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Based on polymer scaling theory and numerical evidence, Orlandini, Tesi, Janse van Rensburg and Whittington conjectured in 1996 that the limiting entropy of knot-type K lattice polygons is the same as that for unknot polygons, and that the entropic critical exponent increases by one for each prime knot in the knot decomposition of K. This Knot Entropy (KE) conjecture is consistent with the idea that for unconfined polymers, knots occur in a localized way (the knotted part is relatively small compared to polymer length). For full confinement (to a sphere or box), numerical evidence suggests that knots are much less localized. Numerical evidence for nanochannel or tube confinement is mixed, depending on how the size of a knot is measured. Here we outline the proof that the KE conjecture holds for polygons in the ∞ × 2 × 1 lattice tube and show that knotting is localized when a connected-sum measure of knot size is used. Similar results are established for linked polygons. This is the first model for which the knot entropy conjecture has been proved. [ABSTRACT FROM AUTHOR]

We consider the question of when the operation of contact surgery with positive surgery coefficient, along a knot K in a contact 3-manifold Y, gives rise to a weakly fillable contact structure. We show that this happens if and only if Y itself is weakly fillable, and K is isotopic to the boundary of a properly embedded symplectic disk inside a filling of Y. Moreover, if Y - is a contact manifold arising from positive contact surgery along K, then any filling of Y - is symplectomorphic to the complement of a suitable neighborhood of such a disk in a filling of Y . Using this result we deduce several necessary conditions for a knot in the standard 3-sphere to admit a fillable positive surgery, such as quasipositivity and equality between the slice genus and the 4-dimensional clasp number, and we give a characterization of such knots in terms of a quasipositive braid expression. We show that knots arising as the closure of a positive braid always admit a fillable positive surgery, as do knots that have lens space surgeries, and suitable satellites of such knots. In fact the majority of quasipositive knots with up to 10 crossings admit a fillable positive surgery. On the other hand, in general, (strong) quasipositivity, positivity, or Lagrangian fillability need not imply a knot admits a fillable positive contact surgery. [ABSTRACT FROM AUTHOR]

We use the degree of the colored Jones knot polynomials to show that the crossing number of a (p,q)$(p,q)$‐cable of an adequate knot with crossing number c$c$ is larger than q2c$q^2\, c$. As an application, we determine the crossing number of 2‐cables of adequate knots. We also determine the crossing number of the connected sum of any adequate knot with a 2‐cable of an adequate knot. [ABSTRACT FROM AUTHOR]

The text is a summary of the Proceedings of the 17th International Workshop on Real and Complex Singularities, which took place in 2022. The workshop is a renowned event in singularity theory and related fields, bringing together experts and young researchers to share their achievements and discuss research trends. The proceedings contain 29 research papers that have been selected and refereed according to the standards of the journal Research in the Mathematical Sciences. The collection aims to provide readers, including graduate students and researchers, with an opportunity to explore the field of singularities. The text also highlights the contributions and achievements of Professor Osamu Saeki, a leading researcher in singularity theory and topology, who was honored during the workshop. [Extracted from the article]

The sums S N ω , ζ = ∑ n - 0 N - 1 1 n 1 + e - 2 π i ω n + ω 2 + ζ , where ω and ζ are parameters, are related to trigonometric products from the theory of quasi-periodic operators as well as to a special function kindred to the Malyuzhinets function from the diffraction theory, the hyperbolic Ruijsenaars G-function, which arose in connection with the theory of integrable systems, and the Faddeev quantum dilogarithm, which plays an important role in the knot theory, Teichmuller quantum theory and the complex Chern–Simons theory. Assuming that ω ∈ (0, 1) and ζ ∈ ℂ−, we describe the behavior of logarithmic sums for large N using renormalization formulas similar to those well-known in the theory of Gaussian exponential sums. [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]

For a virtual knot K and an integer r with r ≥ 2, we introduce a method of constructing an r-component virtual link L(K; r), which we call the r-multiplexing of K. Every invariant of L(K; r) is an invariant of K. We give a way of calculating three kinds of invariants of L(K; r) using invariants of K. As an application of our method, we also show that Manturov’s virtual n-colorings for K can be interpreted as certain classical n-colorings for L(K; 2). [ABSTRACT FROM AUTHOR]

