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Two rational functions are defined logically for special type of knotted trivalent networks as state models of planar trivalent networks. The restriction of these two rational functions reduce to the Jones and Hom y polynomials for non oriented links. Also, these two models are used to define two invariants for this special type of knotted trivalent networks embedded in R3. Finally, we study some congruences of these two polynomials for periodic knotted trivalent networks this generalize the work of periodicity of the Jones and Hom y polynomials on knots to these two rational functions of knotted trivalent networks.

We introduce a certain class of link diagrams, which includes all closed braid diagrams. We show a generalized version of K\'alm\'an's full-twist formula for the HOMFLY polynomial in the class., Comment: 6 pages, 4 figures

We present a novel symmetry of the colored HOMFLY polynomial. It relates pairs of polynomials colored by different representations at specific values of N and generalizes the previously known “tug-the-hook” symmetry of the colored Alexander polynomial (Mishnyakov et al. in Annales Henri Poincare, 2021. https://doi.org/10.1007/s00023-020-00980-8 , arXiv:2001.10596 ). As we show, the symmetry has a superalgebra origin, which we discuss qualitatively. Our main focus are the constraints that such a property imposes on the general group-theoretical structure, namely the $$\mathfrak {sl}(N)$$ weight system, arising in the perturbative expansion of the invariant. Finally, we demonstrate its tight relation to the eigenvalue conjecture.

We give an obstruction for genus one knots $K$, $K'$ to have the Gordian distance one by using the $0$th coefficient of the HOMFLT polynomials. As an application, we give a new constraint for genus one knot to admit a (generalized) cosmetic crossing. Combining known results, we prove the (generalized) cosmetic crossing conjecture for genus one pretzel knots., 5 pages, no Figure

International audience; Assuming the variety of a polynomial set is invariant under a group action, we construct a set of invariants that define the same variety. Our construction can be seen as a generalization of the previously known construction for finite groups. The result though has to be understood outside an invariant variety which is independent of the polynomial set considered. We introduce the symmetrizations of a polynomial that are polynomials in a generating set of rational invariants. The generating set of rational invariants and the symmetrizations are constructed w.r.t. a section to the orbits of the group action.

A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with $SU(N)$ gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra $ZU(\mathfrak{sl}_N)$ is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved., 10 pages

In this paper, we compute explicitly the Khovanov polynomials (using the computer program from katlas.org) for the two simplest families of the satellite knots, which are the twisted Whitehead doubles and the two-strand cables. We find that a quantum group decomposition for the HOMFLY polynomial of a satellite knot can be extended to the Khovanov polynomial, whose quantum group properties are not manifest. Namely, the Khovanov polynomial of a twisted Whitehead double or two-strand cable (the two simplest satellite families) can be presented as a naively deformed linear combination of the pattern and companion invariants. For a given companion, the satellite polynomial “smoothly” depends on the pattern but for the “jump” at one critical point defined by the [Formula: see text]-invariant of the companion knot. A similar phenomenon is known for the knot Floer homology and [Formula: see text]-invariant for the same kind of satellites.

The bond fluctuation method is used to simulate both non-concatenated entangled and interpenetrating melts of ring polymers. We find that the swelling of interpenetrating rings upon dilution follows the same laws as for linear chains. Knotting and linking probabilities of ring polymers in semi-dilute solution are analyzed using the HOMFLY polynomial. We find an exponential decay of the knotting probability of rings. The correlation length of the semi-dilute solution can be used to superimpose knotting data at different concentrations. A power law dependence $f_{n}\sim\phi R^{2}\sim\phi^{0.77}N$ for the average number $f_{n}$ of linked rings per ring at concentrations larger than the overlap volume fraction of rings $\phi^{*}$ is determined from the simulation data. The fraction of non-concatenated rings displays an exponential decay $P_{OO}\sim\exp(-f_{n})$, which indicates $f_{n}$ to provide the entropic effort for not forming concatenated conformations. Based upon this results we find four different regimes for the conformations of rings in melts that are separated by a critical lengths $N_{OO}$, $N_{C}$ and $N^{*}$. $N_{OO}$ describes the onset of the effect of non-concatenation below which topological effects are not important, $N_{C}$ is the cross-over between weak and strong compression of rings, and $N^{*}$ is defined by the cross-over from a non-concatenation contribution $f_{n}\sim\phi R^{2}$ to an overlap dominated concatenation contribution $f_{n}\sim\phi N^{1/2}$ at $N>N^{*}$. For $N_{OO}

