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We present the C W R invariant, a new invariant for alternating links, which builds upon and generalizes the W R P invariant. The C W R invariant is an array of two-variable polynomials that provides a stronger invariant compared to the W R P invariant. We compare the strength of our invariant with the classical HOMFLYPT, Kauffman 3 -variable, and Kauffman 2 -variable polynomials on specific knot examples. Additionally, we derive general recursive "skein" relations, and also specific formulas for the initial components of the C W R invariant using weighted adjacency matrices of modified Tait graphs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an invariant of alternating knots and links (called here WRP), namely, a pair of integer polynomials associated with their two checkerboard planar graphs from their minimal diagram. We prove that the invariant is well-defined and give its values obtained from calculations for some knots in the tables. This invariant is strong enough to distinguish all knots in the tables with up to 10 crossings (including their mirror images). We compare the strength of the new invariant with classical invariants, including the three-variable Kauffman bracket. [ABSTRACT FROM AUTHOR]

Using the skew-Hopf pairing, we obtain ℛ -matrix for the two-parameter quantum algebra U v , t . We further construct a strict monoidal functor from the tangle category (OTa , ⊗ , ∅) to the category (Mod , ⊗ , ℚ (v , t)) of U v , t -modules. As a consequence, the quantum knot invariant of the tangle L of type (n , n) is obtained by the action of on the closure L ̃ of L. [ABSTRACT FROM AUTHOR]

To each rail knotoid we associate two unoriented knots along with their oriented counterparts, thus deriving invariants for rail knotoids based on these associations. We then translate them to invariants of rail isotopy for rail arcs. [ABSTRACT FROM AUTHOR]

In this paper, the authors obtain a numerical invariant d L of oriented links. It satisfies a non-homogeneous skein relation, and d L − 1 is additive under the connected sum operation. We give an obstruction for two links to be 6-equivalent, derived from this numerical invariant. [ABSTRACT FROM AUTHOR]

We show that the isomorphism problems for left distributive algebras, racks, quandles and kei are as complex as possible in the sense of Borel reducibility. These algebraic structures are important for their connections with the theory of knots, links and braids. In particular, Joyce showed that a quandle can be associated with any knot, and this serves as a complete invariant for tame knots. However, such a classification of tame knots heuristically seemed to be unsatisfactory, due to the apparent difficulty of the quandle isomorphism problem. Our result confirms this view, showing that, from a set-theoretic perspective, classifying tame knots by quandles replaces one problem with (a special case of) a much harder problem. [ABSTRACT FROM AUTHOR]

In this article we will construct a knot invariant which is complementary to HOMFLY knot invariant by proving that the skein relation is not compatible with HOMFLY’s. [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]

We study $$\mathrm {U}(N|M)$$ character expectation value with the supermatrix Chern-Simons theory, known as the ABJM matrix model, with emphasis on its connection to the knot invariant. This average just gives the half-BPS circular Wilson loop expectation value in ABJM theory, which shall correspond to the unknot invariant. We derive the determinantal formula, which gives $$\mathrm {U}(N|M)$$ character expectation values in terms of $$\mathrm {U}(1|1)$$ averages for a particular type of character representations. This means that the $$\mathrm {U}(1|1)$$ character expectation value is a building block for the $$\mathrm {U}(N|M)$$ averages and also, by an appropriate limit, for the $$\mathrm {U}(N)$$ invariants. In addition to the original model, we introduce another supermatrix model obtained through the symplectic transform, which is motivated by the torus knot Chern-Simons matrix model. We obtain the Rosso-Jones-type formula and the spectral curve for this case. [ABSTRACT FROM AUTHOR]

We construct the complete invariant for fused links. It is proved that the set of equivalence classes of -component fused links is in one-to-one correspondence with the set of elements of the abelization up to conjugation by elements from the symmetric group . [ABSTRACT FROM AUTHOR]

Using a combination of calculational and theoretical approaches, we establish results that relate two knot invariants, the Alexander polynomial, and the number of quandle colourings using any finite linear Alexander quandle. Given such a quandle, specified by two coprime integers n and m, the number of colourings of a knot diagram is given by counting the solutions of a matrix equation of the form AX = 0 mod n, where A is the m-dependent colouring matrix. We devised an algorithm to reduce A to echelon form, and applied this to the colouring matrices for all prime knots with up to 10 crossings, finding just three distinct reduced types. For two of these types, both upper triangular, we found general formulae for the number of colourings. This enables us to prove that in some cases the number of such quandle colourings cannot distinguish knots with the same Alexander polynomial, whilst in other cases knots with the same Alexander polynomial can be distinguished by colourings with a specific quandle. When two knots have different Alexander polynomials, and their reduced colouring matrices are upper triangular, we find a specific quandle for which we prove that it distinguishes them by colourings. [ABSTRACT FROM AUTHOR]

We give explicit resolutions of all finite-dimensional simple -modules. We use these resolutions to categorify the colored -invariant of framed links via a complex of complexes of graded -modules. [ABSTRACT FROM AUTHOR]

