Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such

Let Rf=ℤ[A±1] be the algebra of Laurent polynomials in the variable A and let Ra=ℤ[A±1,z1,z2,…] be the algebra of Laurent polynomials in the variable A and standard polynomials in the variables z1,z2,…. For n≥1 we denote by VBn the virtual braid group on n strands. We define two towers of algebras {VTLn(Rf)}n=1∞ and {ATLn(Ra)}n=1∞ in terms of diagrams. For each n≥1 we determine presentations for both

We introduce the notion of bicolored diagrams which are closely related to the region crossing changes. Moreover, we refine Cheng’s results on the region crossing changes and propose a certain way to calculate the Arf invariant of a proper link using a bicolored diagram.

We use the Bar-Natan Ж-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Ж-map is functorial under concordance, and also that Satoh’s Tube map (from welded links to ribbon knotted tori in S4) is functorial under concordance. In addition, we extend classical results of Chen, Milnor and Hillman

Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots K(−m,−p) and K(−m,p) where m and p are positive integers. In the (−m,−p) case, this leads to new families of q-hypergeometric series generalizing the Kontsevich–Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of K(m,p) gives a generalization of a duality at

Let Mreg be the topological moduli space of long knots up to regular isotopy, and for any natural number n>1 let Mnreg be the moduli space of all n-cables of framed long knots which are twisted by a string link to a knot in the solid torus V3. We upgrade the Vassiliev invariant v2 of a knot to an integer valued combinatorial 1-cocycle for Mnreg by a very simple formula. This 1-cocycle depends on a

Given an oriented minimal genus Seifert surface F′ for a (1,1)-knot K it is possible to surger F′ along annuli to obtain a simple minimal Seifert surfaceF. Such a surface can be put in a very nice position with respect to the (1,1)-position of the knot K. Using this kind of surfaces we give a description of a (1,1)-knot of genus g as a vertical banding of (1,1)-knots of genus smaller than g. In addition

We calculate the Ohtsuki invariants λi(M)(i=0,1,2,3) of every Brieskorn–Hamm manifold M which is a rational homology 3-sphere. We denote the order of H1(M;ℤ) by H. By the result, we show that the third Ohtsuki invariant λ3(M) of the Brieskorn–Hamm manifolds M with H=1 is negative, and the third Ohtsuki invariant λ3(M) of most Brieskorn–Hamm manifolds M with H≥2 is positive.

This paper concerns the braid index of an alternating link. It is well known that the braid index of any link equals the minimum number of Seifert circles among all link diagrams representing it. For an alternating link represented by a reduced alternating diagram D, it is known that s(D), the number of Seifert circles in D, equals the braid index b(D) of D if D contains no lone crossings, where a

The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial

We introduce a new very large family of transformations of rectangular diagrams of links that preserve the isotopy class of the link. We provide an example when two diagrams of the same complexity are related by such a transformation and are not obtained from one another by any sequence of “simpler” moves not increasing the complexity of the diagram along the way.

The aim of the present paper is to construct series of invariants of free knots (flat virtual knots, virtual knots) valued in free groups (and also free products of cyclic groups).

In this paper, we give a combinatorial description of the concordance invariant 𝜀 defined by Hom, prove some properties of this invariant using grid homology techniques. We compute the value of 𝜀 for (p,q) torus knots and prove that 𝜀(𝔾+)=1 if 𝔾+ is a grid diagram for a positive braid. Furthermore, we show how 𝜀 behaves under (p,q)-cabling of negative torus knots.

In this work, we compute the representation of the mapping class group of the sphere with 4 punctures arising from the non-semi-simple TQFT [C. Blanchet, F. Costantino, N. Geer and B. Patureau, Non-semi-simple TQFTs, Reidemeister torsion and Kashaev’s invariants, Adv. Math.301 (2016) 1–78]. We show that it is faithful. Lastly, we compare quantum representations of punctured spheres in general with

Following the classification of genus one fibered knots in lens spaces by Baker, we determine hyperbolic genus one fibered knots in lens spaces on whose all integral Dehn surgeries yield closed 3-manifolds with left-orderable fundamental groups.

We define and explore rack objects internal to categories with products. In demonstration, we classify the group-racks, and use homotopy to prove both existence and exclusion theorems for path-connected topological racks.

