of 3 along it, for n ≥ 2. [ABSTRACT FROM AUTHOR]

In this paper we give a complete characterization of those knotted toroidal sets that can be realized as attractors for discrete or continuous dynamical systems globally defined in R 3 . We also see that the techniques used to solve this problem can be used to give sufficient conditions to ensure that a wide class of subcompacta of R 3 that are attractors for homeomorphisms must also be attractors for flows. In addition we study certain attractor-repeller decompositions of S 3 which arise naturally when considering toroidal sets. [ABSTRACT FROM AUTHOR]

This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If M is such a manifold, we show that the type D structure \widehat {CFD}(M) may be viewed as a set of immersed curves decorated with local systems in \partial M. These curves-with-decoration are invariants of the underlying three-manifold up to regular homotopy of the curves and isomorphism of the local systems. Given two such manifolds and a homeomorphism h between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with h is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of \widehat {HF} decreases under a certain class of degree one maps (pinches) and we establish that the existence of an essential separating torus gives rise to a lower bound on the dimension of \widehat {HF}. In particular, it follows that a prime rational homology sphere Y with \widehat {HF}(Y)<5 must be geometric. Other results include a new proof of Eftekhary's theorem that L-space homology spheres are atoroidal; a complete characterization of toroidal L-spaces in terms of gluing data; and a proof of a conjecture of Hom, Lidman, and Vafaee on satellite L-space knots. [ABSTRACT FROM AUTHOR]

In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces L (p , q) , KBSM(L (p , q)), for q ≠ 0. For doing this, we introduce a new concept, that of an unoriented braid. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley–Lieb algebra of type B, TL 1 , n , which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, V , for knots and links in ST, via a unique Markov trace constructed on TL 1 , n . The universal invariant V is equivalent to the KBSM(ST). For passing now to the KBSM(L (p , q)), we impose on V relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in L (p , q) but not in ST, and which reflect the surgery description of L (p , q) , obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM(L (p , q)). We first present the solution for the case q = 1 , which corresponds to obtaining a new basis, ℬ p , for KBSM(L (p , 1)) with (⌊ p / 2 ⌋ + 1) elements. We note that the basis ℬ p is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case q > 1 , we first show how the new basis ℬ p of KBSM(L (p , 1)) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case q > 1. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements. [ABSTRACT FROM AUTHOR]

The entanglement of open curves in 3-space appears in many physical systems and affects their material properties and function. A new framework in knot theory was introduced recently, that enables to characterize the complexity of collections of open curves in 3-space using the theory of knotoids and linkoids, which are equivalence classes of diagrams with open arcs. In this paper, new invariants of linkoids are introduced via a surjective map between linkoids and virtual knots. This leads to a new collection of strong invariants of linkoids that are independent of any given virtual closure. This gives rise to a collection of novel measures of entanglement of open curves in 3-space, which are continuous functions of the curve coordinates and tend to their corresponding classical invariants when the endpoints of the curves tend to coincide. [ABSTRACT FROM AUTHOR]

The space of Gauss diagram formulas that are knot invariants is introduced by Goussarov–Polyak–Viro in 2000; it is extended to nanophrases by Gibson–Ito in 2011. However, known invariants in concrete presentations of Gauss diagram formulas are very limited, even in the one-component case. This paper gives a recipe to obtain explicit forms of Gauss diagram formulas that are invariants of virtual links with base points or tangles. As an application, we introduce a new construction of Gauss diagram formulas of 3 -bouquets and how to give link invariants that do not change with base point moves, including a reconstruction of the Milnor's triple linking number. [ABSTRACT FROM AUTHOR]

