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Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas has been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to present the seven invariants of the degree three (Proposition 4). We further give 2 3 Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak–Viro Gauss diagram formula [M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not.1994 (1994) 445–453] is not a long virtual knot invariant; however, it is included in the list of 2 3 formulas. It has been unknown whether this formula would be available by arrow diagram calculus automatically. In consequence, as it relates to the conjecture of Goussarov-Polyak-Viro [Finite-type invariants of classical and virtual knots, Topology39 (2000) 1045–1068, Conjecture 3.C], for all the degree three finite type long virtual knot invariants, each Gauss diagram formula is represented as those of Vassiliev invariants of classical knots (Theorem 1). [ABSTRACT FROM AUTHOR]

In this paper, we construct a kind of new braided monoidal category over two Hom-Hopf algerbas (H , α) and (B , β) and associate it with two nonlinear equations. We first introduce the notion of an (H , B) -Hom-Long dimodule and show that the Hom-Long dimodule category H B L is an autonomous category. Second, we prove that the category H B L is a braided monoidal category if (H , α) is quasitriangular and (B , β) is coquasitriangular and get a solution of the quantum Yang-Baxter equation. Also, we show that the category H B L can be viewed as a subcategory of the Hom-Yetter-Drinfeld category H ⊗ B H ⊗ B H Y D . Finally, we obtain a solution of the Hom-Long equation from the Hom-Long dimodules. [ABSTRACT FROM AUTHOR]

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 group of a link, Math. Z.200 (1989) 187–208], we prove that classical knots with special rank 2 are 2 -bridge knots. Therefore, a modified version of the so called Cappell and Shaneson conjecture could be considered. [ABSTRACT FROM AUTHOR]

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 extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as c. As an application, we give new examples of quasi-alternating knots of 13 and 14 crossings. Moreover, we prove that the Jones polynomial of a quasi-alternating link that is obtained in this way has no gap if the original link has no gap in its Jones polynomial. This supports a conjecture introduced in [N. Chbili and K. Qazaqzeh, On the Jones polynomial of quasi-alternating links, Topology Appl.264 (2019) 1–11], which states that the Jones polynomial of any prime quasi-alternating link except (2 , p) -torus links has no gap. [ABSTRACT FROM AUTHOR]

A surface-knot is a closed oriented surface smoothly embedded in 4-space and a surface-knot diagram is a projected image of a surface-knot under the orthogonal projection in 3-space with crossing information. Every surface-knot diagram induces a rectangular-cell complex. In this paper, we introduce a covering diagram over a surface-knot diagram. the covering map induces a covering of the rectangular-cell complexes. As an application, a lower bound of triple point numbers for a family of surface-knots is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of relation between knots called the descendant relation. One knot is a descendant of another knot if can be obtained from a minimal crossing diagram of by some number of crossing changes. We explore properties of the descendant relation and study how certain knots are related, paying particular attention to those knots, called fertile knots, that have a large number of descendants. Furthermore, we provide computational data related to various notions of knot fertility and propose several open questions for future exploration. [ABSTRACT FROM AUTHOR]

Ichihara and Matsudo introduced the notions of -colorable links and the minimal coloring number for -colorable links, which is one of invariants for links. They proved that the lower bound of minimal coloring number of a non-splittable -colorable link is 4. In this paper, we show the minimal coloring number of any non-splittable -colorable link is exactly 4. [ABSTRACT FROM AUTHOR]

This paper defines versions of the Jones polynomial and Khovanov homology by using several maps from the set of Gauss diagrams to its variant. Through calculation of some examples, this paper also shows that these versions behave differently from the original ones. [ABSTRACT FROM AUTHOR]

Two welded (respectively virtual) link diagrams are homotopic if one may be transformed into the other by a sequence of extended Reidemeister moves, classical Reidemeister moves, and self crossing changes. In this paper, we extend Milnor's μ and $\bar{\mu}$ invariants to welded and virtual links. We conclude this paper with several examples, and compute the μ invariants using the Magnus expansion and Polyak's skein relation for the μ invariants. [ABSTRACT FROM AUTHOR]

In this paper, we show how generalized quaternions including some 2 × 2 matrices, can be used to find solutions of the equation \[ [B,(A - 1)(A,B)] = 0. \] These solutions can then be used to find polynomial invariants of virtual knots and links. The remaining 2 × 2 matrices will be considered in a later paper. [ABSTRACT FROM AUTHOR]

