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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]

In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary. [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]

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]

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]

This paper, in homage to Patrick Dehornoy, tells the birth of Garside groups. [ABSTRACT FROM AUTHOR]

In this paper we study isotopy classes of closed connected orientable surfaces in the standard 3 -sphere. Such a surface splits the 3 -sphere into two compact connected submanifolds, and by using their Heegaard splittings, we obtain a 2 -component handlebody-link. In this paper, we first show that the equivalence class of such a 2-component handlebody-link up to attaching trivial 1 -handles can recover the original surface. Therefore, we can reduce the study of surfaces in the 3 -sphere to that of 2 -component handlebody-links up to stabilizations. Then, by using G -families of quandles, we construct invariants of 2 -component handlebody-links up to attaching trivial 1 -handles, which lead to invariants of surfaces in the 3 -sphere. In order to see the effectiveness of our invariants, we will also show that our invariants can distinguish certain explicit surfaces in the 3 -sphere. [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]

The goal of this paper is two-fold: we generalize the arithmetic Chern–Simons theory over totally imaginary number fields studied in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3; M. Kim, Arithmetic Chern–Simons theory I, preprint (2015) arXiv:1510.05818v4] to arbitrary number fields (with real places) and provide new examples of non-trivial arithmetic Chern–Simons invariant with coefficient ℤ / n ℤ (n ≥ 2) associated to a non-abelian gauge group. The main idea for the generalization is to use cohomology with compact support (see [17]) to deal with real places. Before the results of this paper, non-trivial examples were limited to some non-abelian gauge group with coefficient ℤ / 2 ℤ in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3] and the abelian cyclic gauge group with coefficient ℤ / n ℤ in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3]. Our non-trivial examples (with non-abelian gauge group and general coefficient ℤ / n ℤ) will be given by a simple twisting argument based on examples of [F. Bleher, T. Chinburg, R. Greenberg, M. Kakde, G. Pappas and M. Taylor, Cup products in the étale cohomology of number fields, New York J. Math. 24 (2018) 514–542]. [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]

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]

Random hypergraphs and random simplicial complexes on finite vertices were studied by [M. Farber, L. Mead and T. Nowik, Random simplicial complexes, duality and the critical dimension, J. Topol. Anal.41(1) (2022) 1–32]. The map algebra on random sub-hypergraphs of a fixed simplicial complex, which detects relations between random sub-hypergraphs and random simplicial sub-complexes, was studied by the authors of this paper. In this paper, we study the map algebra on random sub-hypergraphs of a fixed hypergraph. We give some algorithms generating random hypergraphs and random simplicial complexes by considering the actions of the map algebra on the space of probability distributions. We prove some Künneth-type formulae for random hypergraphs and random simplicial complexes on finite vertices. [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]

Let n be the set of un-oriented and rational links with crossing number n , a precise formula for | n | was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let Λ n be the set of oriented rational links with crossing number n and let Λ n (d) be the set of oriented rational links with crossing number n (n ≥ 2) and deficiency d. In this paper, we derive precise formulas for | Λ n | and | Λ n (d) | for any given n and d and show that Λ n (d) = F n − d − 1 (d) + 1 + (− 1) n d 2 F ⌊ n 2 ⌋ − ⌊ d + 1 2 ⌋ (⌊ d 2 ⌋) , where F n (d) is the d th convolution of the convolved Fibonacci sequences. [ABSTRACT FROM AUTHOR]

In this paper, we observe that the hat version of the Heegaard Floer invariant of Legendrian knots in contact three-manifolds defined by Lisca-Ozsváth-Stipsicz-Szabó can be combinatorially computed. We rely on Plamenevskaya's combinatorial description of the Heegaard Floer contact invariant. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the notion of singquandles introduced in [I. R. U. Churchill, M. Elhamdadi, M. Hajij and S. Nelson, Singular knots and involutive quandles, J. Knot Theory Ramifications 26(14) (2017) 1750099]. This is an algebraic structure that naturally axiomatizes Reidemeister moves for singular links, similarly to what happens for ordinary links and quandles. We present a new axiomatization that shows different algebraic aspects and simplifies applications. We also reformulate and simplify the axioms for affine singquandles (in particular in the idempotent case). [ABSTRACT FROM AUTHOR]