In this paper, we recall the geometric definition of the braid group by Emil Artin and we give a complete, elementary geometric/topological proof of the standard presentation of the braid group on n strings.

We consider the relationship between the crosscap number γ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots K and J, we say K≥J if there exists an epimorphism f:π1(S3−K)→π1(S3−J). We prove that if K and J are 2-bridge knots and K>J, then γ(K)≥3γ(J)−4. We also classify all pairs (K,J) for which the inequality is sharp. A similar result relating the

We investigate several conjectures in geometric topology by assembling computer data obtained by studying weaving knots, a doubly infinite family W(p,n) of examples of hyperbolic knots. In particular, we compute some important polynomial knot invariants, as well as knot homologies, for the subclass W(3,n) of this family. We use these knot invariants to conclude that all knots W(3,n) are fibered knots

Consider a knot K in S3 with charge uniformly distributed on it. From the standpoint of both physics and knot theory, it is natural to try to understand the critical points of the potential and their behavior. We show the number of critical points of the potential is at least 2t(K)+2, where t(K) is the tunnel number, defined as the smallest number of arcs one must add to K such that its complement

We consider a natural generalization of braids which we call shrinking braids. We state the relations of shrinking braids and use them to define algebraically the monoid R. We endow a subset of R with a left distributive monoid structure and use it to extend the Dehornoy order on B∞ to an order on R. By using this order, we prove that R is isomorphic to the monoid which is generated (geometrically)

We study Kauffman’s model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a folded ribbon knot. We show for any knot or link type that there exist constants c1,c2>0 such that the ribbonlength is bounded above by c1Cr(K)2, and also by c2Cr(K)3/2. We use a different method for each bound. The constant c1 is quite

Finding homotopy group of spheres is an old open problem in topology. Berrick et al. derive in [A. J. Berrick, F. Cohen, Y. L. Wong and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006)] an exact sequence that relates Brunnian braids to homotopy groups of spheres. We give an interpretation of this exact sequence based on the combed form for braids over the sphere developed

Negami found an upper bound on the stick number s(K) of a nontrivial knot K in terms of the minimal crossing number c(K): s(K)≤2c(K). Huh and Oh found an improved upper bound: s(K)≤32(c(K)+1). Huh, No and Oh proved that s(K)≤c(K)+2 for a 2-bridge knot or link K with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let K

Let M be a closed oriented 3-manifold and let G be a discrete group. We consider a representation ρ:π1(M)→G. For a 3-cocycle α, the Dijkgraaf–Witten invariant is given by (ρ∗α)[M], where ρ∗:H3(G)→H3(M) is the map induced by ρ, and [M] denotes the fundamental class of M. Note that (ρ∗α)[M]=α(ρ∗[M]), where ρ∗:H3(M)→H3(G) is the map induced by ρ, we consider an equivalent invariant ρ∗[M]∈H3(G), and we

In this paper, we study Dehn colorings of spatial graph diagrams, and classify the vertex conditions, equivalently the palettes. We give some example of spatial graphs which can be distinguished by the number of Dehn colorings with selecting an appropriate palette. Furthermore, we also discuss the generalized version of palettes, which is defined for knot-theoretic ternary-quasigroups and region colorings

A twisted torus knot T(p,q,r,s) is a torus knot T(p,q) with r adjacent strands twisted fully s times. In this paper, we determine the braid index of the knot T(p,q,r,s) when the parameters p,q,r satisfy 1

We introduce a numerical invariant called the braid alternation number that measures how far a link is from being an alternating closed braid. This invariant resembles the alternation number, which was previously introduced by the second author. However, these invariants are not equal, even for alternating links. We study the relation of this invariant with others and calculate this invariant for some

Extending upon our previous work, we verify the Jones Unknot Conjecture for all knots up to 24 crossings. We describe the method of our approach and analyze the growth of the computational complexity of its different components.

