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.

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

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

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

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

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

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

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

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

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

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.

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 n≥18, most graphs of order n are intrinsically knotted and, for n≥2m+9, most of order n are not m-apex. We observe that p(n)=1/n is the threshold for intrinsic knotting and linking in Gilbert’s model.

We compute the factorization homology of the four-punctured sphere and punctured torus over the quantum group 𝒰q(𝔰𝔩2) explicitly as categories of equivariant modules using the framework developed by Ben-Zvi et al. We identify the algebra of 𝒰q(𝔰𝔩2)-invariants (quantum global sections) with the spherical double affine Hecke algebra of type (C1∨,C1), in the four-punctured sphere case, and with

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 non-alternating

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 Garduño to virtual knots, by including an additional tile type to represent virtual crossings

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

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 consider embeddings of 3-manifolds M in S4 such that the two complementary regions X and Y each have nilpotent fundamental group. If β=β1(M) is odd then these groups are abelian and β≤3. In general π1(X) and π1(Y) have 3-generator presentations, and β≤6. We give two examples illustrating our results.

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 in general.

The knots 81, 82, 83, 85, 86, 87, 88, 810, 811, 812, 813, 814, 815, 97, 916, 920, 926, 928, 932, and 933 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 is known from 29 to 49. Appendix A gives the current state of knowledge of superbridge

The Gukov–Manolescu series, denoted by FK, 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 FK for some simple links. Specifically, we give a definition of FK for positive braid knots, and compute FK for various knots and links. As a corollary, we present a class of “strange

In this paper the simplest n-correct sets in the plane — GCn sets are studied. An n-correct node set 𝒳 is called GCn set if the fundamental polynomial of each node is a product of n linear factors. We say that a node uses a line if the line is a factor of the fundamental polynomial of this node. A line is called k-node line if it passes through exactly k nodes of 𝒳. At most n+1 nodes can be collinear

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

Let M be a subset of vector space or projective space. The authors define generalized configuration space of M which is formed by n-tuples of elements of M, where any k elements of each n-tuple are linearly independent. The generalized configuration space gives a generalization of Fadell’s classical configuration space, and Stiefel manifold. Denote generalized configuration space of M by Wk,n(M). For

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

Suppose Γ is a discrete group, and α∈Z3(BΓ;A), with A an abelian group. Given a representation ρ:π1(M)→Γ, with M a closed 3-manifold, put F(M,ρ)=〈(Bρ)∗[α],[M]〉, where Bρ:M→BΓ is a continuous map inducing ρ which is unique up to homotopy, and 〈−,−〉:H3(M;A)×H3(M;ℤ)→A is the pairing. We extend the definition of F(M,ρ) to manifolds with corners, and establish a gluing law. Based on these, we present a

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 2008, Bokut obtained a Gröbner–Shirshov basis S of the braid group Bn+1 in the Artin–Garside generators and showed that S-irreducible words of the Bn+1 coincided with the Garside normal forms of words. Using this basis S, we obtain the concrete expression of the S-irreducible words, i.e. normal forms, of the Bn+1, and hence give a new understanding of the word problem and Garside normal forms of

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 present two practical and widely applicable methods, including some criteria and a general procedure, for detecting Brunnian property of a link, if each component is known to be unknot. The methods are based on observation and handwork. They are used successfully for all Brunnian links known so far. Typical examples and extensive experiments illustrate their efficiency. As an application, infinite

In this study, we describe a method of making an invariant of virtual knots defined in terms of an integer labeling of the flat virtual knot diagram. We give an invariant of flat virtual knots and virtual doodles modifying the previous invariant.

The interior polynomial is a Tutte-type invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to the maximal z-degree part of the HOMFLY polynomial of a naturally associated

Our goal is to systematically compute the ℂP2-genus of as many prime knots up to 8-crossings as possible. We obtain upper bounds on the ℂP2-genus via coherent band surgery. We obtain lower bounds by obstructing homological degrees of potential slice discs. The obstructions are pulled from a variety of sources in low-dimensional topology and adapted to ℂP2. There are 27 prime knots and distinct mirrors

In this paper, we adapt the procedure of the Long-Moody procedure to construct linear representations of welded braid groups. We exhibit the natural setting in this context and compute the first examples of representations we obtain thanks to this method. We take this way also the opportunity to review the few known linear representations of welded braid groups.