In previous work, results were obtained relating two different knot invariants, the Alexander polynomial on the one hand, and the number of quandle colourings using any finite linear Alexander quandle, on the other. Given such a quandle, specified by two coprime integers n and m, the number of colourings of a knot diagram is given by counting the solutions of a matrix equation of the form AX = 0 mod n, where A is the m-dependent colouring matrix. The same matrix A determines the Alexander polynomial of the knot. Our previous results were based in part on computations using an algorithm to reduce A to echelon form, and in part on proving properties of the matrix equations in their reduced form. When two knots have di erent Alexander polynomials, and their reduced colouring matrices are upper triangular, we proved that there exists a specific quandle which distinguishes them by colourings, and conjectured that this would be true without the condition on the reduced form of the colouring matrices. In the present article we address this issue from a new perspective, using the fact that all colouring matrices are triangularizable when certain conditions on m and n are met, and find further support for our conjecture. A description of an improved version of the algorithm is also presented. [ABSTRACT FROM AUTHOR]

We define an invariant of tangles and framed tangles, given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, and central extensions of groups. We prove that our construction includes all rack and quandle cohomology (framed) link invariants, as well as the Eisermann invariant of knots. We construct a class of Reidemeister pairs which constitute a lifting of the Eisermann invariant, and show through an example that this class is strictly stronger than the Eisermann invariant itself. [ABSTRACT FROM AUTHOR]

We study the diagram alphabet of knot moves associated with the character rings of certain matrix groups. The primary object is the Hopf algebra Char-GL of characters of the finite dimensional polynomial representations of the complex groupin the inductive limit, realized as the ring of symmetric functionson countably many variables. Isomorphic as spaces are the character rings Char-O and Char-Sp of the classical matrix subgroups of, the orthogonal and symplectic groups. We also analyse the formal character rings Char-Hof algebraic subgroups of, comprised of matrix transformations leaving invariant a fixed but arbitrary tensor of Young symmetry type, which have been introduced earlier (Fauser et al.) (these include the orthogonal and symplectic groups as special cases). The set of tangle diagrams encoding manipulations of the group and subgroup characters has many elements deriving from products, coproducts, units and counits as well as different types of branching operators. From these elements we assemble for eacha crossing tangle which satisfies the braid relation and which is nontrivial, in spite of the commutative and co-commutative setting. We identify structural elements and verify the axioms to establish that each Char-Hring is a ribbon Hopf algebra. The corresponding knot invariant operators are rather weak, giving merely a measure of the writhe. [ABSTRACT FROM PUBLISHER]

Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set theoretic solutions of the well-known Yang-Baxter equation. The first half of this paper proposes some natural constructions of biquandles from groups and from their simpler counterparts, namely, quandles. We completely determine all words in the free group on two generators that give rise to (bi)quandle structures on all groups. We give some novel constructions of biquandles on unions and products of quandles, including what we refer as the holomorph biquandle of a quandle. These constructions give a wealth of solutions of the Yang-Baxter equation. We also show that for nice quandle coverings a biquandle structure on the base can be lifted to a biquandle structure on the covering. In the second half of the paper, we determine automorphism groups of these biquandles in terms of associated quandles showing elegant relationships between the symmetries of the underlying structures. [ABSTRACT FROM AUTHOR]

In the planar limit of the’ t Hooft expansion, the Wilson-loop vacuum average in the three-dimensional Chern-Simons theory (in other words, the HOMFLY polynomial) depends very simply on the representation (Young diagram), H R(A|q)| q=1 = (σ 1(A) |R|. As a result, the (knot-dependent) Ooguri-Vafa partition function $$\sum\nolimits_R {H_{R\chi R} \left\{ {\bar pk} \right\}}$$ becomes a trivial τ -function of the Kadomtsev-Petviashvili hierarchy. We study higher-genus corrections to this formula for H R in the form of an expansion in powers of z = q − q −1. The expansion coefficients are expressed in terms of the eigenvalues of cut-and-join operators, i.e., symmetric group characters. Moreover, the z-expansion is naturally written in a product form. The representation in terms of cut-and-join operators relates to the Hurwitz theory and its sophisticated integrability. The obtained relations describe the form of the genus expansion for the HOMFLY polynomials, which for the corresponding matrix model is usually given using Virasoro-like constraints and the topological recursion. The genus expansion differs from the better-studied weak-coupling expansion at a finite number N of colors, which is described in terms of Vassiliev invariants and the Kontsevich integral. [ABSTRACT FROM AUTHOR]

We study the 4-move invariant for links in the 3-sphere developed by Dabkowski and Sahi, which is defined as a quotient of the fundamental group of the link complement. We develop techniques for computing this invariant and show that for several classes of knots it is equal to the invariant for the unknot; therefore, in these cases the invariant cannot detect a counterexample to the 4-move conjecture. [ABSTRACT FROM AUTHOR]

We offer an alternative construction of Roberts' totally twisted Khovanov homology and prove that for knots, it agrees with δ-graded reduced characteristic-2 Khovanov homology. [ABSTRACT FROM AUTHOR]

