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This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. Information on determinations of the four-ball genus of some virtual knots is obtained by via the affine index polynomial in conjunction with results on the genus of positive virtual knots using joint work with Dye and Kaestner. [ABSTRACT FROM AUTHOR]

In our previous paper, we studied the braid index of the Kanenobu knot k(n) for n ≥ 0. In this paper, we study the braid index of the Kanenobu knot K(a, b) for a, b ∈ ℤ. In particular, k(n) is K(2n, -2n). The MFW inequality is known for giving a lower bound of the braid index of an oriented link by applying the HOMFLYPT polynomial. The HOMFLYPT polynomial of K(a, b) is given by Professor Taizo Kanenobu. Therefore, we have a lower bound of the braid index of K(a, b). The purpose of this paper is to give an upper bound of the braid index of K(a, b). As a result, we determine the braid indices of infinitely many Kanenobu knots. [ABSTRACT FROM AUTHOR]

In the previous paper, the representativity of a non-trivial knot K was defined as \[ r(K)=\max_{F\in\mathcal{F}} \min_{D\in\mathcal{D}_F} |\partial D\cap K|, \] where $\mathcal{F}$ is the set of all closed surfaces in S3 containing K and $\mathcal{D}_F$ is the set of all compressing disks for F in S3. Then an inequality r(K) ≤ b(K) has shown, where b(K) denotes the bridge number of K. This shows that a (p, q, r)-pretzel knot has the representativity less than or equal to 3. In this paper, we will show that a (p, q, r)-pretzel knot has the representativity 3 if and only if (p, q, r) is either ±(-2, 3, 3) or ±(-2, 3, 5). We also show that an algebraic knot has the representativity less than or equal to 3. [ABSTRACT FROM AUTHOR]

A chord graph in 3 -space is constructed from a ribbon surface-link in 4 -space. In earlier papers, the three moves on the diagrams of chord graphs (namely, the chord diagrams) were introduced to describe the faithful equivalence of a ribbon surface-link. In this paper, it is shown that any two equivalent ribbon surface-links are faithfully equivalent, so that any chord diagrams of any two equivalent ribbon surface-links are connected by a finite number of these three moves. By combining it with an earlier result, it is shown that any two TOP-equivalent ribbon surface-links are equivalent. In other words, there is no exotic ribbon surface-link, generalizing an earlier result on the trivial ribbon surface-knot. In another earlier result, the three moves on the chord diagrams were modified into the 16 moves on the chord diagrams without base crossing. In this paper, further modified moves of the 16 moves on the chord diagrams without base crossing are also introduced to describe how the set of ribbon torus-links is produced from the set of welded virtual links. [ABSTRACT FROM AUTHOR]

The warping sum e (K) of a knot K is the minimal value of the sum of the warping degrees of a minimal diagram of K with both orientations. In this paper, knots K with e (K) ≤ 3 are characterized, and some knots K with e (K) = 4 are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce -braids and, more generally, -braids for an arbitrary group . They form a natural group-theoretic counterpart of -knots, see [V. O. Manturov; Reidemeister moves and groups, preprint (2014), arXiv:1412.8691]. The underlying idea used in the construction of these objects - decoration of crossings with some additional information - generalizes an important notion of parity introduced by the second author (see [V. O. Manturov, Parity in knot theory, Sb. Math. 201(5) (2010) 693-733]) to different combinatorically geometric theories, such as knot theory, braid theory and others. These objects act as natural enhancements of classical (Artin) braid groups. The notion of dotted braid group is introduced: classical (Artin) braid groups live inside dotted braid groups as those elements having presentation with no dots on the strands. The paper is concluded by a list of unsolved problems. [ABSTRACT FROM AUTHOR]

We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo T-relations admit a Hopf algebra structure, where the multiplication is given by any connected sum with respect to the orientation. But in the case of framed chord diagrams a natural way to define a multiplication is not known yet. In this paper, we first define a new module which is generated by chord diagrams on two circles and factored by T-relations. Then we construct a 'parity' map from the module of framed chord diagrams into and a weight system on . Using the map and weight system we show that a connected sum for framed chord diagrams is not a well-defined operation. In the end of the paper we touch linear diagrams, the circle replaced by a directed line. [ABSTRACT FROM AUTHOR]

