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

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

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.

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

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.

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.

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

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

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

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

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.

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 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.

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

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

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

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

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

We introduce labeled singular knots and equivalently labeled 4-valent rigid vertex spatial graphs. Labeled singular knots are singular knots with labeled singularities. These knots are considered subject to isotopies preserving the labelings. We provide a topological invariant schema similar to that of Henrich and Kauffman in [A. Henrich and L. H. Kauffman, Tangle insertion invariants for pseudoknots

A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure, we consider a knot diagram on the xy-plane in xyz-space missing some straight lines parallel to the z-axis. These straight lines correspond to fingers. We study minimal number of crossings

We show that Dehn filling on the manifold v2503 results in a non-orderable space for all rational slopes in the interval (−∞,−1). This is consistent with the L-space conjecture, which predicts that all fillings will result in a non-orderable space for this manifold.

It is known that there exists a surjective map from the set of weak (1, 3) homotopy classes of knot projections to the set of positive knots [N. Ito and Y. Takimura, (1, 2) and weak (1, 3) homotopies on knot projections, J. Knot Theory Ramifications22 (2013) 1350085]. An interesting question whether this map is also injective, which question was formulated independently by S. Kamada and Y. Nakanishi

A finitely generated group G acting on a tree with infinite cyclic edge and vertex stabilizers is called a generalized Baumslag–Solitar group (GBS group). We prove that a one-knot group G is a GBS group if and only if G is a torus knot group, and describe all n-knot GBS groups for n≥3.

In [V. O. Manturov, An almost classification of free knots, Dokl. Math.88(2) (2013) 556–558.] the second author constructed an invariant which in some sense generalizes the quantum sl(3) link invariant of Kuperberg to the case of free links. In this paper, we generalize this construction to free graph-links. As a result, we obtain an invariant of free graph-links with values in linear combinations

In the 1950s Milnor defined a family of higher-order invariants generalizing the linking number. Even the first of these new invariants, the triple linking number, has received fruitful study since its inception. In the case that a link L has vanishing pairwise linking numbers, this triple linking number gives an integer-valued invariant. When the linking numbers fail to vanish, the triple linking

We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by terms of the lower central series of knot groups. Also, we study decomposition of virtual

In 2003, Ozsváth and Szabó defined the concordance invariant τ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of τ for knots in S3 and a combinatorial proof that τ gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory

Ng constructed an invariant of knots in ℝ3, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ℝ4 using marked graph diagrams.

A twisted torus knot is a torus knot with some consecutive strands twisted. More precisely, a twisted torus knot T(p,q,r,s) is a torus knot T(p,q) with r consecutive strands s times fully twisted. We determine which twisted torus knots T(p,q,p−kq,−1) are a torus knot.

For any classical knot k1, we can construct a ribbon 2-knot spun(k1) by spinning an arc removed a small segment from k1 about R2 in R4. A ribbon 2-knot is an embedded 2-sphere in R4. If k1 has an n-crossing presentation, by spinning this, we can naturally construct a ribbon presentation with n ribbon crossings for spun(k1). Thus, we can define naturally a notion on ribbon 2-knots corresponding to the

We extend results of Positive Twist Knots and Thickenings [W. George, Positive twist knots and thickenings, J. Knot Theory Ramifications22 (2013) 1350046] to show that positive twist knots 𝒦m, for m≥3, satisfy the Uniform Thickness Property.

We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics.

We obtain information on torsion in Khovanov cohomology by performing calculations directly over ℤ/pkℤ for p prime and k≥2. In particular, we get that the torus knots T(9,10) and T(9,11) contain torsion of order 9 and 27 in their Khovanov cohomology.

A sequence of F-polynomials {FKn(t,ℓ)}n=1∞ of virtual knots K was defined by Kaur et al. in 2018. These polynomials have been expressed in terms of index value of crossing and n-writhe of K. By the construction, F-polynomials are generalizations of Kauffman’s Affine Index Polynomial, and are invariants of virtual knot K. We present values of F-polynomials of oriented virtual knots having at most four

We prove that a version of the Thurston–Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere (M,ξ), whenever ξ is tight. More specifically, we show that the self-linking number of a transverse link T in (M,ξ), such that the boundary of its tubular neighborhood consists of incompressible tori, is bounded by the Thurston norm ∥T∥T of T. A similar inequality

We show an infinite family of satellite knots that can be unknotted by a single band move, but such that there is no band unknotting the knots which is disjoint from the satellite torus.

Laplacian matrices of signed graphs in surfaces S are used to define module and polynomial invariants of ℤ/2-homologically trivial links in S×[0,1]. Information about virtual genus is obtained.

We determine the minimal number of colors for nontrivial ℤ-colorings on the standard minimal diagrams of ℤ-colorable torus links. Also included is a complete classification of such ℤ-colorings, which are shown by using rack colorings on link diagrams.

The warping degree of an oriented knot diagram is the minimal number of crossing changes which are required to obtain a monotone diagram from the diagram. The minimal warping degree of a knot is the minimal value of the warping degree for all oriented minimal diagrams of the knot. In this paper, all prime alternating knots with minimal warping degree two are determined.

Motivated by problem of determining the unknotted routes for the scaffolding strand in DNA origami self-assembly, we examine the existence and knottedness of A-trails in graphs embedded on the torus. We show that any A-trail in a checkerboard-colorable torus graph is unknotted and characterizes the existence of A-trails in checkerboard-colorable torus graphs in terms of pairs of quasitrees in associated

Reshetikhin–Turaev (a.k.a. Chern–Simons) TQFT is a functor that associates vector spaces to two-dimensional genus g surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald q,t-deformation — refinement — of these operators that preserves the defining relations of the mapping class groups beyond genus 1. For this, we explicitly

A spatial surface is a compact surface embedded in the 3-sphere. In this paper, we provide several typical examples of spatial surfaces and construct a coloring invariant to distinguish them. The coloring is defined by using a multiple group rack, which is a rack version of a multiple conjugation quandle.