The warping degree of an oriented knot diagram is the minimal number of crossings which we meet as an under-crossing first when we travel along the diagram from a fixed point. The warping degree of a knot projection is the minimal value of the warping degree for all oriented alternating diagrams obtained from the knot projection. In this paper, we consider the maximal number of regions which share no crossings for a knot projection with a fixed crossing, and give lower bounds for the warping degree. [ABSTRACT FROM AUTHOR]

In this paper, we discuss applications of neural networks to recognizing knots and, in particular, to the unknotting problem. One of the motivations for this study is to understand how neural networks work on the example of a problem for which rigorous mathematical algorithms for its solution are known. We represent knots by rectangular Dynnikov diagrams and apply neural networks to distinguish a given diagram's class from a finite family of topological types. The data presented to the program is generated by applying Dynnikov moves to initial samples. The significance of using these diagrams and moves is that in this context the problem of determining whether a diagram is unknotted is a finite search of a bounded combinatorial space. In this way, this paper provides a foundation for further work where the neural network itself will learn to use the Dynnikov moves for knot recognition. Source code of the programs is available at https://github.com/nerusskikh/deepknots. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new periodicity criterion. For that purpose, we study the HOMFLY-PT polynomial and the Kauffman polynomial of cables of periodic links. Furthermore, we exhibit a couple of examples for which our criterion is stronger than many previously known criteria, among which is the Khovanov homology criterion of Borodzik and the second author. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate a natural family of metrics on connected components of a rack. The metrics are closely related to certain bi-invariant metrics on the group of inner automorphisms of the rack. We also introduce a bounded cohomology of racks and quandles, relate them to the above metrics and prove a vanishing result for racks and quandles with amenable group of inner automorphisms. [ABSTRACT FROM AUTHOR]

In 2019, Cho and Nelson introduced an enhancement of the quandle counting invariant, which is quandle coloring quivers. In 2021, Basi and Caprau used it to study (p, 2)-torus links. Recently, Zhou and Liu used it to study (p, 3)-torus links. In this paper, we study the quandle coloring space of (p, 4)- and (p, 5)-torus links by dihedral quandles and make a conjecture about the quandle coloring quiver of (p,q)-torus links, where q is an odd prime number. [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]

Over-then-Under (OU) tangles are oriented tangles whose strands travel through all of their over crossings before any under crossings. In this paper, we discuss the idea of gliding: an algorithm by which tangle diagrams could be brought to OU form. By analyzing cases in which the algorithm converges, we obtain a braid classification result, which we also extend to virtual braids, and provide a Mathematica implementation. We discuss other instances of successful "gliding ideas" in the literature — sometimes in disguise — such as the Drinfel'd double construction, Enriquez's work on quantization of Lie bialgebras, and Audoux and Meilhan's classification of welded homotopy links. [ABSTRACT FROM AUTHOR]

It is known that the category H of nilpotent weight modules over the quantum group associated with the super Lie algebra (2 | 1) is a relative pre-modular G -category. Its modified trace enables one to define an invariant of 3 -manifolds. In this paper, we show that the category H is a relative modular G -category which allows one to constructing a family of non-semi-simple extended topological quantum field theories, and the family of TQFTs is anomaly free. The quantum group associated with (2 | 1) is considered at odd roots of unity. [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 embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let Γ be a chart, and we denote by Γ m the union of all the edges of label m. A chart Γ is of type (7) if there exists a label m such that w (Γ) = 7 , w (Γ m ∩ Γ m + 1) = 7 where w (G) is the number of white vertices in G. In this paper, we prove that there is no minimal chart of type (7). [ABSTRACT FROM AUTHOR]

In this paper, we study the Riley polynomial of double twist knots with higher genus. Using the root of the Riley polynomial, we compute the range of rational slope r such that r -filling of the knot complement has left-orderable fundamental group. Further more, we make a conjecture about left-orderable surgery slopes of two-bridge knots. [ABSTRACT FROM AUTHOR]