In this paper, we study doubly transitive property of a connected quandle of a prime order. In 2011, Ferman, Nowik and Teicher proved that the group of all quandle automorphisms of a connected quandle of a prime order acts sharply doubly transitively on itself. In this paper, we give a simple alternative proof. [ABSTRACT FROM AUTHOR]

For coprime integers p(> 0) and q, the (p, q)-cable Γ-polynomial of a knot is the Γ- polynomial of the (p, q)-cable knot of the knot, where the Γ-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we show that there exist infinitely many knots with the trivial (2, 1)-cable Γ- polynomial, that is, the (2, 1)-cable Γ-polynomial of the trivial knot. Moreover, we see that the knots have the trivial Γ-polynomial, the trivial first coefficient HOMFLYPT and Kauffman polynomials. [ABSTRACT FROM AUTHOR]

The Kuperberg bracket is a well-known invariant of classical links. Recently, the second named author and Kauffman constructed the graph-valued generalization of the Kuperberg bracket for the case of virtual links: unlike the classical case, the invariant in the virtual case is valued in graphs which carry a significant amount of information about the virtual knot. The crucial difference between virtual knot theory and classical knot theory is the rich topology of the ambient space for virtual knots. In a paper by Chrisman and the second named author, two-component classical links with one fibered component were considered; the complement to the fibered component allows one to get highly non-trivial ambient topology for the other component. In this paper, we combine the ideas of the above mentioned papers and construct the "virtual" Kuperberg bracket for two-component links L = J ⊔ K with one component (J) fibered. We consider a new geometrical complexity for such links and establish minimality of diagrams in a strong sense. Roughly speaking, every other "diagram" of the knot in question contains the initial diagram as a subdiagram. We prove a sufficient condition for minimality in a strong sense where minimality cannot be established as introduced in the paper by Chrisman and the second named author. [ABSTRACT FROM AUTHOR]

For an effectively -colorable knot , the palette number is the minimum number of distinct colors for all effective -colorings of . It is known that for any effectively -colorable knot . In this paper, we show that for any odd and effectively -colorable torus knot it holds that namely, any effectively -colorable torus knot has an effectively -colored diagram with colors. [ABSTRACT FROM AUTHOR]

In this paper and its sequel, we give a necessary and sufficient condition for two essential simple loops on a -bridge sphere in an even Heckoid orbifold for a -bridge link to be homotopic in the orbifold. We also give a necessary and sufficient condition for an essential simple loop on a -bridge sphere in an even Heckoid orbifold for a -bridge link to be peripheral or torsion in the orbifold. This paper treats the case, when the -bridge link is a -torus link, and its sequel will treat the remaining cases. [ABSTRACT FROM AUTHOR]

In this paper, we investigate twist sequences for Kauffman finite-type invariants and Goussarov–Polyak–Viro finite-type invariants. It is shown that one obtains a Kauffman or GPV type of degree ≤ n if and only if an invariant is a polynomial of degree ≤ n on every twist lattice of the right form. The main result of this paper is an application of this technique to the coefficients of the Jones–Kauffman polynomial. It is shown that the Kauffman finite-type invariants obtained from these coefficients are not GPV finite-type invariants of any degree by explicitly showing they can never be polynomials. This generalizes a result of Kauffman [8], where it is known for degree k = 2. [ABSTRACT FROM AUTHOR]

In this paper, we present a construction of a family of surface-knot diagrams with cross-exchangeable curves, along which we can change the crossing information to obtain trivial diagrams. These diagrams also satisfy a kind of minimality, called -minimal surface-knot diagrams. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nullification numbers of the 249 prime knots up to 10 crossings to an unlink with components. This extends an existing table of knot nullification numbers up to nine crossings in [C. Ernst, A. Montemayor and A. Stasiak, Nullification of small knots, Prog. Theor. Phys. Suppl. 191 (2011) 66-77]. [ABSTRACT FROM AUTHOR]