In 1979, Hartley and Kawauchi proved that the Conway polynomial of a strongly negative amphichiral knot factors as f (z)f (-z). In this paper, we normalize the factor f (z) to define the half-Conway polynomial. First, we prove that the half-Conway polynomial satisfies an equivariant skein relation, giving the first feasible computational method, which we use to compute the half-Conway polynomial for knots with 12 or fewer crossings. This skein relation also leads to a diagrammatic interpretation of the degree-one coefficient, from which we obtain a lower bound on the equivariant unknotting number. Second, we completely characterize polynomials arising as half-Conway polynomials of knots in S3, answering a problem of Hartley-Kawauchi. As a special case, we construct the first examples of non-slice strongly negative amphichiral knots with determinant one, answering a question of Manolescu. The double branched covers of these knots provide potentially non-trivial torsion elements in the homology cobordism group. [ABSTRACT FROM AUTHOR]

A diquandle is a set equipped with two quandle operations interacting via a kind of distributive laws which come from Reidemeister moves on dichromatic links. This algebraic systems provide coloring invariants for dichromatic links. In this paper, we give explicit constructions of free diquandles and diquandle presentations, and then discuss Tietze transformations for the diquandle presentations. We also introduce the fundamental diquandles for dichromatic links. Particularly, we describe the fundamental diquandles and diquandle counting invariants for knots and links in the solid torus via annulus diagrams. We append the tables of diquandles and dikei's of orders ≤ 5. [ABSTRACT FROM AUTHOR]

In [B. Kwon and S. Kang, Rectangle conditions and families of 3-bridge prime knots, Topol. Appl. 291 (2021) 107453], using the set E A T k of all essential alternating rational 3-tangles for positive integer k , the authors showed that all knot diagrams in the numerator closure set C N (E A T 2 l + 1) and the denominator closure set C D (E A T 2 l + 2) with l > 0 are 3-bridge prime knot diagrams. In this paper, for n > 4 we construct a set A A T 4 n of additions of alternating rational tangles in E A T 4 . The set A A T 4 n generalizes E A T k and contains it as a subset for some k. We show that any closure set C (A A T 4 n) on A A T 4 n so that the resulting diagrams are reduced and alternating knot diagrams represent alternating 3-bridge prime knot diagrams. Since a tangle diagram in A A T 4 n + 1 is constructed inductively from a tangle diagram in A A T 4 n by adding only one crossing positively, the result of this paper supports the conjecture that 3-bridge property is preserved under one-crossing alternating addition positively to alternating 3-bridge knots in 3-bridge representations. [ABSTRACT FROM AUTHOR]

This paper discusses some geometric ideas associated with knots in real projective 3-space ℝ P 3 . These ideas are borrowed from classical knot theory. Since knots in ℝ P 3 are classified into three disjoint classes: affine, class- 0 non-affine and class- 1 knots, it is natural to wonder in which class a given knot belongs to. In this paper we attempt to answer this question. We provide a structure theorem for these knots which helps in describing their behavior near the projective plane at infinity. We propose a procedure called space bending surgery, on affine knots to produce several examples of knots. We later show that this operation can be extended on an arbitrary knot in ℝ P 3 . We then study the notion of companionship of knots in ℝ P 3 and using it we provide geometric criteria for a knot to be affine. We also define a notion of "genus" for knots in ℝ P 3 and study some of its properties. We prove that this genus detects knottedness in ℝ P 3 and gives some criteria for a knot to be affine and of class- 1. We also prove a "non-cancellation" theorem for space bending surgery using the properties of genus. Then we show that a knot can have genus 1 if and only if it is a cable knot with a class-1 companion. We produce examples of class- 0 non-affine knots with genus 1. Thus we highlight that, ℝ P 3 admits a knot theory with a truly different flavor than that of S 3 or ℝ 3 . [ABSTRACT FROM AUTHOR]

Although the connected sum of trivial knots is trivial in classical knot theory, there is a nontrivial connected sum of trivial knots in virtual knot theory. In this paper, we give connected sum formulae of the intersection polynomials of virtual knots introduced in the preceding paper [6]. As an application, we show that there are infinitely many connected sums of any pair of virtual knots. [ABSTRACT FROM AUTHOR]