Bu çalışma dört bölümden oluşmaktadır. Birinci bölümde giriş kısmına yer verilmiştir. İkinci bölümde düğümler için bazı temel kavramlar tanıtılmıştır. Üçüncü bölümde yönlendirilmiş ve yönlendirilmemiş düğüm polinomları hakkında temel bilgiler ve onların bazı özellikleri verilmiştir. Bu çalışmanın orijinal kısmını oluşturan dördüncü bölümde yönlendirilmiş burulmalı düğümlerin HOMFLY polinomları ile yönlendirilmemiş burulmalı düğümlerin Kauffman L ve F polinomları için formüller elde edilmiştir ve burulmalı düğümler için bahsedilen düğüm polinomları dördüncü mertebeden yineleme bağıntıları olarak tanımlanmıştır. Burada bazı sonuçları elde ederken (2,n)-torus halkalarının düğüm polinomları ile ilgili önceki formüller ve bağıntılardan yararlanılmıştır. Ayrıca bulunan bağıntıların bazı yinelemeli özellikleri incelenerek üreten fonksiyonları, genel çözümleri ve açık formları verilmiştir. This study consists of four chapters. In the first chapter, it is given the introduction part. In the second chapter, some basic concepts for knots are introduced. In the third chapter, basic information about oriented and unoriented knot polynomials and some of their properties are discussed. In the fourth chapter, which is the original part of this study, the formulas for HOMFLY polynomials of oriented twist knots and Kauffman L and F polynomials of unoriented twist knots are obtained, and the mentioned knot polynomials for twist knots are defined as fourth-order recurrence relations. Here, the previous formulas and relations related to the knot polynomials of (2,n)-torus links are used to obtain some results. In addition, some recursive properties of the relations found are examined and their generating functions, general solutions and explicit forms are given.

We discuss the relation between knot polynomials and the KP hierarchy. Mainly, we study the scaling 1-hook property of the coloured Alexander polynomial: $\mathcal{A}^\mathcal{K}_R(q)=\mathcal{A}^\mathcal{K}_{[1]}(q^{\vert R\vert})$ for all 1-hook Young diagrams $R$. Via the Kontsevich construction, it is reformulated as a system of linear equations. It appears that the solutions of this system induce the KP equations in the Hirota form. The Alexander polynomial is a specialization of the HOMFLY polynomial, and it is a kind of a dual to the double scaling limit, which gives the special polynomial, in the sense that, while the special polynomials provide solutions to the KP hierarchy, the Alexander polynomials provide the equations of this hierarchy. This gives a new connection with integrable properties of knot polynomials and puts an interesting question about the way the KP hierarchy is encoded in the full HOMFLY polynomial., 10 pages

In 1994, J. Chen, J. Gross, and R. Rieper demonstrated how to use the rank of Mohar's overlap matrix to calculate the crosscap-number distribution, that is, the distribution of the embeddings of a graph in the nonorientable surfaces. That has ever since been by far the most frequent way that these distributions have been calculated. This article introduces a way to calculate the Euler-genus polynomial of a graph, which combines the orientable and the nonorientable embeddings, without using the overlap matrix. The crosscap-number polynomial for the nonorientable embeddings is then easily calculated from the Euler-genus polynomial and the genus polynomial.

We construct a new polynomial invariant of maps (graphs embedded in closed surfaces, not necessarily orientable). Our invariant is tailored to contain as evaluations the number of local flows and local tensions taking non-identity values in any given finite group. Moreover, it contains as specializations the Krushkal polynomial, the Bollobás-Riordan polynomial, the Las Vergnas polynomial, and their extensions to non-orientable surfaces, and hence in particular the Tutte polynomial of the under-lying graph of the map.

