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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]

In this paper, we consider the Reshetikhin-Turaev invariants of knots in the three-sphere obtained from a twisted Drinfeld double of a Hopf algebra, or equivalently, the relative Drinfeld center of the crossed product \text {Rep}(H)\rtimes \text {Aut}(H). These are quantum invariants of knots endowed with a homomorphism of the knot group to \text {Aut}(H). We show that, at least for knots in the three-sphere, these invariants provide a non-involutory generalization of the Fox-calculus-twisted Kuperberg invariants of sutured manifolds introduced previously by the author, which are only defined for involutory Hopf algebras. In particular, we describe the SL(n,\mathbb {C})-twisted Reidemeister torsion of a knot complement as a Reshetikhin-Turaev invariant. [ABSTRACT FROM AUTHOR]

The deep connection among braids, knots and topological physics has provided valuable insights into studying topological states in various physical systems. However, identifying distinct braid groups and knot topology embedded in non-Hermitian systems is challenging and requires significant efforts. Here, we demonstrate that an unsupervised learning with the representation basis of su(n) Lie algebra on n-fold extended non-Hermitian bands can fully classify braid group and knot topology therein, without requiring any prior mathematical knowledge or any pre-defined topological invariants. We demonstrate that the approach successfully identifies different topological elements, such as unlink, unknot, Hopf link, Solomon ring, trefoil, and so on, by employing generalized Gell-Mann matrices in non-Hermitian models with n=2 and n=3 energy bands. Moreover, since eigenstate information of non-Hermitian bands is incorporated in addition to eigenvalues, the approach distinguishes the different parity-time symmetry and breaking phases, recognizes the opposite chirality of braids and knots, and identifies out distinct topological phases that were overlooked before. Our study shows significant potential of machine learning in classification of knots, braid groups, and non-Hermitian topological phases. The topology of braids and knots plays a central role in the understanding of many physical systems. In this paper, the authors demonstrate that unsupervised learning can be used to fully classify the braid group and knot topology associated with the bands of non-Hermitian systems, without requiring any prior information such as mathematical knowledge of topological invariants [ABSTRACT FROM AUTHOR]

The present paper is devoted to a lower bound for the number of critical points of the Lyapunov function for Morse–Smale 3-diffeomorphisms with fixed points with pairwise distinct indices. It is known that, in the presence of a single noncompact heteroclinic curve, the supporting manifold of the diffeomorphisms under consideration is a 3-sphere, and the class of topological conjugacy of such a diffeomorphism is completely determined by the equivalence class (there exist infinitely many of them) of the Hopf knot , which is a knot in the generating class of the fundamental group of the manifold . Moreover, any Hopf knot is realized by some diffeomorphism of the class under consideration. It is known that the diffeomorphisms defined by the standard Hopf knot have an energy function, which is a Lyapunov function whose set of critical points coincides with the chain recurrent set. However, the set of critical points of any Lyapunov function of a diffeomorphism with a nonstandard Hopf knot is strictly greater than the chain recurrent set of the diffeomorphism. In the present paper, for the diffeomorphisms defined by generalized Mazur knots, a quasi-energy function has been constructed, which is a Lyapunov function with a minimum number of critical points. [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]

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]

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]

Suppose K is a knot in a 3-manifold Y, and that Y admits a pair of distinct contact structures. Assume that K has Legendrian representatives in each of these contact structures, such that the corresponding Thurston-Bennequin framings are equivalent. This paper provides a method to prove that the contact structures resulting from Legendrian surgery along these two representatives remain distinct. Applying this method to the situation where the starting manifold is - Σ (2 , 3 , 6 m + 1) and the knot is a singular fiber, together with convex surface theory we can classify the tight contact structures on certain families of Seifert fiber spaces. [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]

In this paper, we consider two properties on the braid index of a two-bridge knot. We prove an inequality on the braid indices of two-bridge knots if there exists an epimorphism between their knot groups. Moreover, we provide the average braid index of all two-bridge knots with a given crossing number. [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]

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]

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]

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]

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]

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]

The concept of a knot mosaic was introduced by Lomonaco and Kauffman as a means to construct a quantum knot system. The mosaic number of a given knot K is defined as the minimum integer n that allows the representation of K on an n × n mosaic board. Building upon this, the first author and Nelson extended the knot mosaic system to encompass surface-links through the utilization of marked graph diagrams and established both lower and upper bounds for the mosaic number of the surface-links presented in Yoshikawa's table. In this paper, we establish a mosaic system for immersed surface-links by using singular marked graph diagrams. We also provide the definition and discussion on the mosaic number for immersed surface-links. [ABSTRACT FROM AUTHOR]

Take a sequence of contactomorphisms of a contact three-manifold that C^0-converges to a homeomorphism. If the images of a Legendrian knot limit to a smooth knot under this sequence, we show that it is contactomorphic to the original knot. We prove this by establishing that, on one hand, non–Legendrian knots admit a type of contact-squashing (similar to squeezing) onto transverse knots while, on the other hand, Legendrian knots do not admit such a squashing. The non-trivial input from contact topology that is needed is (a local version of) the Thurston–Bennequin inequality. [ABSTRACT FROM AUTHOR]