In this paper, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot with Alexander polynomial , I then use these lassos as patterns to construct families of satellite knots that have Alexander polynomial where . In particular, I prove that if these satellite knots have different Jones polynomials. [ABSTRACT FROM AUTHOR]

This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. We give examples of non-trivial rotational virtuals that are undectable by quantum invariants. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of Hopf module structures on the spaces of framed chord diagrams and framed graphs. We also introduce a framed version of the chromatic polynomial and propose two methods to construct framed weight systems. [ABSTRACT FROM AUTHOR]

We compute integer valued knot concordance invariants of a family of general pretzel knots if the invariants are equal to the negative values of signatures for alternating knots. Examples of such invariants are Rasmussen s-invariants and twice Ozsváth-Szabó knot Floer homology τ-invariants. We use the crossing change inequalities of Livingston and the fact that pretzel knots are almost alternating. As a consequence, for the family of pretzel knots given in this paper, s-invariants are twice τ-invariants. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the (co)homology theory of biquandles, derived biquandle cocycle invariants for oriented surface-links using broken surface diagrams and how to compute the biquandle cocycle invariants from marked graph diagrams. We also develop the shadow (co)homology theory of biquandles and construct the shadow biquandle cocycle invariants for oriented surface-links. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the (co)homology group of a multiple conjugation biquandle. It is the (co)homology group of the prismatic chain complex, which is related to the homology of foams introduced by J. S. Carter, modulo a certain subchain complex. We construct invariants for S 1 -oriented handlebody-links using 2 -cocycles. When a multiple conjugation biquandle X × ℤ type X Y is obtained from a biquandle X using n -parallel operations, we provide a 2 -cocycle (or 3 -cocycle) of the multiple conjugation biquandle X × ℤ type X Y from a 2 -cocycle (or 3 -cocycle) of the biquandle X equipped with an X -set Y. [ABSTRACT FROM AUTHOR]

A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. Normal virtual links have some properties similar to classical links. In this paper, we introduce a method of converting a virtual link diagram to a normal virtual link diagram. We show that the normal virtual link diagrams obtained by this method from two equivalent virtual link diagrams are equivalent. We relate this method to some invariants of virtual links. [ABSTRACT FROM AUTHOR]

Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. The mosaic number m(K) of a knot K is the smallest integer n for which K is representable as a knot n-mosaic. In this paper, we establish an upper bound on the mosaic number of a knot or a link K in terms of the crossing number c(K). Let K be a nontrivial knot or a non-split link except the Hopf link. Then m(K) ≤ c(K) + 1. Moreover if K is prime and non-alternating except link, then m(K) ≤ c(K) - 1. [ABSTRACT FROM AUTHOR]

In this paper we present an explicit isomorphism between Khovanov-Rozansky sl2-homology and ordinary Khovanov homology. This result was originally claimed in Khovanov and Rozansky's paper [Matrix factorizations and link homology, Fund. Math. 199(1) (2008) 1-91, MR 2391017 (2010a:57011)], though the proof was never presented. The main missing detail is providing a coherent choice of signs when identifying variables in the sl2-homology. Along with the behavior of the signs and local orientations in the sl2-homology, both theories behave differently when we try to extend their definitions to virtual links, which seemed to suggest that the sl2-homology may instead correspond to a different variant of Khovanov homology. In this paper we describe both theories and prove that they are in fact isomorphic by showing that a coherent choice of signs can be made. In doing so we emphasize the interpretation of the sl2-complex as a cube of resolutions. [ABSTRACT FROM AUTHOR]

In this paper, we prove the validity of a formula for computing the Alexander invariant, which was originally conjectured by Bar-Natan and Dancso in [2]. [ABSTRACT FROM AUTHOR]