Let G be a connected bipartite graph with colour classes E and V and root polytope Q. Regarding the hypergraph H=(V,E) induced by G, we prove that the interior polynomial of H is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It follows that the interior polynomials of H and its transpose H=(E,V) agree. When G is a complete bipartite graph, our result recovers a well-known hypergeometric identity due to Saalschütz. It also implies that certain extremal coefficients in the Homfly polynomial of a special alternating link can be read off of an associated Floer homology group., National Science Foundation (U.S.) (Grant DMS‐1100147), National Science Foundation (U.S.) (Grant DMS‐1362336)

International audience; In this article, we conjecture that the Links–Gould invariant of links, which we know is a generalization of the Alexander–Conway polynomial, shares some of its classical features. In particular, it seems to give a lower bound for the genus of links and to provide a criterion for fibredness of knots. We give some evidence for these two assumptions.

Работа посвящена построению одного оператора Янга Бакстера на двумерном векторном пространстве и изучению свойств соответствующего инварианта ориентированных узлов и зацеплений. Найден явный вид скейн-соотношения, которому удовлетворяет этот инвариант, и доказано, что этот инвариант не является следствием полинома HOMFLY. В конце работы приводится таблица значений инварианта для всех ориентированных узлов и зацепленй, допускающих диаграммы с не более чем семью двойными точками., The paper is devoted to construction of a Yang Baxter operator over two-dimensional vector space. Properties of the corresponding invariant of oriented knot and links are studied. An explicit form of the skein relation of this invariant is presented. It’s proved, that this invariant is not a consequence of the HOMFLY polynomial. At the end of the paper the table of invariant’s values for all oriented knots and links that admit diagrams with at most seven crossing points is given.

The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups $$\vec {F}$$ and $$\vec {T}$$ of the Thompson groups F and T, respectively, together with a procedure that associates an oriented link diagram to any element of these subgroups. Moreover, several specializations of some well-known polynomial link invariants can be seen as functions of positive type on the Thompson groups or the Jones–Thompson subgroups. One important example is provided by suitable evaluations of the Jones polynomial, which are thus associated with certain unitary representations of the groups $$\vec {F}$$ and $$\vec {T}$$ . Within this framework, we discuss an alternative approach that relies on some partition function interpretation of the Jones polynomial, and also exhibit more examples associated with other link invariants, notably the two-variable Kauffman polynomial and the HOMFLY polynomial. In the unoriented case, extending our previous results, we also show by similar methods that certain evaluations of the Tutte polynomial and of the Kauffman bracket, suitably renormalized, yield functions of positive type on T.

The author discusses 2-adjacency of two-component links and study the relations between the signs of the crossings to realize 2-adjacency and the coefficients of the Conway polynomial of two related links. By discussing the coefficient of the lowest m power in the Homfly polynomial, the author obtains some results and conditions on whether the trivial link is 2-adjacent to a nontrivial link, whether there are two links 2-adjacent to each other, etc. Finally, this paper shows that the Whitehead link is not 2-adjacent to the trivial link, and gives some examples to explain that for any given two-component link, there are infinitely many links 2-adjacent to it. In particular, there are infinitely many links 2-adjacent to it with the same Conway polynomial.

In this paper, we fix a polynomial with complex coefficients and determine the eigenforms for SL2(Z) which can be expressed as the fixed polynomial evaluated at other eigenforms. In particular, we show that when one excludes trivial cases, only finitely many such identities hold for a fixed polynomial., 14 pages

The HOMFLY polynomial P L ( v , z ) of a link L is a well known link invariant with two variables v and z , which generalizes the Jones polynomial. The v -span of the HOMFLY polynomial is the difference between the maximum and the minimum degrees of the polynomial in the variable v . We show the HOMFLY polynomial of the 3-component link with v -span 4 is determined by the Jones polynomial.

The Γ-polynomial is an invariant of an oriented link in the 3-sphere, which is the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials. It is known that the HOMFLYPT and Kauffman polynomials, their 2-cable versions, and the satellite versions of the Alexander and Jones polynomials are invariant under mutation. On the other hand, there exists a mutant knot pair which is distinguished by the 3-cable version of the HOMFLYPT polynomial. In this paper, we show that the 3-cable version of the Γ-polynomial is invariant under mutation.