We obtain a localized version of the configuration space integral for the Casson knot invariant, where the standard symmetric Gauss form is replaced with a locally supported form. An interesting technical difference between the arguments presented here and the classical arguments is that the vanishing of integrals over hidden and anomalous faces does not require the well-known "involution tricks". The integral formula easily yields the well-known arrow diagram expression for regular knot diagrams, first presented in the work by Polyak and Viro. Moreover, it yields an arrow diagram count for the multicrossing knot diagrams, such as petal diagrams and gives a new lower bound for the übercrossing number. Previously, the known arrow diagram formulas were applicable only to the regular knot diagrams. [ABSTRACT FROM AUTHOR]

Summary Analysing protein structure similarities is an important step in protein engineering and drug discovery. Methodologies that are more advanced than simple RMSD are available but often require extensive mathematical or computational knowledge for implementation. Grouping and optimizing such tools in an efficient open-source library increases accessibility and encourages the adoption of more advanced metrics. Melodia is a Python library with a complete set of components devised for describing, comparing and analysing the shape of protein structures using differential geometry of 3D curves and knot theory. It can generate robust geometric descriptors for thousands of shapes in just a few minutes. Those descriptors are more sensitive to structural feature variation than RMSD deviation. Melodia also incorporates sequence structural annotation and 3D visualizations. Availability and implementation Melodia is an open-source Python library freely available on https://github.com/rwmontalvao/Melodia%5fpy , along with interactive Jupyter Notebook tutorials. [ABSTRACT FROM AUTHOR]

The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still presents us with tantalizing questions, such as Fox's conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial |$\Delta _{L}(t)$| of an alternating link |$L$| are unimodal. Fox's conjecture remains open in general with special cases settled by Hartley (1979) for two-bridge knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsváth and Szabó (2003) for the case of genus |$2$| alternating knots, among others. We settle Fox's conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of |$\Delta _{L}(t)$|⁠ , where |$L$| is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal. [ABSTRACT FROM AUTHOR]

We explicitly identify the algebra generated by symplectic Fourier–Deligne transforms (i.e. convolution with Kazhdan–Laumon sheaves) acting on the Grothendieck group of perverse sheaves on the basic affine space |$G/U$|⁠ , answering a question originally raised by A. Polishchuk. We show it is isomorphic to a distinguished subalgebra, studied by I. Marin, of the generalized algebra of braids and ties (defined in Type |$A$| by F. Aicardi and J. Juyumaya and generalized to all types by Marin), providing a connection between geometric representation theory and an algebra defined in the context of knot theory. Our geometric interpretation of this algebra entails some algebraic consequences: we obtain a short and type-independent geometric proof of the braid relations for Juyumaya's generators of the Yokonuma–Hecke algebra (previously proved case-by-case in types |$A, D, E$| by Juyumaya and separately for types |$B, C, F_{4}, G_{2}$| by Juyumaya and S. S. Kannan), a natural candidate for an analogue of a Kazhdan–Lusztig basis, and finally an explicit formula for the dimension of Marin's algebra in Type |$A_{n}$| (previously only known for |$n \leq 4$|⁠). [ABSTRACT FROM AUTHOR]

While finite graphs have tree-decompositions that efficiently distinguish all their tangles, locally finite graphs with thick ends need not have such tree-decompositions. We show that every locally finite graph without thick ends admits such a tree-decomposition, in fact a canonical one. Our proof exhibits a thick end at any obstruction to the existence of such tree-decompositions and builds on new methods for the analysis of the limit behaviour of strictly increasing sequences of separations. [ABSTRACT FROM AUTHOR]

Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in D 4 with knot group Z 2. In particular we show that if two such surfaces have the same normal Euler number, the same fixed knot boundary K such that | det ⁡ (K) | = 1 , and the same nonorientable genus 4 or 5, then they are ambiently isotopic rel. boundary. This implies that closed, nonorientable, locally flatly embedded surfaces in the 4-sphere with knot group Z 2 of nonorientable genus 4 and 5 are topologically unknotted. The proof relies on calculations, implemented in Sage, which imply that an obstruction to modified surgery is elementary. Furthermore we show that this method fails for nonorientable genus 6 and 7. [ABSTRACT FROM AUTHOR]

We show that there are no distance one surgeries on non-null-homologous knots in M that yield -M (M with opposite orientation) if M is a 3-manifold obtained by a Dehn surgery on a knot K in S^{3}, such that the order of its first homology is divisible by 9 but is not divisible by 27. As an application, we show several knots, including the (2,9) torus knot, do not have chirally cosmetic bandings. This simplifies the proof of a result first proven by Yang that the (2,k) torus knot (k>1) has a chirally cosmetic banding if and only if k=5. [ABSTRACT FROM AUTHOR]