Using a combinatorial argument, we prove the well-known result that the Wirtinger and Dehn presentations of a link in 3-space describe isomorphic groups. The result is not true for links ℓ in a thickened surface S×[0,1]. Their precise relationship, as given in [R. E. Byrd, On the geometry of virtual knots, M.S. Thesis, Boise State University (2012)], is established here by an elementary argument. When

In this paper, we show that every spinal open book decomposition of a closed oriented 3-manifold M spinal open book embeds into a certain spinal open book decomposition of #kS2×̃S3, the connected sum of k copies of the twisted S3-bundle over S2, where k depends on the spinal open book decomposition of M. We also discuss spinal open book embeddings of a huge class of spinal open books of closed oriented

By considering unknotting operations, we obtain ways of measuring how knotted a knot is. Unknotting phenomena can be seen not only in knot theory, but also in various settings such as DNA knots, mind knots and so on ([C. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (American Mathematical Society, Providence, RI, 2004); A. Kawauchi, K. Kishimoto and A. Shimizu

Fox conjectured the Alexander polynomial of an alternating knot is trapezoidal, i.e. the absolute values of the coefficients first increase, then stabilize and finally decrease in a symmetric way. Recently, Hirasawa and Murasugi further conjectured a relation between the number of the stable coefficients in the Alexander polynomial and the signature invariant. In this paper we prove the Hirasawa–Murasugi

In this paper, we consider local moves on classical and welded diagrams of string links, and the notion of welded extension of a classical move. Such extensions being non-unique in general, the idea is to find a topological criterion which could isolate one extension from the others. To that end, we turn to the relation between welded string links and knotted surfaces in ℝ4, and the ribbon subclass

We improve the upper bound on superbridge index sb[K] in terms of bridge index b[K] from sb[K]≤5b[K]−3 to sb[K]≤3b[K]−1.

A flat plumbing basket is a Seifert surface consisting of a disk and bands contained in distinct pages of the disk open book decomposition of the 3-sphere. In this paper, we examine close connections between flat plumbing baskets and the contact structure supported by the open book. As an application we give lower bounds for the flat plumbing basket numbers and determine the flat plumbing basket numbers

Cho and Nelson introduced the notion of a quandle coloring quiver, which is a quiver-valued link invariant. This invariant is in general a stronger link invariant than the quandle coloring number. In this paper, we study quandle coloring quiver using dihedral quandle. We show that when we use a dihedral quandle of prime order, the quandle coloring quivers are equivalent to the quandle coloring numbers

It is known that the Kauffman–Murasugi–Thislethwaite type inequality becomes an equality for any (possibly virtual) adequate link diagram. We refine this condition. As an application we obtain a criterion for virtual link diagram with exactly one virtual crossing to represent a properly virtual link.

Quandle coloring quivers are directed graph-valued invariants of oriented knots and links, defined using a choice of finite quandle X and set S⊂Hom(X,X) of endomorphisms. From a quandle coloring quiver, a polynomial knot invariant known as the in-degree quiver polynomial is defined. We consider quandle coloring quiver invariants for oriented surface-links, represented by marked graph diagrams. We provide

We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and show that this new singular link invariant generalizes the singquandle counting invariant. In particular, using the new polynomial invariant, we can distinguish

It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce

We show that, for any integers, g≥3 and n≥2, there exists a link in S3 such that its complement has a genus g Heegaard splitting with distance n.

We introduce the concept of tied links in the solid torus, which generalizes naturally the concept of tied links in S3 previously introduced by Aicardi and Juyumaya. We also define an invariant of these tied links by using skein relations, and we then recover this invariant by using Jones’ method over the bt-algebra of type B and the Markov trace defined on this.

Joyce showed that for a classical knot K, the involutory medial quandle IMQ(K) is isomorphic to the core quandle of the homology group H1(X2), where X2 is the cyclic double cover of 𝕊3, branched over K. It follows that |IMQ(K)|=|detK|. In this paper, the extension of Joyce’s result to classical links is discussed. Among other things, we show that for a classical link L of μ≥2 components, the order

We give a classification of 2-twist-spun spherical Montesinos knots.

In this paper, we study several complex manifolds by using the following idea. First, we construct a certain moduli space and study the fundamental group of this space. This fundamental group is naturally mapped to the groups Gnk and Γnk. This is the step towards “complexification” of the Gnk and Γnk approach first developed in [V. O. Manturov, D. A. Fedoseev, S. Kim and I. M. Nikonov, On groups Gnk

We give a brief survey of virtual knot invariants derived from chord parity or chord index. These invariants have grown into an area in its own right due to rapid developing in the last decade. Several similar invariants of flat virtual knots and free knots are also discussed.