The interior polynomial is an 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 a part of the HOMFLY polynomial of a naturally associated link. We also establish some other, more basic properties of this new notion. This leads to new identities involving the original interior polynomial., 24 pages, 16figures

Given any pair of positive integers m and n, we construct a new Hopf algebra, which may be regarded as a degenerate version of the quantum group of gl(m+n). We study its structure and develop a highest weight representation theory. The finite dimensional simple modules are classified in terms of highest weights, which are essentially characterised by m+n-2 nonnegative integers and two arbitrary nonzero scalars. In the special case with m=2 and n=1, an explicit basis is constructed for each finite dimensional simple module. For all m and n, the degenerate quantum group has a natural irreducible representation acting on C(q)^(m+n). It admits an R-matrix that satisfies the Yang-Baxter equation and intertwines the co-multiplication and its opposite. This in particular gives rise to isomorphisms between the two module structures of any tensor power of C(q)^(m+n) defined relative to the co-multiplication and its opposite respectively. A topological invariant of knots is constructed from this R-matrix, which reproduces the celebrated HOMFLY polynomial. Degenerate quantum groups of other classical types are briefly discussed.

We give a quotient of the ring so that the Miyazawa polynomial is a non-trivial invariant of welded links. Furthermore we show that this is also an invariant under the other forbidden move , and so it is a fused isotopy invariant. Also, we give some quotient ring so that the index polynomial can be an invariant for welded links.

Starting from the free field realization of Kac–Moody Lie algebra, we define a generalized Yang–Yang function. Then for the Lie algebra of type $$A_{n}$$ , we derive braiding and fusion matrix by braiding the thimble from the generalized Yang–Yang function. One can construct a knots invariant H(K) from the braiding and fusion matrix. It is an isotropy invariant and obeys a skein relation. From them, we show that the corresponding knots invariant is HOMFLY polynomial.

It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Murasugi had conjectured that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link. This conjecture turned out to be false. In this paper we prove the next best thing that one could hope for: we characterize exactly those alternating links for which their braid indices equal to the numbers of Seifert circles in their corresponding reduced alternating link diagrams. More specifically, we prove that if $D$ is a reduced alternating link diagram of an alternating link $L$, then $b(L)$, the braid index of $L$, equals the number of Seifert circles in $D$ if and only if $G_S(D)$ contains no edges of weight one. Here $G_S(D)$, called the Seifert graph of $D$, is an edge weighted simple graph obtained from $D$ by identifying each Seifert circle of $D$ as a vertex of $G_S(D)$ such that two vertices in $G_S(D)$ are connected by an edge if and only if the two corresponding Seifert circles share crossings between them in $D$ and that the weight of the edge is the number of crossings between the two Seifert circles. This result is partly based on the well known MFW inequality, which states that the $a$-span of the HOMFLY polynomial of $L$ is a lower bound of $2b(L)-2$, as well as a result due to Yamada, which states that the minimum number of Seifert circles over all link diagrams of $L$ equals $b(L)$., Minor corrections

We explain how existing results (such as categorical sl(n) actions, associated braid group actions and infinite twists) can be used to define a triply graded link invariant which categorifies the HOMFLY polynomial of links coloured by arbitrary partitions. The construction uses a categorified HOMFLY clasp defined via cabling and infinite twists. We briefly discuss differentials and speculate on related structures., 21 pages

We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using earlier work, this establishes a connection between contact topology and the Homfly polynomial. We also show that the contact invariants of our tight contact structures form a basis for sutured Floer homology. Finally, we relate our methods and results to Kauffman's formal knot theory., Comment: 45 pages, 19 figures

A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern–Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers. If the knot is hyperbolic, the representation can be regarded as a deformation of the holonomy representation that determines the complete hyperbolic structure. In this article, we study a similar phenomenon when the knot is a twice-iterated torus knot. In this case, the asymptotic expansion of the colored Jones polynomial splits into sums, and each summand is related to the Chern–Simons invariant and the Reidemeister torsion associated with a representation.