A well-known conjecture of Rasmussen states that for any knot K in S^{3}, the rank of the reduced Khovanov homology of K is greater than or equal to the rank of the reduced knot Floer homology of K. This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Floer homology. Using an oriented cube of resolutions construction for a homology theory related to knot Floer homology, we prove this conjecture. [ABSTRACT FROM AUTHOR]

In this paper, which is the first paper from a trio on important developments of low dimensional topology in the past 100 years, we review the history and major developments in knot theory. This historic account includes the initial attempts at formulating some of the main questions about knots in a mathematical language, putting the definitions and arguments in a rigorous mathematical framework, and employing tools from other fields of mathematics to extract interesting and intriguing results in knot theory. The study starts from Tait's work in nineteenth century and reviews the important steps taken before the introduction of gauge theory. In particular, we will review the prime decomposition of knots and various polynomial invariants constructed for knots and links. We finish the paper by discussing some of the important conjectures in knot theory which have surprisingly simple statement. We also review some of the recent developments around the aforementioned conjecture, including some theorems including contributions from the 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]

Reconstructing evolutionary trajectories and transitions that have shaped floral diversity relies heavily on the phylogenetic framework on which traits are modelled. In this study, we focus on the angiosperm order Ranunculales, sister to all other eudicots, to unravel higher-level relationships, especially those tied to evolutionary transitions in flower symmetry within the family Papaveraceae. This family presents an astonishing array of floral diversity, with actinomorphic, disymmetric (two perpendicular symmetry axes), and zygomorphic flowers. We generated nuclear and plastid datasets using the Angiosperms353 universal probe set for target capture sequencing (of 353 single-copy nuclear ortholog genes), together with publicly available transcriptome and plastome data mined from open-access online repositories. We relied on the fossil record of the order Ranunculales to date our phylogenies and to establish a timeline of events. Our phylogenomic workflow shows that nuclear-plastid incongruence accompanies topological uncertainties in Ranunculales. A cocktail of incomplete lineage sorting, post-hybridization introgression, and extinction following rapid speciation most likely explain the observed knots in the topology. These knots coincide with major floral symmetry transitions and thus obscure the order of evolutionary events. [ABSTRACT FROM AUTHOR]

Spline Quasi-Interpolation (QI) of even degree 2 R on general partitions is introduced, where derivative information up to order R ≥ 1 at the spline breakpoints is required and maximal convergence order can be proved. Relying on the B-spline basis with possible multiple inner knots, a family of quasi-interpolating splines with smoothness of order R is associated with each R ≥ 1 , since there is the possibility of using different local sequences of breakpoints to define each QI spline coefficient. By using suitable finite differences approximations of the necessary discrete derivative information, each QI spline in this family can be associated also with a twin approximant belonging to the same spline space, but requiring just function information at the breakpoints. Among possible different applications of the introduced QI scheme, a smooth continuous extension of the numerical solution of Gauss-Lobatto and Gauss-Legendre Runge-Kutta methods is here considered. When R > 1 , such extension is based on the use of the variant of the QI scheme with derivative approximation which preserves the approximation power of the original Runge-Kutta scheme. [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]

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]

Smart materials provide a means by which we can create engineered mechanisms that artificially mimic the adaptability, flexibility, and responsiveness found in biological systems. Previous studies have developed material-based actuators that could produce targeted shape changes. Here, we extend this capability by introducing a novel computational and experimental method for design and synthesis of a material-based mechanism capable of achieving complex pre-programmed motion. By combining active and passive materials, the algorithm can encode the desired movement into the material distribution of the mechanism. We use multimaterial, multiphysics topology optimization to design a set of kinematic elements that exhibit basic bending and torsional deflection modes. We then use a genetic algorithm to optimally arrange these elements into a sequence that produces the desired motion. We also use experimental measurements to accurately characterize the angular deflection of the 3D-printed kinematic elements in response to thermomechanical loading. We demonstrate this new capability by de novo design of a 3D-printed self-tying knot. This method advances a new paradigm in mechanism design that could enable a new generation of material-driven machines that are lightweight, adaptable, robust to damage, and easily manufacturable by 3D printing. [ABSTRACT FROM AUTHOR]