Let A, B be invertible, non-commuting elements of a ring R. Suppose that A - 1 is also invertible and that the equation \[ [B,(A - 1)(A,B)] = 0 \] called the fundamental equation is satisfied. Then an invariant R-module is defined for any diagram of a (virtual) knot or link. Solutions in the classic quaternion case have been found by Bartholomew, Budden and Fenn. Solutions in the generalized quaternion case have been found by Fenn in an earlier paper. These latter solutions are only partial in the case of 2 × 2 matrices and the aim of this paper is to provide solutions to the missing cases. [ABSTRACT FROM AUTHOR]

In this paper we prove a Markov theorem for virtual braids and for analogs of this structure including flat virtual braids and welded braids. The virtual braid group is the natural companion to the category of virtual knots, just as the Artin braid group is the natural companion to classical knots and links. In this paper we follow L-move methods to prove the Virtual Markov theorems. One benefit of this approach is a fully local algebraic formulation of the theorems in each category. [ABSTRACT FROM AUTHOR]

Let m,n ∈ ℕ. We define $B_{n,m}(\mathbb{S}^{2})$ to be the set of (n+m)-braids of the sphere whose associated permutation lies in the subgroup Sn × Sm of the symmetric group Sn+m on n+m letters. In a previous paper [13], we showed that if n ≥ 3, then there exists the following generalisation of the Fadell–Neuwirth short exact sequence: \[ 1\to B_m(\mathbb{S}^{2}\setminus\{x_1,\ldots,x_n\})\to B_{n,m}(\mathbb{S}^{2})\stackrel{p_{\ast}}{\longrightarrow} B_n(\mathbb{S}^{2})\to 1, \] where ${p_{\ast}}{:}\, B_{n,m}(\mathbb{S}^{2}) \to B_n(\mathbb{S}^{2})$ is the group homomorphism (defined for all n ∈ ℕ) given geometrically by forgetting the last m strings. In this paper we study the splitting of this short exact sequence, as well as the existence of a cross-section for the fibration $p{:}\, D_{n,m}(\mathbb{S}^{2}) \to D_n(\mathbb{S}^{2})$ of the quotients of the corresponding configuration spaces. Our main results are as follows: if n = 1 (respectively, n = 2) then the homomorphism p* and the fibration p admit (respectively, do not admit) a section. If n = 3, then p* and p admit a section if and only if m ≡ 0,2 (mod 3). If n ≥ 4, we show that if p* and p admit a section then m ≡ ε1(n - 1)(n - 2) - ε2n(n - 2) (mod n(n - 1)(n - 2)), where ε1,ε2 ∈ {0,1}. Finally, we show that $B_n(\mathbb{S}^{2})$ is generated by two of its torsion elements. [ABSTRACT FROM AUTHOR]

In the present paper, we show that the Morimoto Conjecture on the super additivity of the tunnel numbers of knots in S3 is true for knots K1,K2 in S3 in which each Ki is either a tunnel number one or m-small, i = 1, 2. This extends two known results by Morimoto. [ABSTRACT FROM AUTHOR]

In a recent work "Arc-presentation of links: Monotonic simplification", Dynnikov shows that each rectangular diagram of the unknot, composite link, or split link can be monotonically simplified into a trivial, composite, or split diagram, respectively. The following natural question arises: Is it always possible to simplify monotonically a rectangular diagram of a satellite knot or link into one where the satellite structure is seen? Here we give a negative answer to that question both for knot and link cases. An example of a torus embedding that cannot be obtained from ordinary "thin" torus by methods of paper [2] is also constructed. [ABSTRACT FROM AUTHOR]

In this paper, we prove that an adequate virtual link diagram of an adequate virtual link has minimal real crossing number. [ABSTRACT FROM AUTHOR]

It is well known that any virtual link is described as the closure of a virtual braid. Therefore, we can define the virtual braid index. Ohyama proved an inequality for the crossing number and the braid index of a classical link. In this paper, we prove an analogous inequality for the (total) crossing number and the braid index of a virtual link. [ABSTRACT FROM AUTHOR]