We formulate a refinement of SU(N) Chern–Simons theory on a three-manifold M via an index in the (2, 0) theory on N M5 branes. The refined Chern–Simons theory is defined on any M with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refinement of the S and T matrices of Chern–Simons theory, related to the theory of Macdonald polynomials. The ordinary and refined Chern–Simons theory are similar in many ways; for example, the Verlinde formula holds in both. Refined Chern–Simons theory gives rise to new topological invariants of Seifert three-manifolds and torus knots inside them. We conjecture that the invariants are certain indices on knot homology groups. For torus knots in S3 colored by fundamental representation, the index equals the Poincare polynomials of the knot homology theory categorifying the HOMFLY polynomial. As a byproduct, we show that our theory on S3 has a large-N dual which is the refined topological string on \({X=\mathcal{O}(-1) \oplus \mathcal{O}(-1) \rightarrow {\rm I\!P}^1}\) ; this supports the conjecture by Gukov, Schwarz and Vafa relating the spectrum of BPS states on X to SLN knot homology. We also provide a matrix model description of some amplitudes of the refined Chern–Simons theory on S3.

We give, using an explicit expression obtained in [V. Jones, Ann. of Math. 126, 335 (1987)], a basic hypergeometric representation of the HOMFLY polynomial of $(n,m)$ torus knots, and present a number of equivalent expressions, all related by Heine's transformations. Using this result the $(m,n)\leftrightarrow (n,m)$ symmetry and the leading polynomial at large $N$ are explicit. We show the latter to be the Wilson loop of 2d Yang-Mills theory on the plane. In addition, after taking one winding to infinity, it becomes the Wilson loop in the zero instanton sector of the 2d Yang-Mills theory, which is known to give averages of Wilson loops in $\mathcal{N}$=4 SYM theory. We also give, using matrix models, an interpretation of the HOMFLY polynomial and the corresponding Jones-Rosso representation in terms of $q$-harmonic oscillators., 17 pages, v2: More concise (published) version; typos corrected

Virtual knots are associated with knot diagrams, which are not obligatory planar. The recently suggested generalization from N=2 to arbitrary N of the Kauffman-Khovanov calculus of cycles in resolved diagrams can be straightforwardly applied to non-planar case. In simple examples we demonstrate that this construction preserves topological invariance -- thus implying the existence of HOMFLY extension of cabled Jones polynomials for virtual knots and links., Comment: 12 pages

We study the relations between the related crossings and the second and fourth coefficients of Conway polynomials of 2-adjacent knots, the relations between the crossings and the extreme coefficients of the related Homfly polynomials and explain how these coefficients determine the signs of the crossings. We give some interesting properties of the related Homfly polynomials. In fact, we show that 17 knots of the 19 exceptions given in A. Stoimenow's paper are not 2-adjacent.

Cochran defined the nth-order integral Alexander module of a knot in the three-sphere as the first homology group of the knot’s (n + 1)st iterated abelian cover. The case n = 0 gives the classical Alexander module (and polynomial). After a localization, one can obtain a finitely presented module over a principal ideal domain, from which one can extract a higher-order Alexander polynomial. We present an algorithm to compute the first-order Alexander module for any knot. As applications, we show that these higher-order Alexander polynomials provide a better bound on the knot genus than does the classical Alexander polynomial, and that they distinguish mutant knots. Included in this algorithm is a solution to the word problem in finitely presented -modules.

Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decompositions of a sphere, this modified polynomial satisfies the same duality identity as the original polynomial. We find that evaluating the Tutte-Krushkal-Renardy along a certain line gives the Bott polynomial. Finally we prove skein relations for the Tutte-Krushkal-Renardy polynomial.

We show that roots of log-concave Alexander knot polynomials are dense in C. This in particular implies that the log-concavity and Hoste's conjecture on the Alexander polynomial of alternating knots are (essentially) independent.

We give a general formula of the quantum sl2-invariant of a family of braid knots. To compute the quantum invariant of the links we use the Lie algebra g=sl2 in its standard two-dimensional representation. We also recover the Jones polynomial of these knots as a special case of this quantum invariant.