We investigate the capability of symplectic quandles to detect causality for (2+1)-dimensional globally hyperbolic spacetimes (X). Allen and Swenberg showed that the Alexander–Conway polynomial is insufficient to distinguish connected sum of two Hopf links from the links in the family of Allen–Swenberg 2-sky like links, suggesting that it cannot always detect causality in X. We find that symplectic quandles, combined with Alexander–Conway polynomial, can distinguish these two types of links, thereby suggesting their ability to detect causality in X. The fact that symplectic quandles can capture causality in the Allen–Swenberg example is intriguing since the theorem of Chernov and Nemirovski, which states that Legendrian linking equals causality, is proved using Contact Geometry methods. [ABSTRACT FROM AUTHOR]

We prove that all rational slopes are characterizing for the knot 52$5_2$, except possibly for positive integers. Along the way, we classify the Dehn surgeries on knots in S3$S^3$ that produce the Brieskorn sphere Σ(2,3,11)$\Sigma (2,3,11)$, and we study knots on which large integral surgeries are almost L‐spaces. [ABSTRACT FROM AUTHOR]

The fundamental quandle is a powerful invariant of knots, links and spatial graphs, but it is often difficult to determine whether two quandles are isomorphic. One approach is to look at quotients of the quandle, such as the n-quandle defined by Joyce [8]; in particular, Hoste and Shanahan [5] classified the knots and links with finite n-quandles. Mellor and Smith [12] introduced the N-quandle of a link as a generalization of Joyce's n-quandle, and proposed a classification of the links with finite N-quandles. We generalize the N-quandle to spatial graphs, and investigate which spatial graphs have finite N-quandles. We prove basic results about N-quandles for spatial graphs, and conjecture a classification of spatial graphs with finite N-quandles, extending the conjecture for links in [12]. We verify the conjecture in several cases, and also present a possible counterexample. [ABSTRACT FROM AUTHOR]

The technique of distinguishing one knot from another by coloring arcs and applying some basic modular arithmetic is part of most standard undergraduate knot theory classes. When we study n-colorability, we are usually only interested when n is a prime number. But what if n is composite? What can we say then? [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]

Let |$K$| be a knot type for which the quadratic term of the Conway polynomial is nontrivial, and let |$\gamma : {\mathbb {R}}\to {\mathbb {R}}^{3}$| be an analytic |${\mathbb {Z}}$| -periodic function with non-vanishing derivative that parameterizes a knot of type |$K$| in space. We prove that there exists a sequence of numbers |$0\leq t_{1} < t_{2} <... < t_{6} < 1$| so that the polygonal path obtained by cyclically connecting the points |$\gamma (t_{1}), \gamma (t_{2}),... \gamma (t_{6})$| by line segments is a trefoil knot. [ABSTRACT FROM AUTHOR]

We re‐build the quantum sl(2)${\mathfrak {sl}(2)}$ unified invariant of knots F∞$F_{\infty }$ from braid groups' action on tensors of Verma modules. It is a two variables series having the particularity of interpolating both families of colored Jones polynomials and ADO polynomials, that is, semisimple and non‐semisimple invariants of knots constructed from quantum sl(2)${\mathfrak {sl}(2)}$. We prove this last fact in our context that re‐proves (a generalization of) the famous Melvin–Morton–Rozansky conjecture first proved by Bar‐Natan and Garoufalidis. We find a symmetry of F∞$F_{\infty }$ nicely generalizing the well‐known one of the Alexander polynomial, ADO polynomials also inherit this symmetry. It implies that quantum sl(2)${\mathfrak {sl}(2)}$ non‐semisimple invariants are not detecting knots' orientation. Using the homological definition of Verma modules we express F∞$F_{\infty }$ as a generating sum of intersection pairing between fixed Lagrangians of configuration spaces of disks. Finally, we give a formula for F∞$F_{\infty }$ using a generalized notion of determinant, that provides one for the ADO family. It generalizes that for the Alexander invariant. [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]

We determine the crossing numbers of (prime) amphicheiral knots. This problem dates back to the origin of knot tables by Tait and Little at the end of the nineteenth century. The proof is the most substantial application of the semiadequacy formulas for the edge coefficients of the Jones polynomial. [ABSTRACT FROM AUTHOR]