We say a null-homologous knot K in a 3-manifold Y has Property G, if the Thurston norm and fiberedness of the complement of K is preserved under the zero surgery on K. In this paper, we will show that, if the smooth 4-genus of K\times \{0\} (in a certain homology class) in (Y\times [0,1])\#N\overline {\mathbb CP^2}, where Y is a rational homology sphere, is smaller than the Seifert genus of K, then K has Property G. When the smooth 4-genus is 0, Y can be taken to be any closed, oriented 3-manifold. [ABSTRACT FROM AUTHOR]

A petal diagram of a knot is a projection with a single multi-crossing such that there are no nested loops. The petal number p (K) of a knot K is the minimum number of loops among all petal diagrams of K. Let T n , s denote the (n , s) -torus knot for relatively prime integers 2 ≤ n < s. Recently, Kim et al. proved that p (T n , s) ≤ 2 s − 2 ⌊ s n ⌋ + 1 whenever s ≡ ± 1 mod n. They conjectured that the inequality holds without the assumption s ≡ ± 1 mod n. They also showed that p (T n , s) = 2 s − 1 whenever 2 ≤ n < s < 2 n and n ≡ 1 mod s − n. Their proofs construct petal grid diagrams for those torus knots. In this paper, we prove the conjecture that p (T n , s) ≤ 2 s − 2 ⌊ s n ⌋ + 1 holds for any 2 ≤ n < s. We also show that p (T n , s) = 2 s − 1 holds for any 2 ≤ n < s < 2 n. Our proofs construct petal grid diagrams for any torus knots. [ABSTRACT FROM AUTHOR]

We study the structure of the augmented fundamental quandle of a knot whose complement contains an incompressible torus. We obtain the relationship between the fundamental quandle of a satellite knot and the fundamental quandles/groups of its companion and pattern knots. General presentations of the fundamental quandles of a link in a solid torus, a link in a lens space and a satellite knot are described. In the last part of this paper, an algebraic approach to the study of affine quandles is presented and some known results about the Alexander module and quandle colorings are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the properties of timelike and spacelike shifted-knots Bézier surfaces in Minkowski space- E13. These surfaces are commonly used in mathematical models for surface formation in computer science for computer-aided geometric design and computer graphics, as well as in other fields of mathematics. Our objective is to analyze the characteristics of timelike and spacelike shifted-knots Bézier surfaces in Minkowski space- E13. To achieve this, we compute the fundamental coefficients of shifted-knots Bézier surfaces, including the Gauss-curvature, mean-curvature, and shape-operator of the surface. Furthermore, we present numerical examples of timelike and spacelike bi-quadratic (m = n = 2) and bi-cubic (m = n = 3) shifted-knots Bézier surfaces in Minkowski space- E13 to demonstrate the applicability of the technique in Minkowski space. [ABSTRACT FROM AUTHOR]

Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group S n has three natural extensions, namely the braid group B n , the twin group T n and the triplet group L n . The latter two groups are small Coxeter groups, and play the role of braid groups under the Alexander–Markov correspondence for appropriate knot theories, with their pure subgroups admitting suitable hyperplane arrangements as Eilenberg-MacLane spaces. In this paper, we prove that the congruence subgroup property fails for infinite small Coxeter groups which are not virtually abelian. As an application, we deduce that the congruence subgroup property fails for both T n and L n when n ≥ 4 . We also determine subquotients of principal congruence subgroups of T n , and identify the pure twin group P ⁢ T n and the pure triplet group P ⁢ L n with suitable principal congruence subgroups. Further, we investigate crystallographic quotients of these two families of small Coxeter groups, and prove that T n / P ⁢ T n ′ , T n / T n ′′ and L n / P ⁢ L n ′ are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of T n / T n ′′ . [ABSTRACT FROM AUTHOR]