A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper, we show that if a generic immersion of a planar graph is knotted then the number of double points of the immersion is more than or equal to three. To prove this, we also show that an embedding of a graph obtained from a generic immersion of the graph (does not need to be planar) with at most three double points is totally free if it contains neither a Hopf link nor a trefoil knot. [ABSTRACT FROM AUTHOR]

Abstract: The Turaev genus of a knot is an obstruction to the knot being alternating. An adequate knot is a generalization of an alternating knot. A natural problem is a characterization of the Turaev genus of an adequate knot. In this paper, we show that the Turaev genus of an adequate knot is realized by the genus of the Turaev surface associated to an adequate diagram of the knot using the Khovanov homology. As a result, we obtain the additivity of the Turaev genus of adequate knots, and show that the Turaev genus of an adequate knot is “often” preserved under mutation. We also show that an n-semi-alternating knot is of Turaev genus n. This is the first examples of adequate knots of Turaev genus two or more. [Copyright &y& Elsevier]

We completely determine which Dehn surgeries on 2-bridge links yield reducible 3-manifolds. Further, we consider which surgery on one component of a 2-bridge link yields a torus knot, a cable knot and a satellite knot in this paper. [ABSTRACT FROM AUTHOR]

The stick index of a knot K is defined to be the least number of line segments needed to construct a polygonal embedding of K. We define the projection stick index of K to be the least number of line segments in any projection of a polygonal embedding of K. In this paper, we establish bounds on the projection stick index for various torus knots. We then show that the stick index of a (p, 2p + 1)-torus knot is 4p, and the projection stick index is 2p + 1. This provides examples of knots such that the projection stick index is one greater than half the stick index. We show that for all other torus knots for which the stick index is known, the projection stick index is larger than this. We conjecture that a projection stick index of half the stick index is unattainable for any knot. [ABSTRACT FROM AUTHOR]

In this paper, based on a talk given at the Oberwolfach research centre in May 2008 I will describe how biquandles and their big brother, biracks, can be used to differentiate isotopy classes of virtual (and welded) knots and links. [ABSTRACT FROM AUTHOR]

The present paper is an introduction to a combinatorial theory arising as a natural generalization of classical and virtual knot theory. There is a way to encode links by a class of "realizable" graphs. When passing to generic graphs with the same equivalence relations we get "graph-links". On one hand graph-links generalize the notion of virtual link, on the other hand they do not detect link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalization of the Kauffman–Murasugi–Thistlethwaite theorem on "minimal diagrams" for graph-links. [ABSTRACT FROM AUTHOR]

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Yd,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SBn into the algebra Yd,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Yd,n(u). [ABSTRACT FROM AUTHOR]

Our longterm plan is to classify knot modules and pairings by utilizing the power of computational number theory. The first step in this is to define invariants for which any given value arises from only finitely many modules: this is the purpose of the present paper. [ABSTRACT FROM AUTHOR]

Isomorphism classes of Alexander quandles of order 16 are determined, and classes of connected quandles are identified. This paper extends the list of distinct connected finite Alexander quandles. [ABSTRACT FROM AUTHOR]

Motivated by Massey–Milnor linking and Chern–Simon–Witten perturbative quantum field theory, we developed some combinatorial and diagrammatic study in this paper, aiming at knot theory in the combinatoric aspect. [ABSTRACT FROM AUTHOR]

In his initial paper on braids, E. Artin gave a presentation with two generators for an arbitrary braid group. We give analogs of Artin’s presentation for various generalizations of braids. [ABSTRACT FROM AUTHOR]

In this paper, we explore the issue of minimizing the degree sequence for torus knots. We find the minimal degree sequence for torus knots of type (p, 2p-1) for any integer p ≥2. We use some results from algebraic geometry to prove our main result. [ABSTRACT FROM AUTHOR]

In this paper, we will explore the properties of knots that lie in a surface cross an interval. We show that any alternating knot nontrivially embedded in S × I is chiral. This demonstrates that knots in S × I behave differently from knots in the three sphere, where alternating knots (like the figure eight) can be amphichiral. We use a generalized Kauffman bracket polynomial, as well as geometric and covering space techniques. [ABSTRACT FROM AUTHOR]