Yokota suggested an optimistic limit method of the Kashaev invariants of hyperbolic knots and showed it determines the complex volumes of the knots. His method is very effective and gives almost combinatorial method of calculating the complex volumes. However, to describe the triangulation of the knot complement, he restricted his method to knot diagrams with certain conditions. Although these restrictions are general enough for any hyperbolic knots, we have to select a good diagram of the knot to apply his theory. In this paper, we suggest more combinatorial way to calculate the complex volumes of hyperbolic links using the modified optimistic limit method. This new method works for any link diagrams, and it is more intuitive, easy to handle and has natural geometric meaning. [ABSTRACT FROM AUTHOR]

The involutory birack counting invariant is an integer-valued invariant of unoriented tangles defined by counting homomorphisms from the fundamental involutory birack of the tangle to a finite involutory birack over a set of framings modulo the birack rank of the labeling birack. In this first of an anticipated series of several papers, we enhance the involutory birack counting invariant with quantum weights, which may be understood as tangle functors of involutory birack-labeled unoriented tangles. [ABSTRACT FROM AUTHOR]

This paper gives an alternate definition of the Affine Index Polynomial (called the Wriggle Polynomial) using virtual linking numbers and explores applications of this polynomial. In particular, it proves the Cosmetic Crossing Change Conjecture for odd virtual knots and pure virtual knots. It also demonstrates that the polynomial can detect mutations by positive rotation and proves it cannot detect mutations by positive reflection. Finally it exhibits a pair of mutant knots that can be distinguished by a type 2 vassiliev invariant coming from the polynomial. [ABSTRACT FROM AUTHOR]

Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms. In this paper, the author introduces several new ways to smooth a crossings, and uses a system of skein equations to construct link invariants. This invariant can also be modified by writhe to get a more powerful invariant. The modified invariant is a generalization of both the HOMFLYPT polynomial and the two-variable Kauffman polynomial. Using the diamond lemma, a simplified version of the modified invariant is given. It is easy to compute and is a generalization of the two-variable Kauffman polynomial. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application, we distinguish several singular knots and links. [ABSTRACT FROM AUTHOR]

Let be a perfect -manifold, be a component of , be a homeomorphic map, and . In this paper, we show that if and , then . As a corollary, if is a component of , is a homeomorphic map, , , and , then . [ABSTRACT FROM AUTHOR]

In this paper, we investigate surface braids obtained from minimal charts with exactly four white vertices. [ABSTRACT FROM AUTHOR]

We introduce the warping polynomial of an oriented knot diagram. In this paper, we characterize the warping polynomial, and define the span of a knot to be the minimal span of the warping polynomial for all diagrams of the knot. We show that the span of a knot is one if and only if it is non-trivial and alternating, and we give an inequality between the span and the dealternating number. [ABSTRACT FROM AUTHOR]

In this paper, the support genera of all Legendrian right-handed trefoil knots and some other Legendrian knots are computed. We give examples of Legendrian knots in the three-sphere with the standard contact structure which have positive support genera with arbitrarily negative Thurston-Bennequin invariant. This answers a question in [S. Onaran, Invariants of Legendrian knots from open book decompositions, Int. Math. Res. Not. 10 (2010) 1831-1859]. [ABSTRACT FROM AUTHOR]

This paper generalizes Van Cott's bounds on the Heegaard-Floer τ-invariant and Rasmussen's s-invariant of cabled knots to apply to all satellite knots. [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]

Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering manifolds given by surgery on framed links in the 3-sphere. We study the lowest degree case — the celebrated Casson–Walker invariant of rational homology spheres. This paper is dedicated to a detailed treatment of 2-component links; a general case will be considered in a forthcoming paper. We present simple Gauss diagram formulas for the Casson–Walker invariant. This enables us to understand/separate its dependence on the unframed link and on the framings. We also obtain skein relations for the Casson–Walker invariant under crossing changes, and study its asymptotic behavior when framings tend to infinity. Finally, we present results of extensive computer calculations. [ABSTRACT FROM AUTHOR]

This paper establishes that sutured annular Khovanov homology is not invariant for braid closures under axis-preserving mutations. This follows from an explicit relationship between sutured annular Khovanov homology and the classical Burau representation for braid closures. [ABSTRACT FROM AUTHOR]