In the paper, we construct a representation : FVB n → Aut (F 2 n) of the flat virtual braid group FVB n on n strands by automorphisms of the free group F 2 n with 2 n generators which does not preserve the forbidden relations in the flat virtual braid group. This representation gives a positive answer to the problem formulated by Bardakov in the list of unsolved problems in virtual knot theory and combinatorial knot theory by Fenn et al. Also we find the set of normal generators of the groups VP n ∩ H n in VB n , FVP n ∩ FH n in FVB n , GVP n ∩ GH n in GVB n , which play an important role in the study of the kernel of the representation  . [ABSTRACT FROM AUTHOR]

We create an invariant of virtual Y -oriented trivalent spatial graphs using colorings by virtual Niebrzydowski algebras. This paper generalizes the color invariants using virtual tribrackets and Niebrzydowski algebras by Nelson–Pico, and Graves-Nelson-T. We computed all tribrackets, Niebrzydowski algebras and virtual Niebrzydowski algebras of orders 3 and 4, and provide generative code for all data sets. [ABSTRACT FROM AUTHOR]

The 3-loop invariant (or, the 3-loop polynomial) of a knot is a rational form (or, a polynomial) presenting the 3-loop part of the Kontsevich invariant of knots. In this paper, we calculate the 3-loop polynomial of knots obtained by plumbing the doubles of two knots; this class of knots includes untwisted Whitehead doubles. We construct the 3-loop invariant by calculating the rational version of the Aarhus integral of a surgery presentation. As a consequence, we obtain an explicit presentation of the 3-loop polynomial for the knots. [ABSTRACT FROM AUTHOR]

Starting from the work by Jones on representations of Thompson's group F , subgroups of F with interesting properties have been defined and studied. One of these subgroups is called the 3-colorable subgroup F , which consists of elements whose "regions" given by their tree diagrams are 3-colorable. On the other hand, in his work on representations, Jones also gave a method to construct knots and links from elements of F. Therefore it is a natural question to explore a relationship between elements in F and 3-colorable links in the sense of knot theory. In this paper, we show that all elements in F give 3-colorable links. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a polynomial invariant of virtual knots by using weights of crossings. It is motivated from the writhe polynomial and the affine index polynomial for virtual knots. The new polynomial invariant is distinct from these polynomials and is a Vassiliev invariant of degree 1. By using the polynomial invariant we give a lower bound for the Gordian distance of two homotopic virtual knots. [ABSTRACT FROM AUTHOR]

One strategy for distinguishing smooth structures on closed 4‐manifolds is to produce a knot K$K$ in S3$S^3$ that is slice in one smooth filling W$W$ of S3$S^3$ but not slice in some homeomorphic smooth filling W′$W^{\prime }$. In this paper, we explore how 0‐surgery homeomorphisms can be used to potentially construct exotic pairs of this form. To systematically generate a plethora of candidates for exotic pairs, we give a fully general construction of pairs of knots with the same zero surgeries. By computer experimentation, we find five topologically slice knots such that, if any of them were slice, we would obtain an exotic 4‐sphere. We also investigate the possibility of constructing exotic smooth structures on #nCP2$\#^n \mathbb {CP}^2$ in a similar fashion. [ABSTRACT FROM AUTHOR]

Consider two parallel lines ℓ 1 and ℓ 2 in ℝ 3 . A rail arc is an embedding of an arc in ℝ 3 such that one endpoint is on ℓ 1 , the other is on ℓ 2 , and its interior is disjoint from ℓ 1 ∪ ℓ 2 . Rail arcs are considered up to rail isotopies, ambient isotopies of ℝ 3 with each self-homeomorphism mapping ℓ 1 and ℓ 2 onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs are called stick rail arcs. The stick number of a rail arc class is the minimum number of sticks, line segments in a p.l. arc, needed to create a representative. This paper calculates the stick number of rail arcs classes with a crossing number at most 2 and uses a winding number invariant to calculate the stick numbers of infinitely many rail arc classes. Each rail arc class has two canonically associated knot classes, its under and over companions. This paper also introduces the rail stick number of knot classes, the minimum number of sticks needed to create a rail arcs whose under or over companion is the knot class. The rail stick number is calculated for 29 knot classes with crossing number at most 9. The stick number of multi-component rail arcs classes is considered as well as the lattice stick number of rail arcs. [ABSTRACT FROM AUTHOR]