Let K be a rationally null-homologous knot in a 3-manifold Y, equipped with a non-zero framing λ, and let Yλ​(K) denote the result of λ-framed surgery on Y. Ozsváth and Szabó gave a formula for the Heegaard Floer homology groups of Yλ​(K) in terms of the knot Floer complex of (Y,K). We strengthen this formula by adding a second filtration that computes the knot Floer complex of the dual knot Kλ​ in Yλ​, i.e., the core circle of the surgery solid torus. In the course of proving our refinement we derive a combinatorial formula for the Alexander grading which may be of independent interest. [ABSTRACT FROM AUTHOR]

We discuss principles of measuring a topological charge or representation that travels in a set of anyons. We describe the procedure and analyze how it works for different values of theory parameters. We also show how it can be modified to be more efficient. [ABSTRACT FROM AUTHOR]

Lomonaco and Kauffman introduced knot mosaics in their work on quantum knots. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. In this paper, we consider the alternating mosaic number of an alternating knot K which is defined as the smallest integer n for which K is representable as a reduced alternating knot n -mosaic. We define a signed mosaic graph and a diagonal grid graph and construct Hamiltonian cycles derived from the diagonal grid graphs. Using the cycles, we completely determine the alternating mosaic number of torus knots of type (2 , q) for q ≥ 2 , which grows in an order of q 1 / 2. [ABSTRACT FROM AUTHOR]

In this article by means of shifted knots properties, we introduce a new type of coupled Bernstein operators for Bézier basis functions. First, we construct the operators based on shifted knots properties of Bézier basis functions then investigate the Korovkin's theorem, establish a local approximation theorem, and provide a convergence theorem for Lipschitz continuous functions and Peetre's K-functional. In addition, we also obtain an asymptotic formula of the type Voronovskaja. [ABSTRACT FROM AUTHOR]

We establish a connection between tangles, a concept from structural graph theory that plays a central role in Robertson and Seymour's graph minor project, and hierarchical clustering. Tangles cannot only be defined for graphs, but in fact for arbitrary connectivity functions, which are functions defined on the subsets of some finite universe. In typical clustering applications, these universes consist of points in some metric space. Connectivity functions are usually required to be submodular. It is our first contribution to show that the central duality theorem connecting tangles with hierarchical decompositions (so-called branch decompositions) also holds if submodularity is replaced by a different property that we call maximum-submodular. We then define a connectivity function on finite data sets in an arbitrary metric space and prove that its tangles are in one-to-one correspondence with the clusters obtained by applying the well-known single linkage clustering algorithms to the same data set. Last, we generalize this correspondence for any hierarchical clustering. We show that the data structure that represents hierarchical clustering results, called dendograms, are equivalent to maximum-submodular connectivity functions and their tangles. The idea of viewing tangles as clusters was first proposed by Diestel and Whittle in 2016 as an approach to image segmentation. [ABSTRACT FROM AUTHOR]

Fair division has been studied in both continuous and discrete contexts. One strand of the continuous literature seeks to award each agent with a single connected piece—a subinterval. The analogue for the discrete case corresponds to the fair division of a graph, where allocations must be contiguous so that each bundle of vertices is required to induce a connected subgraph. With envy-freeness up to one item (EF1) as the fairness criterion, however, positive results for three or more agents have mostly been limited to traceable graphs. We introduce tangles as a new context for fair division. A tangle is a more complicated cake—a connected topological space constructed by gluing together several copies of the unit interval [0, 1]—and each single tangle T corresponds in a natural way to an infinite topological class G (T) of graphs, linking envy-free fair division of tangles to EFk fair division of graphs. In addition to the unit interval itself, we show that only five other stringable tangles guarantee the existence of envy-free and connected allocations for arbitrarily many agents, with the corresponding topological classes containing only traceable graphs. Any other tangle T has a bound r on the number of agents for which such allocations necessarily exist, and our Negative Transfer Principle then applies to the graphs in T 's class; for any integer k ≥ 1 , almost all graphs in this class are non-traceable and fail to guarantee EFk contiguous allocations for r + 1 or more agents, even when very strict requirements are placed on the valuation functions for the agents. With bounds on the number of agents, however, we obtain positive results for some non-stringable classes. An elaboration of Stromquist's moving knife procedure shows that the non-stringable lips tangle L guarantees envy-free allocations of connected shares for three agents. We then modify the discrete version of Stromquist's procedure in Bilò et al. (Games Econ Behav 131:197–221, 2022) to show that all graphs in the topological class G (L) (most of which are non-traceable) guarantee EF1 allocations for three agents. [ABSTRACT FROM AUTHOR]

We introduce new techniques for working with presentations for a large class of (strict) tensor categories. We then apply the general theory to obtain presentations for partition, Brauer and Temperley-Lieb categories, as well as several categories of (partial) braids, vines, and mappings. [ABSTRACT FROM AUTHOR]

This document is a foreword for a volume celebrating two prominent Romanian mathematicians, Laurentiu Paunescu and Alexander Suciu. The contributions to the volume were delivered by speakers and participants in a conference held in their honor in Bucharest. The foreword provides brief biographical information about both mathematicians, including their educational background, research interests, and academic achievements. It also mentions their personal lives and interests outside of mathematics. [Extracted from the article]