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

We enhance the quandle coloring quiver invariant of oriented knots and links with quandle modules. This results in a two-variable polynomial invariant which specializes to the previous quandle module polynomial invariant as well as to the quandle counting invariant. We provide example computations to show that the enhancement is proper in the sense that it distinguishes knots and links with the same

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

Boyer, Gordon and Watson have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since Dehn surgeries on knots in S3 can produce large families of L-spaces, it is natural to examine the conjecture on these 3-manifolds. Greene, Lewallen and Vafaee have proved that all 1-bridge braids are L-space knots. In this paper, we

We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second homology of the 4-manifold. This naturally extends r-shake slice, a generalization of slice that has previously only been studied for knots, to links

In this paper we introduce the special rank for virtual knots and some properties of this number are studied. Although we do not know if it can be considered as a nontrivial extension of the meridional rank given by [H. U. Boden and A. I. Gaudreau, Bridge number for virtual and welded knots, J. Knot Theory Ramifications24 (2015), Article ID: 1550008] and by [M. Boileau and B. Zimmermann, The π-orbifold

In this paper, we define an explicit basis for the 𝔤𝔩n-web algebra Hn(k) (the 𝔤𝔩n generalization of Khovanov’s arc algebra) using categorified q-skew Howe duality. Our construction is a 𝔤𝔩n-web version of Hu–Mathas’ graded cellular basis and has two major applications: it gives rise to an explicit isomorphism between a certain idempotent truncation of a thick calculus cyclotomic KLR algebra and

Let R be a compact, connected, orientable surface of genus g with n boundary components with g≥2, n≥0. Let 𝒩(R) be the nonseparating curve graph, 𝒞(R) be the curve graph and ℋ𝒯(R) be the Hatcher–Thurston graph of R. We prove that if λ:𝒩(R)→𝒩(R) is an edge-preserving map, then λ is induced by a homeomorphism of R. We prove that if 𝜃:𝒞(R)→𝒞(R) is an edge-preserving map, then 𝜃 is induced by

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.

Suppose α and R are disjoint simple closed curves in the boundary of a genus two handlebody H such that H[R] (i.e. a 2-handle addition along R) embeds in S3 as the exterior of a hyperbolic knot k (thus, k is a tunnel-number-one knot), and α is Seifert in H (i.e. a 2-handle addition H[α] is a Seifert-fibered space) and not the meridian of H[R]. Then for a slope γ of k represented by α, γ-Dehn surgery

A knot K is a parent of a knot H if there exists a minimal crossing diagram D of K such that a subset of the crossings of D can be changed to produce a diagram of H. A knot K with crossing number n is fertile if for any prime knot H with crossing number less than n, K is a parent of H. It is known that only 01,31,41,52,62,63,76 are fertile for knots up to 10 crossings. However it is unknown whether

We introduce a graphical calculus for computing morphism spaces between the categorified spin networks of Cooper and Krushkal. The calculus, phrased in terms of planar compositions of categorified Jones–Wenzl projectors and their duals, is then used to study the module structure of spin networks over the colored unknots.

In a group, a generalized torsion element is a non-identity element whose some non-empty finite product of its conjugates yields the identity. Such an element is an obstruction for a group to be bi-orderable. We show that the Weeks manifold, the figure-eight sister manifold, and the complement of Whitehead sister link admit generalized torsion elements in their fundamental groups. In particular, the

The arc index of a knot is the minimal number of arcs in all arc presentations of the knot. An arc presentation of a knot can be shown in the form of a grid diagram which is a closed plane curve consisting of finitely many horizontal line segments and the same number of vertical line segments. The arc index of an alternating knot is its minimal crossing number plus two. In this paper, we give a list

Champanerkar and Kofman [Twisting quasi-alternating links, Proc. Amer. Math. Soc.137(7) (2009) 2451–2458] introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been