In a recent paper, the first author and his collaborator developed a method to compute an upper bound of the dimension of instanton Floer homology via Heegaard diagrams of 3 -manifolds. In this paper, for a knot inside S 3 , we further introduce an algorithm that computes an upper bound of the dimension of instanton knot homology from knot diagrams. We test the algorithm with all knots up to seven crossings as well as a more complicated knot 1 0 1 5 3 . In the second half of the paper, we show that if the instanton knot Floer homology of a knot has a specified form, then the knot must be an instanton L-space knot. [ABSTRACT FROM AUTHOR]

Quandles with good involutions, which are called symmetric quandles, can be used to define invariants of unoriented knots and links. In this paper, we determine the necessary and sufficient condition for good involutions of a generalized Alexander quandle to exist. Moreover, we classify all good involutions of a connected generalized Alexander quandle up to symmetric quandle isomorphisms. [ABSTRACT FROM AUTHOR]

Knot theory is a distinct branch of topology. In recent years, exciting new application of knot theory to biology and chemistry, especially to DNA structures are being developed. Alexander polynomial is very closely connected with the topological properties of knots. In this paper we have determined the solution of Ak and & knots using the condition of Alexander polynomial and Jones polynomial. [ABSTRACT FROM AUTHOR]

We consider the Lorenz equations, a system of three-dimensional ordinary differential equations modeling atmospheric convection. These equations are chaotic and hard to study even numerically, and so a simpler "geometric model" has been introduced in the seventies. One of the classical problems in dynamical systems is to relate the original equations to the geometric model. This has been achieved numerically by Tucker for the classical parameter values and remains open for general values. In this paper, we establish analytically a relation to the geometric model for a different set of parameter values that we prove must exist. This is facilitated by finding a way to apply topological tools developed for the study of surface dynamics to the more intricate case of three-dimensional flows. [ABSTRACT FROM AUTHOR]

In the theory of topology of words, the move to swap two adjacent letters in words is a simple operator that makes an S -homotopy class of a nanoword into the S -homotopy class of the empty word. In this paper, we study the relation between the swap move of letters and the forbidden moves of virtual diagrams, that is, local moves of virtual diagrams which unknot any virtual knot. [ABSTRACT FROM AUTHOR]

In this paper we discuss the applications of knotoids to modelling knots in open curves and produce new knotoid invariants. We show how invariants of knotoids generally give rise to well-behaved measures of how much an open curve is knotted. We define biframed planar knotoids, and construct new invariants of these objects that can be computed in polynomial time. As an application of these invariants we improve the classification of planar knotoids with up to five crossings by distinguishing two pairs of prime knotoids that were conjectured to be distinct by Goundaroulis et al. [ABSTRACT FROM AUTHOR]

Originally, tangles were invented as an abstract tool in mathematical graph theory to prove the famous graph minor theorem. In this paper, we showcase the practical potential of tangles in machine learning applications. Given a collection of cuts of any dataset, tangles aggregate these cuts to point in the direction of a dense structure. As a result, a cluster is softly characterized by a set of consistent pointers. This highly flexible approach can solve clustering problems in various setups, ranging from questionnaires over community detection in graphs to clustering points in metric spaces. The output of our proposed framework is hierarchical and induces the notion of a soft dendrogram, which can help explore the cluster structure of a dataset. The computational complexity of aggregating the cuts is linear in the number of data points. Thus the bottleneck of the tangle approach is to generate the cuts, for which simple and fast algorithms form a sufficient basis. In our paper we construct the algorithmic framework for clustering with tangles, prove theoretical guarantees in various settings, and provide extensive simulations and use cases. Python code is available on github. [ABSTRACT FROM AUTHOR]