We briefly review problems arising in the study of the beta deformation, which turns out to be the most difficult element in a number of modern problems: the deviation of β from unity is connected with the 'exit from the free-fermion point' in two-dimensional conformal theories, from the symmetric graviphoton field with ∈ = −∈ in instanton sums in four-dimensional supersymmetric Yang-Mills theories, with the transition from matrix models to beta ensembles, from HOMFLY polynomials to superpolynomials in the Chern-Simons theory, from quantum groups to elliptic and hyperbolic algebras, and so on. We mainly attend to issues related to the Alday-Gaiotto-Tachikawa correspondence and its possible generalizations. [ABSTRACT FROM AUTHOR]

An extension of the two-dimensional (2d) Alday-Gaiotto-Tachikawa (AGT) relation to three dimensions starts from relating the theory on the domain wall between some two S-dual supersymmetric Yang-Mills (SYM) models to the 3d Chern-Simons (CS) theory. The simplest case of such a relation would presumably connect traces of the modular kernels in 2d conformal theory with knot invariants. Indeed, the two quantities are very similar, especially if represented as integrals of quantum dilogarithms. But there are also various differences, especially in the 'conservation laws' for the integration variables holding for the monodromy traces but not for the knot invariants. We also consider another possibility: interpreting knot invariants as solutions of the Baxter equations for the relativistic Toda system. This implies another AGT-like relation: between the 3d CS theory and the Nekrasov-Shatashvili limit of the 5d SYM theory. [ABSTRACT FROM AUTHOR]

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We study the structure of the periodic orbits on a simple branched manifold which is a subtemplate of the branched manifold of the chaotic attractor that is obtained from a cancer model. We indicate the conditions for the knotted and linked periodic orbits by using symbolic dynamics, then we extend the results to the periodic solutions of the chaotic attractor. Furthermore, we compare the knots obtained by using same symbol sequences for the simple branched manifold and the attractor's branched manifold by using a knot invariant, specifically, we calculate the Kauffman bracket polynomial. In order to count the number of closed curves which is required to calculate the bracket polynomial we propose a new method which uses cyclic permutations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a new algorithm to evaluate the Kauffman bracket polynomial. The algorithm uses cyclic permutations to count the number of states obtained by the application of ‘A’ and ‘B’ type smoothings to the each crossing of the knot. We show that our algorithm can be implemented easily by computer programming. [Copyright &y& Elsevier]

We give an alternative presentation of Khovanov homology of links. The original construction rests on the Kauffman bracket model for the Jones polynomial, and the generators for the complex are enhanced Kauffman states. Here we use an oriented sl(2) state model allowing a natural definition of the boundary operator as twisted action of morphisms belonging to a TQFT for trivalent graphs and surfaces. Functoriality in original Khovanov homology holds up to sign. Variants of Khovanov homology fixing functoriality were obtained by Clark–Morrison–Walker [7] and also by Caprau [6]. Our construction is similar to those variants. Here we work over integers, while the previous constructions were over gaussian integers. [ABSTRACT FROM AUTHOR]

Abstract: We construct combinatorial formulas of Fiedler type (i.e. composed of oriented Gauss diagrams arranged by homotopy classes of loops in the base manifold, see [T. Fiedler, Gauss Diagram Invariants for Knots and Links, Math. Appl., vol. 552, Kluwer Academic Publishers, 2001; M. Polyak, O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not. 11 (1994) 445–453]) for an infinite family of finite type invariants of knots in ( orientable), introduced in [S.A. Grishanov, V.A. Vassiliev, Two constructions of weight systems for invariants of knots in non-trivial 3-manifolds, Topology Appl. 155 (2008) 1757–1765]. [Copyright &y& Elsevier]

This paper is the second in the series on topological classification of textile structures. The classification problem can be resolved with the aid of invariants used in knot theory for classification of knots and links. Various numerical and polynomial invariants are considered in application to textile structures. A new Kauffman-type polynomial invariant is constructed for doubly-periodic textile structures. The values of the numerical and polynomial invariants are calculated for some simplest doubly-periodic interlaced structures and for some woven and knitted textiles. [ABSTRACT FROM AUTHOR]

A knot diagram can be divided by a circle into two parts, such that each part can be coded by a planar tree with integer weights on its edges. A half of the number of intersection points of this circle with the knot diagram is called the girth. The girth of a knot is the minimal girth of all diagrams of this knot. The girth of a knot minus one is an upper bound of the Heegaard genus of the 2-fold branched covering of that knot. We will use Topological Quantum Field Theory (TQFT) coming from the Kauffman bracket to determine the girth of some knots. Consequently, our method can be used to determine the Heegaard genus of the 2-fold branched covering of some knots. [ABSTRACT FROM AUTHOR]

Abstract: A new family of weight systems of finite type knot invariants of any positive degree in orientable 3-manifolds with non-trivial first homology group is constructed. The principal part of the Casson invariant of knots in such manifolds is split into the sum of infinitely many independent weight systems. Examples of knots separated by corresponding invariants and not separated by any other known finite type invariants are presented. [Copyright &y& Elsevier]