In the previous paper, we showed that the Riley polynomial ℛ K (λ) of each 2-bridge knot K is split into ℛ K (− u 2) = ± g (u) g (− u) , for some integral coefficient polynomial g (u) ∈ ℤ [ u ]. In this paper, we study this splitting property of the Riley polynomial. We show that the Riley polynomial can be expressed by ' -Chebyshev polynomials', which is a generalization of Chebyshev polynomials containing the information of i -sequence ( i = (− 1) [ i β α ]) of the 2-bridge knot K = S (α , β) , and then we give an explicit formula for the splitting polynomial g (u) also as -Chebyshev polynomials. As applications, we find a sufficient condition for the irreducibility of the Riley polynomials and show the unimodal property of the symmetrized Riley polynomial. [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]

A relative trisection of a compact 4 -manifold with connected boundary is said to be a Trivially Extendable (TE)-relative trisection if the boundary monodromy of the relative trisection is the identity map. We call a relative trisection diagram associated to a T E -relative trisection of a compact 4 -manifold, a T E -relative trisection diagram. Due to a work of Castro and Ozbagci, there is a natural way to associate a trisection diagram to a T E -relative trisection diagram. In fact, every closed oriented 4 -manifold can be realized as a T E -relative trisection. In this paper, we study the notion of Murasugi sum of T E -relative trisection diagrams. We show that a trisection diagram associated to a Murasugi sum of two T E -relative trisection diagrams and ′ is diffeomorphism and handle slide equivalent to the connected sum of the trisection diagrams associated to and ′ . This implies that the closed oriented 4 -manifold associated to the Murasugi sum of T E -relative trisection diagrams and ′ is the connected sum of the closed 4 -manifolds associated to and ′ . Thus, the notion of Murasugi sum can be considered for closed 4 -manifolds in terms of T E -relative trisections. Finally, we discuss an external stabilization of a T E -relative trisection of a closed oriented 4 -manifold. We prove that the trisection diagram associated to a stabilization of a T E -relative trisection diagram is diffeomorphism and handle slide equivalent to a stabilization of the trisection diagram associated to the T E -relative trisection diagram. [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]

Gay and Meier asked whether or not a trisection diagram constructed by the Gluck twist on a spun or a twist spun 2-knot obtained from some method is standard. In this paper, we depict the trisection diagrams explicitly when the 2-knot is the spun (2 n + 1 , 2) -torus knot, where n ≥ 1 , and show that the trisection diagram is standard when n = 1. Moreover, we introduce a notion of homologically standard for trisection diagrams and show that the trisection diagram is homologically standard for all n. [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]

Plucking polynomial of a plane rooted tree with a delay function α was introduced in 2014 by Przytycki. As shown in this paper, plucking polynomial factors when α satisfies additional conditions. We use this result and Θ A -state expansion introduced in our previous work to derive new properties of coefficients C (A) of Catalan states C resulting from an (m × n) -lattice crossing L (m , n). In particular, we show that C (A) factors when C has arcs with some special properties. In many instances, this yields a more efficient way for computing C (A). As an application, we give closed-form formulas for coefficients of Catalan states of L (m , 3). [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 surface-link is a closed surface embedded in the 4-space, possibly disconnected or non-orientable. Every surface-link can be presented by the plat closure of a braided surface, which we call a plat form presentation. The knot symmetric quandle of a surface-link F is a pair of a quandle and a good involution determined from F. In this paper, we compute the knot symmetric quandle for surface-links using a plat form presentation. As an application, we show that for any integers g ≥ 0 and m ≥ 2 , there exist infinitely many distinct surface-knots of genus g whose plat indices are m. [ABSTRACT FROM AUTHOR]

Let T be a tree with vertices 2 m -dimensional oriented vector bundles over S 2 m and M (T) be the boundary of the oriented manifold obtained by plumbing according to T. In this paper, we calculate linking numbers between the fiber spheres, provided M (T) is a rational homology sphere, and thereby provides a correction and completion of a work of tom Dieck. [ABSTRACT FROM AUTHOR]