Let p$p$ be a prime number and m$m$ an integer coprime to p$p$. In the spirit of arithmetic topology, we introduce the notions of the twisted Iwasawa invariants λ,μ,ν$\lambda , \mu , \nu$ of GLN${\rm GL}_N$‐representations and Z/mZ×Zp${\mathbb {Z}}/m{\mathbb {Z}}\times {\mathbb {Z}}_{p}$‐covers of knots. We prove among other things that the set of Iwasawa invariants determines the genus and the fiberedness of a knot, yielding their profinite rigidity. Several intuitive examples are attached. We further prove the μ=0$\mu =0$ theorem for SL2${\rm SL}_2$‐representations of twist knot groups and give some remarks. [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]

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]

We define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends to virtual knots and can obstruct a virtual knot from being classical. Furthermore, it can distinguish a knot in a thickened surface from its mirror up to isotopy. We also propose several constructions of Heegaard Floer homology for knots in thickened surfaces, and give examples why they are not stabilization invariant. However, we can define Floer homology for virtual knots by taking a minimal genus representative. Finally, we use the Behrens–Golla δ -invariant to obstruct a knot from being a stabilization of another. [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]

For some important knots, closed-form solutions are presented for the holding forces which are needed to keep a knot in equilibrium for given pulling forces. If the holding forces become zero for finite pulling forces, the knot is self-locking and is called stable. This is only possible when, first, the friction coefficient exceeds a critical value and, second, when there is additional pressure on some knot segments sandwiched by surrounding knot segments. The number of these segments depends on the topology of the knot and is characteristic for it. The other important parameter is the total curvature of the knot. In this way, the complete frictional contact inside the knot is taken into account. The presented model can explain the available experiments. [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]

We prove a simple necessary and sufficient condition for a two-bridge knot K (p , q) to be quasipositive, based on the continued fraction expansion of p / q. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in Appendix A, by Stepan Orevkov. [ABSTRACT FROM AUTHOR]

We show that the cobordism maps on Khovanov homology can distinguish smooth surfaces in the 4-ball that are exotically knotted (i.e., isotopic through ambient homeomorphisms but not ambient diffeomorphisms). We develop new techniques for distinguishing cobordism maps on Khovanov homology, drawing on knot symmetries and braid factorizations. We also show that Plamenevskaya's transverse invariant in Khovanov homology is preserved by maps induced by positive ascending cobordisms. [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]

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]

Highly entangled gels have gained extensive attention due to their excitingly large deformation and high toughness. To understand the toughening mechanism of these highly entangled gels, an entanglement constraint model has been established, based on the spatially prismatic constraint and Gaussian distribution models. A free-energy function is formulated to study the conformational dynamics, rubbery elasticity and sliding effect of topological knots in the entangled chains. Monte Carlo, molecular dynamics and finite element analysis were conducted to verify the coupling effect of inter-chain entanglement and intra-chain knot topology on the toughness behavior of highly entangled gels. Finally, experimental data available in the literature were used to verify the proposed models, providing a physical insight into the toughening mechanism of inter-chain entanglement constraint and intra-chain knot topology in the highly entangled gel. [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 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]

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]

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]

For a balanced sutured manifold |$(M,\gamma)$|⁠ , we construct a decomposition of |$SHI(M,\gamma)$| with respect to torsions in |$H=H_{1}(M;\mathbb {Z})$|⁠ , which generalizes the decomposition of |$I^{\sharp }(Y)$| in a previous work of the authors. This decomposition can be regarded as a candidate for the counterpart of the torsion spin |$^{c}$| decompositions in |$SFH(M,\gamma)$|⁠. Based on this decomposition, we define an enhanced Euler characteristic |$\chi _{\textrm {en}}(SHI(M,\gamma))\in \mathbb {Z}[H]/\pm H$| and prove that |$\chi _{\textrm {en}}(SHI(M,\gamma))=\chi (SFH(M,\gamma))$|⁠. This provides a better lower bound on |$\dim _{\mathbb {C}}SHI(M,\gamma)$| than the graded Euler characteristic |$\chi _{\textrm {gr}}(SHI(M,\gamma))$|⁠. As applications, we prove instanton knot homology detects the unknot in any instanton L-space and show that the conjecture |$KHI(Y,K)\cong \widehat {HFK}(Y,K)$| holds for all |$(1,1)$| -L-space knots and constrained knots in lens spaces, which include all torus knots and many hyperbolic knots in lens spaces. [ABSTRACT FROM AUTHOR]