In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander polynomial and a polynomial associated to a cyclic presentation of the fundamental group of an n-fold strongly-cyclic covering branched over the knot K, which we call the n-cyclic polynomial of K. In this way, we generalize to all (1,1)-knots, with the only exception of those lying in S2×S1, a result obtained by Minkus for 2-bridge knots and extended by the author and M. Mulazzani to the case of (1,1)-knots in S3. As corollaries some properties of the Alexander polynomial of knots in S3 are extended to the case of (1,1)-knots in lens spaces. [ABSTRACT FROM AUTHOR]

Habiro found in his thesis a topological interpretation of finite type invariants of knots in terms of local moves called Habiro's Ck-moves. Ck-moves are defined by using his claspers. In this paper we define "oriented" claspers and RCk-moves among ribbon 2-knots as modifications of Habiro's notions to give a similar interpretation of Habiro–Kanenobu–Shima's finite type invariants of ribbon 2-knots. It works also for ribbon 1-knots. Furthermore, by using oriented claspers for ribbon 1-knots, we can prove Habiro–Shima's conjecture in the case of ℚ-valued invariants, saying that ℚ-valued Habiro–Kanenobu–Shima finite type invariant and ℚ-valued Vassiliev–Goussarov finite type invariant are the same thing. [ABSTRACT FROM AUTHOR]

In this paper, we list in explicit form the factoring relations of the Kauffman bracket skein module (KBSM for short) of a twist knot exterior. This is done using curves decorated by characters of irreducible SL(2, ℂ)-representations. In the process, we exhibit a relation which holds in the KBSM of the knot exterior, called the minimal relation. In the final section we prove that, when specializing the variable of the Kauffman bracket at t = -1, the minimal relation becomes the defining polynomial of the SL(2, ℂ)-character variety of the twist knot. [ABSTRACT FROM AUTHOR]

In this paper the authors use a modified Wirtinger presentation to give a lower bound on the unknotting number of a knot in S 3. [ABSTRACT FROM AUTHOR]

For a hyperbolic knot in the 3-sphere, the distance between toroidal surgeries is at most 5, except the figure eight knot. In this paper, we determine all hyperbolic knots that admit two toroidal surgeries with distance 5. [ABSTRACT FROM AUTHOR]

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S3 by using "a crossing change". In this paper, we define the Ck-Gordian complex of knots which is an extension of the Gordian complex of knots. Let k be a natural number more than 2 and we show that for any knot K0 and any given natural number n, there exists a family of knots {K0, K1,..., Kn} such that for any pair (Ki, Kj) of distinct elements of the family, the Ck-distance dCk(Ki, Kj) = 1. [ABSTRACT FROM AUTHOR]

For a knot or link K, let L(K) denote the rope length of K and let Cr(K) denote the crossing number of K. An important problem in geometric knot theory concerns the bound on L(K) in terms of Cr(K). It is well-known that there exist positive constants c1, c2 such that for any knot or link K, c1 · (Cr(K))3/4 ≤ L(K) ≤ c2 · (Cr(K))3/2. It is also known that for any real number p such that 3/4 ≤ p ≤ 1, there exists a family of knots {Kn} with the property that Cr(Kn) → ∞ (as n → ∞) such that L(Kn) = O(Cr(Kn)p). However, it is still an open question whether there exists a family of knots {Kn} with the property that Cr(Kn) → ∞ (as n → ∞) such that L(Kn) = O(Cr(Kn)p) for some 1 < p ≤ 3/2. In this paper, we show that there are many families of prime alternating Conway algebraic knots {Kn} with the property that Cr(Kn) → ∞ (as n → ∞) such that L(Kn) can grow no faster than linearly with respect to Cr(Kn). [ABSTRACT FROM AUTHOR]

In this paper, we extend a Howards–Luecke's theorem on strongly n-trivial knots to link case. We show that a link is strongly n-trivial for all n if and only if it is a trivial link. [ABSTRACT FROM AUTHOR]

Akbulut generalized Fintushel–Stern knot surgery in a certain way. He showed that an operation C(K#(-K)) → {diffeo type of X} is not injective, where C(K#(-K)) is self-concordance set of K#(-K) and K is trefoil knot. In the present paper we prove that the same phenomenon occurs for any knot K and for some other knots L instead of K#(-K). [ABSTRACT FROM AUTHOR]