This is the first paper in a series of three devoted to studying twisted linking forms of knots and three-manifolds. Its function is to provide the algebraic foundations for the next two papers by describing how to define and calculate signature invariants associated to a linking form M × M → F (t) / F [ t ± 1 ] for F = R , C , where M is a torsion F [ t ± 1 ] -module. Along the way, we classify such linking forms up to isometry and Witt equivalence and study whether they can be represented by matrices. [ABSTRACT FROM AUTHOR]

This paper continues a program due to Motegi regarding universal bounds for the number of nonisotopic essential n -punctured tori in the complement of a hyperbolic knot in S 3. For n = 1 , Valdez-Sánchez showed that there are at most five nonisotopic Seifert tori in the exterior of a hyperbolic knot. In this paper, we address the case n = 2. We show that there are at most six nonisotopic, nested, essential 2-holed tori in the complement of every hyperbolic knot. [ABSTRACT FROM AUTHOR]

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics. [ABSTRACT FROM AUTHOR]

Dielectric function representation by B-splines became quite popular and widely used in the context of spectroscopic ellipsometry data interpretation. B-splines are defined by a polynomial degree and a sequence of knots (i.e., the number and positions of knots). Defining the knot sequence is non-trivial, and this task has a significant effect on the actual effectiveness of the B-spline parameterization in spectroscopic ellipsometry data analysis. In this paper, we propose a simple, practical, and systematic knot placement scheme that improves ordinary trial-and-error technique in establishing the knot spacing. The approach suggested here is based on an integral span, a measure introduced in this work. The proposed procedure provides a possibility to determine the knot locations automatically (or, at least, semi-automatically) and excludes widespread modeling ambiguities associated with uncertain knot vector. Moreover, our approach guarantees the absence of non-monotonic behavior of the mean-squared-error function and thereby improves the accuracy of our analysis. The performance of the proposed method has been tested for several real cases of the study. [ABSTRACT FROM AUTHOR]

In the first of this series of papers, we looked at various generalized knot theories. In this follow up paper, we look at the theory of generalized braids and show how they are related to generalized knots. The twin groups are related to the theory of planar doodles and we prove a Markov-type theorem for them. An example and invariants are given to show that this is best possible. We also give conditions for when an exchange move is dependent on the other moves. [ABSTRACT FROM AUTHOR]

In this paper, a localized boundary knot method is adopted to solve two-dimensional inverse Cauchy problems, which are controlled by a second-order linear differential equation. The localized boundary knot method is a numerical method based on the local concept of the localization method of the fundamental solution. The approach is formed by combining the classical boundary knot method with the localization method. It has the potential to solve many complex engineering problems. Generally, in an inverse Cauchy problem, there are no boundary conditions in specific boundaries. Additionally, in order to be close to the actual engineering situation, a certain level of noise is added to the known boundary conditions to simulate the measurement error. The localized boundary knot method can be used to solve two-dimensional Cauchy problems more stably and is truly free from mesh and numerical quadrature. In this paper, the stability of the method is verified by using multi-connected domain and simply connected domain examples in Laplace equations. [ABSTRACT FROM AUTHOR]

The Gordian complex of knots is a simplicial complex whose vertices consist of all knot types in 3 . Local moves play an important role in defining knot invariants. There are many local moves known as unknotting operations for knots. In this paper, we discuss the 4-move operation. We show that for any knot K 0 and for any given natural number n , there exists a family of knots { K 0 , K 1 , ... , K n } such that for any pair (K i , K j) of distinct elements of the family, the Gordian distance of knots by 4-move is d f (K i , K j) = 1. We also show the existence of an arbitrarily high dimensional simplex in the Gordian complexes. [ABSTRACT FROM AUTHOR]