Two links are link-homotopic if they are transformed into each other by a sequence of self-crossing changes and ambient isotopies. The link-homotopy classes of links with up to three components were classified by the Milnor homotopy invariants. Levine investigated the indeterminacy of Milnor homotopy invariants and gave a classification of the link-homotopy classes of 4-component links. In this paper, we give another classification of the link-homotopy classes of 4-component links by using the clasper theory. This classification is obtained through a tetrahedron standard form of the link-homotopy classes of 4-component links. As an application, we give several new subsets of the link-homotopy classes of 4-component links which are classified by computable invariants. Moreover, the classification allows us to run the Habegger–Lin algorithm which determines whether two given links are link-homotopic or not. [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]

Yoshikawa made a table of knotted surfaces in ℝ 4 with ch-index 10 or less. This remarkable table is the first to enumerate knotted surfaces analogous to the classical prime knot table. A broken sheet diagram of a surface-link is a generic projection of the surface in ℝ 3 with crossing information along its singular set. The minimal number of triple points among all broken sheet diagrams representing a given surface-link is its triple point number. This paper compiles the known triple point numbers of the surface-links represented in Yoshikawa's table and calculates or provides bounds on the triple point number of the remaining surface-links. [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]

A weave is the lift to the thickened Euclidean plane of a particular type of quadrivalent planar connected graph with an over or under crossing information to each vertex and such that the lifted components are non-intersecting simple open curves. In this paper, we introduce a formal topological definition of weaves as three-dimensional entangled structures and characterize the equivalence classes of doubly periodic untwisted (p , q) -weaves by introducing a new invariant, called crossing matrix. Finally, we suggest a combinatorial approach to classify this specific class of weaves by their crossing number. [ABSTRACT FROM AUTHOR]

In this paper, we give a new proof of Vassilievs planarity criterion of a graph with vertices of degree 4 and an additional cross structure. It is also established that all height atoms are orientable. The criterion for an atom to be an orientable one is established in terms of its f -graph. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of Lefschetz open book decompositions for closed oriented smooth 4 -manifolds. We define the notion of a Murasugi sum of Lefschetz open book decompositions. We show that the closed oriented 4 -manifold associated to a Murasugi sum of Lefschetz open books of two closed oriented 4 -manifolds X 1 and X 2 is the connected sum of X 1 and X 2 . We also introduce the notion of a stabilization of a Lefschetz open book. We introduce the notion of a Lefschetz fibered link in a closed oriented 4 -manifold and using the stabilization of Lefschetz open book, we show that in a simply connected closed oriented 4 -manifold X , every oriented link B in X is Lefschetz fibered. We also discuss the existence of a folded Kähler structure on a closed oriented 4 -manifold associated to a given Lefschetz open book which essentially follows by the work of Baykur. [ABSTRACT FROM AUTHOR]

A ribbon is a two-dimensional object with one-dimensional properties, which is related with geometry, robotics and molecular biology. A folded ribbon structure provides a complex structure through a series of folds. We focus on a folded ribbon with knotted core. The folded ribbonlength Rib (K) of a knot K is the infimum of the quotient of the length by the width among the ribbons representing a knot type of K. This quantity tells how efficiently the folded ribbon is realized. Kusner conjectured that folded ribbonlength is bounded above by a linear function of the minimal crossing number c (K). In this paper, we confirm that the folded ribbonlength of a 2-bridge knot K is bounded above by 2 c (K) + 2. [ABSTRACT FROM AUTHOR]

Feller et al. showed the homology and the intersection form of a closed trisected four-manifold are described in terms of trisection diagram. In this paper, it is confirmed that we are able to calculate those of a trisected four-manifold with boundary in a similar way. Moreover, we describe a representative of the second Stiefel–Whitney class by the relative trisection diagram. [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]

The problem whether the purely cosmetic surgery conjecture holds for any knots is open. Using recent work of Hanselman, the conjecture was proved for any pretzel knots [A. I. Stipsicz and Z. Szabo, Purely cosmetic surgeries and pretzel knots (2020), arXiv:2006.06765]. In this paper, we prove the conjecture for any three strand pretzel knots in a way that does not depend on Heegaard Floer homology. [ABSTRACT FROM AUTHOR]