Polyak proved that the set is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining different types of moves, which we call directed oriented Reidemeister moves. In this paper, we prove that the set of eight directed Polyak moves is a minimal generating set of directed oriented Reidemeister moves. We also specialize the problem, introducing the notion of a -generating set for a link . The same set is proven to be a minimal -generating set for any link with at least two components. Finally, we discuss knot diagram invariants arising in the study of -generating sets for an arbitrary knot , emphasizing the distinction between ascending and descending moves of type . [ABSTRACT FROM AUTHOR]

In this paper, we classify, up to rigid isotopy, real rational knots of degrees less than or equal to in a real quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots of low degree in and classify real rational curves of degree 6 in the 3-sphere with exactly one ordinary double point. [ABSTRACT FROM AUTHOR]

A well-known conjecture states that for any -component link in , the rank of the knot Floer homology of (over any field) is less than or equal to times the rank of the reduced Khovanov homology of . In this paper, we describe a framework that might be used to prove this conjecture. We construct a modified version of Khovanov homology for links with multiple basepoints and show that it mimics the behavior of knot Floer homology. We also introduce a new spectral sequence converging to knot Floer homology whose page is conjecturally isomorphic to our new version of Khovanov homology; this would prove that the conjecture stated above holds over the field . [ABSTRACT FROM AUTHOR]

This paper introduces a number of new intrinsically 3-linked graphs through five new constructions. We prove that intrinsic 3-linkedness is not preserved by moves. We will see that the graph , which is obtained through a move on , is not intrinsically 3-linked. [ABSTRACT FROM AUTHOR]

This paper contains two remarks about the application of the -invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being smoothly slice. The second carries out a suggestion by Stefan Friedl to replace the use of Heegaard Floer homology by Donaldson's theorem in the proof of the main result of [J. E. Greene, Lattices, graphs, and Conway mutation, Invent. Math. 192(3) (2013) 717-750] concerning Conway mutation of alternating links. [ABSTRACT FROM AUTHOR]

In the present paper, we develop the parity theory invented in [V. O. Manturov, Mathematics 201(5) (2010) 693733]; we construct new parities for two-component (virtual and free) links. New parities significantly depend on geometrical properties of diagrams; in particular, they are mutation-sensitive. New parities can be used practically in all problems, where parities were previously applied. [ABSTRACT FROM AUTHOR]

Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the orientation double covering of an abstract link or alternatively by constructing a diagram called a double covering diagram. We also discuss links in thickened surfaces, their diagrams and their stable equivalence classes. Bourgoin's twisted knot group is understood as the virtual knot group of the double covering. [ABSTRACT FROM AUTHOR]

Let be a -manifold, be an essential planar surface which cuts into two 3-manifolds and . Suppose is a Heegaard splitting of , is the dual Heegaard splitting of and along , where and . In this paper, we give a condition of unstabilized dual Heegaard splittings of -manifolds by using Hempel's distance and the method of proof of Gordon's Conjecture. Also, we give a counterexample for stabilized dual Heegaard splittings of -manifolds for any Hempel's distance. [ABSTRACT FROM AUTHOR]

Milnor defined two kinds of link-homotopy invariants and . By definition it would seem that the -invariant is weaker than the -invariant. In this paper, we show that this is indeed the case by giving an example of length greater than 4 where and . For non-repeated sequences of length not greater than 4, Milnor has shown that . [ABSTRACT FROM AUTHOR]

In the present paper, we proceed with the study of framed 4-graph minor theory initiated in [V. O. Manturov, Framed 4-valent graph minor theory I: Intoduction planarity criterion, arxiv: 1402.1564v1 [Math.Co]] and justify the planarity theorem for arbitrary framed 4-graphs; besides, we prove analogous results for embeddability in . [ABSTRACT FROM AUTHOR]

In this paper, we discuss the region unknotting number of different classes of 2-bridge knots. In particular, we provide region unknotting number for the classes of 2-bridge knots whose Conway notations are C(m, n), C(m, 2, m), C(m, 2, m ± 1) and C(2, m, 2, n). By generalizing, we also provide a sharp upper bound for all the remaining classes of 2-bridge knots. [ABSTRACT FROM AUTHOR]