We resume work on telling embeddings of codimension two apart by counting colorings of the corresponding diagrams by given quandles. Previously, we illustrated the efficiency of this approach on classical knots. In the present paper we apply it to knotted surfaces. We recover work of Kamada in telling ribbon knots apart and we distinguish all elements of a class of twist-spun torus knots. [ABSTRACT FROM AUTHOR]

Carter, Jelsovsky, Kamada, Langford and Saito introduced the quandle cocycle invariants of 2-knots, and calculated the cocycle invariant of a 2-twist-spun trefoil knot associated with a 3-cocycle of the dihedral quandle of order 3. Asami and Satoh calculated the cocycle invariants of twist-spun torus knots τrT(m,n) associated with 3-cocycles of some dihedral quandles. They used tangle diagrams of the torus knots. In this paper, we calculate the cocycle invariants of twist-spun 2-bridge knots τrS(α,β) by a similar method. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce three geometric invariants of knots K: t(K), g1(K), h(K), and study the relationship among these invariants, connected sum and tangle decompositions. [Copyright &y& Elsevier]

In this paper, we study lens knots and periodic knots by using integral Vassiliev invariants. Knot polynomials such as the Jones, HOMFLY, Kauffman polynomials give infinitely many integral Vassiliev invariants and we get some necessary conditions for a link to be a lens knot or a periodic link by using these polynomial invariants. [ABSTRACT FROM AUTHOR]

It has long been conjectured that the crossing numbers of links are additive under the connected sum of links. This is a difficult problem in knot theory and has been open for more than 100 years. In fact, many questions of much weaker conditions are still open. For instance, it is not known whether Cr(K1#K2)≥Cr(K1) or Cr(K1#K2)≥Cr(K2) holds in general, here K1#K2 is the connected sum of K1 and K2 and Cr(K) stands for the crossing number of the link K. However, for alternating links K1 and K2, Cr(K1#K2)=Cr(K1)+Cr(K2) does hold. On the other hand, if K1 is an alternating link and K2 is any link, then we have Cr(K1#K2)≥Cr(K1). In this paper, we show that there exists a wide class of links over which the crossing number is additive under the connected sum operation. This class is different from the class of all alternating links. It includes all torus knots and many alternating links. Furthermore, if K1 is a connected sum of any given number of links from this class and K2 is a non-trivial knot, we prove that Cr(K1#K2)≥Cr(K1)+3. [ABSTRACT FROM AUTHOR]

The first aim of this paper is to give an example of a virtual knot which vanishes all the known invariants and to show that it is a non-classical (in particular, non-trivial) knot. The second is to study some properties of the Z-polynomial of a virtual knot with virtual crossing number one and to show that there are infinitely many virtual knots with virtual crossing number two. [ABSTRACT FROM AUTHOR]

The transforms of two oriented parallel strands to a k-half twist of two strands are called tk-move and &tmacr;k-move respectively depending on the orientations of the two strands. In this paper we give criterions to detect whether a knot K can be transformed to a knot K' by t2k-moves and t2k-moves respectively and if so, we give some results on how many moves are needed in these transformations respectively, by using some Vassiliev invariants. Moreover we give a relation between the Δ-move and the t2k-move by considering the coefficient of z2 in the Conway polynomial of a knot, which is a Vassiliev invariant of degree 2. [ABSTRACT FROM AUTHOR]

The paper computes the noncommutative A-ideal of the figure-eight knot, a noncommutative generalization of the A-polynomial. It is shown that if a knot has the same A-ideal as the figure-eight knot, then all colored Kauffman brackets are the same as those of the figure eight knot. [ABSTRACT FROM AUTHOR]

A polygonal chain is the union of a finite number of straight line segments in ℝ3 that are connected end-to-end. Two chains are considered to be equivalent if there is an isotopy of ℝ3 that moves one chain to the other while keeping the segments rigid. Each segment must remain straight during the isotopy and the lengths of the segments may not change, but bending and twisting are allowed at the joints between the segments. Chains may be knotted and stuck in this category even though all chains are topologically trivial. Cantarella and Johnston have classified polygonal chains with five or fewer segments. In this paper we classify polygonal chains of six segments. [ABSTRACT FROM AUTHOR]

In this paper we show that a system containing any knot or link can be directly constructed in a simple way. The system is not chaotic and can even contain wild knots. [ABSTRACT FROM AUTHOR]