Im et al. [Signature, nullity and determinant of checkerboard colorable virtual links, J. Knot Theory Ramifications 19(8) (2010) 1093–1114] introduced how to define Goeritz matrices for checkerboard colorable virtual links. In this paper, we extend this for the Goeritz matrices of virtual knots. And we consider its signature and determinant and show they are invariants for virtual knots. [ABSTRACT FROM AUTHOR]

A slope p q is called a characterizing slope for a given knot K 0 ⊂ S 3 if whenever the p q -surgery on a knot K ⊂ S 3 is homeomorphic to the p q -surgery on K 0 via an orientation preserving homeomorphism, then K = K 0 . In a previous paper, we showed that, outside a certain finite set of slopes, only the negative integers could possibly be non-characterizing slopes for the torus knot T 5 , 2 . More explicitly besides all negative integer slopes there are 2 4 7 slopes which were unknown to be characterizing for T 5 , 2 , including 8 9 nontrivial L -space slopes. Applying recent work of Baldwin–Hu–Sivek, we improve our result by showing that a nontrivial slope p q is a characterizing slope for T 5 , 2 if p q > − 1 and p q ∉ { 0 , 1 , ± 1 2 , ± 1 3 }. In particular every nontrivial L -space slope of T 5 , 2 is characterizing for T 5 , 2 . More explicitly this work yields 1 2 1 new characterizing slopes for T 5 , 2 . Another interesting consequence of this work is that if a nontrivial p q -surgery on a non-torus knot in S 3 yields a manifold of finite fundamental group, then | p | > 9. [ABSTRACT FROM AUTHOR]

A knot shadow is a diagram without over/ under information at all crossings. Many knot types are obtained from a knot shadow by assigning over/ under information to each crossing. The purpose of the paper is to observe numbers of knot types obtained from closed 3-braid knot shadows. As a result, we discover that given an arbitary odd integer n ≥ 3 , there exists a closed 3-braid knot shadow S such that the number of T (m , 2) (3 ≤ m ≤ n) torus knots is more than that of trivial knots in the knots obtained from S by assigning over/ under information to each crossing. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new index polynomial invariants for virtual knots. They are defined by using invariants of flat virtual knots. And we give some properties of these invariants and examples. [ABSTRACT FROM AUTHOR]

In this paper, we enumerate the number of oriented rational knots and the number of oriented rational links with any given crossing number and minimum genus. This allows us to obtain a precise formula for the average minimal genus of oriented rational knots and links with any given crossing number. [ABSTRACT FROM AUTHOR]

In this paper, mosaic tiles have been used to build knots and define invariants since they were introduced by Lomonaco and Kauffman to describe quantum knots. In this paper, we propose a modified set of tiles to describe the front projections of Legendrian knots. We explore the effect of stabilization on the mosaic number of a Legendrian knot by defining and classifying "two-cell" stabilizations. Inspired by a construction of Ludwig et al. in 2013, we provide an infinite family of Legendrian unknots whose mosaic numbers are realized only in non-reduced projections. Finally, we provide a census of known mosaic numbers for Legendrian unknots and trefoils. [ABSTRACT FROM AUTHOR]

The aim of this paper is to realize the techniques of picture-valued invariants and invariants valued in free groups for long knots in the full torus. Such knots and links are of particular interest because of their relation to Legendrian knots, knotoids, 3 -manifolds and many other objects. Invariants constructed in the paper are powerful and easy to compare. This paper is a sequel of [V. O. Manturov, A free-group valued invariant of free knots, preprint (2020), arXiv:2012.15571v2]. Long knots naturally appear in the study of classical knots [T. Fiedler, More 1 -cocycles for classical knots, preprint (2020), arXiv:2004.04624; A. Mortier, A Kontsevich integral of order 1 , preprint (2018), arXiv:1810.05747]. [ABSTRACT FROM AUTHOR]