In this paper, we study representations of Gn3-like groups. The group Gn3 itself appeared in works of the third named author on non-Reidemeister knot (and braid) theory. This group is closely related to dynamical systems of points and their invariants. Representations of Gn3-like groups are useful both for the study of those groups themselves, and constructing invariants of knots and braids based on

In this paper, we give the users an introduction about the framework of persistent homology methods for hypergraphs. We list the steps for standard computations of the persistent homology and discuss about the algorithms. We also give some potential mathematical tools for efficient computing.

We show that the disk complex of a closed, connected surface of genus $g>1$, properly embedded in the 3-sphere is homotopy equivalent to a wedge of $(2g-2)$-dimensional spheres. This implies that genus $g>1$ Heegaard surfaces of the 3-sphere are topologically minimal with index $2g-1$.

We consider embeddings of 3-manifolds $M$ in $S^4$ such that the two complementary regions $X$ and $Y$ each have nilpotent fundamental group. If $\beta=\beta_1(M)$ is odd then these groups are abelian and $\beta\leq3$. In general, $\pi_1(X)$ and $\pi_1(Y)$ have 3-generator presentations, and $\beta\leq6$. We determine all such nilpotent groups which are torsion-free and have Hirsch length $\leq5$.

The Gukov-Manolescu series, denoted by $F_K$, is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color $R$-matrix to study $F_K$ for some simple links. Specifically, we give a definition of $F_K$ for positive braid knots, and compute $F_K$ for various knots and links. As a corollary, we present a class

It is known that the unknotting number $u(L)$ of a link $L$ is less than or equal to half the crossing number $c(L)$ of $L$. We show that there are a planar graph $G$ and its spatial embedding $f$ such that the unknotting number $u(f)$ of $f$ is greater than half the crossing number $c(f)$ of $f$. We study relations between unknotting number and crossing number of spatial embedding of a planar graph

For a Heegaard splitting of a 3-manifold, Casson–Gordon’s rectangle condition, simply rectangle condition, is a condition on its Heegaard diagram that guarantees the strong irreducibility of the splitting; it requires nine types of rectangles for every combination of two pairs of pants from opposite sides. The rectangle condition is also applied to bridge decompositions of knots. We give examples of

Square mosaic knots have many applications in algebra, such as modeling quantum states. In this paper, we extend mosaic knot theory to a theory of hexagonal mosaic links, which are links embedded in a plane tiling of regular hexagons. We investigate hexagonal mosaic links created from particular patches of hextiles with a high number of crossings, which we describe as saturated diagrams. Considering

Mosaic diagrams for knots were first introduced in 2008 by Lomanoco and Kauffman for the purpose of building a quantum knot system. Since then, many others have explored the structure of these knot mosaic diagrams, as they are interesting objects of study in their own right. Knot mosaics have been generalized by Garduno to virtual knots, by including an additional tile type to represent virtual crossings

We present four models for a random graph and show that, in each case, the probability that a graph is intrinsically knotted goes to one as the number of vertices increases. We also argue that, for $k \geq 18$, most graphs of order $k$ are intrinsically knotted and, for $k \geq 2n+9$, most of order $k$ are not $n$-apex. We observe that $p(n) = 1/n$ is the threshold for intrinsic knotting and linking

We construct an infinite family of links which are both almost alternating and quasi-alternating from a given either almost alternating diagram representing a quasi-alternating link, or connected and reduced alternating tangle diagram. To do that we use what we call a dealternator extension which consists in replacing the dealternator by a rational tangle extending it. We note that all not alternating

This is a preface to the special issue on Proceedings of the Sixth Russian-Chinese Conference on Knot Theory and Related Topics which was held in Novosibirsk in June 2019.

The knots $8_1$, $8_2$, $8_3$, $8_5$, $8_6$, $8_7$, $8_8$, $8_{10}$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{15}$, $9_7$, $9_{16}$, $9_{20}$, $9_{26}$, $9_{28}$, $9_{32}$, and $9_{33}$ all have superbridge index equal to 4. This follows from new upper bounds on superbridge index not coming from the stick number and increases the number of knots from the Rolfsen table for which superbridge index