We study the expressions of p 1 − c ( L ) ( l ) , the coefficient of the lowest power of m of the HOMFLY polynomial of a link L with c ( L ) components, give what these coefficients are if they include exactly two terms for knots and the properties of p 1 − c ( L ) ( 1 ) for knots as well as properties of higher derivatives of p 1 − c ( L ) evaluate at i. We show that there are infinitely many different links (knots), especially infinitely many different pretzel links (knots) to have such coefficients and infinitely many different pretzel links (knots) to have same p 1 − c ( L ) ( l ) .

We show, using Mellit's recent results, that K\'alm\'an's full twist formula for the HOMFLY polynomial can be generalized to a formula for superpolynomials in the case of positive toric braids., Comment: 7 pages, 2 figures

In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus corrections to this formula in the form of expansion in powers of z = q-q^{-1}. Expansion coefficients are expressed through the eigenvalues of the cut-and-join operators, i.e. symmetric group characters. Moreover, the z-expansion is naturally exponentiated. Representation through cut-and-join operators makes contact with Hurwitz theory and its sophisticated integrability properties. Our formulas describe the shape of genus expansion for the HOMFLY polynomials, which for their matrix model counterparts is usually controlled by Virasoro like constraints and AMM/EO topological recursion. The genus expansion differs from the better studied weak coupling expansion at finite number of colors N, which is described in terms of the Vassiliev invariants and Kontsevich integral., Comment: 34 pages

We observe the twisted Alexander polynomial for metabelian representations of knot groups into SL 2 ( C ) and study relations to the characterizations of metabelian representations in the character varieties. We give a factorization of the twisted Alexander polynomial for irreducible metabelian representations with the adjoint action on sl 2 ( C ) , in which the Alexander polynomial and the twisted Alexander polynomial appear as factors. We also show several explicit examples.

Motivated by double crossover DNA polyhedra (He et al. in Nature 452:198, 2008; Lin et al. in Biochemistry 48:1663, 2009; Zhang et al. in J Am Chem Soc 131:1413, 2009; Zhang et al. in Proc Natl Acad Sci USA 105:10665, 2008; He et al. in Angew Chem Int Ed 49:748, )2010, in this paper, we construct a new type of link, called the double crossover link, formed by utilizing the “ $$n$$ -point star” to cover each vertex of a connected graph $$G$$ . The double crossover link can be used to characterize the topological properties of double crossover DNA polyhedra. We show that the Homfly polynomial of the double crossover link can be obtained from the chain polynomial of the truncated graph of $$G$$ with two distinct labels. As an application, by using computer algebra (Maple) techniques, the Homfly polynomial of a double crossover tetrahedral link is obtained. Our result may be used to characterize and analyze the topological structure of DNA polyhedra.

This paper extends the methodology for the construction of odd polyhedral links. Building blocks are odd chain tangles, each of which consists of finitely many $$2n+1$$ -twist tangles for any nonnegative integer $$n$$ . For any polyhedral graph $$G$$ , replacing each edge with an odd chain tangle results in an infinite collection of odd polyhedral links. The relationship between the HOMFLY polynomials of these odd links and the $$Q^{d}$$ -polynomial of $$G$$ is established. It leads to the determination of the span of the HOMFLY polynomial, the bound on the braid index and the genus of each odd link. Our results show that these indices depend not only on the building blocks but also on the graph $$G.$$

By making simple, heuristic assumptions, a new method based on the derivation of the Jones polynomial invariant of knot theory to tackle and quantify structural complexity of vortex filaments in ideal fluids is presented. First, we show that the topology of a vortex tangle made by knots and links can be described by means of the Jones polynomial expressed in terms of kinetic helicity. Then, for the sake of illustration, explicit calculations of the Jones polynomial for the left-handed and right-handed trefoil knot and for the Whitehead link via the figure-of-eight knot are considered. The resulting polynomials are thus function of the topology of the knot type and vortex circulation and we provide several examples of those. While this heuristic approach extends the use of helicity in terms of linking numbers to the much richer context of knot polynomials, it gives also rise to new interesting problems in mathematical physics and it offers new tools to perform real-time numerical diagnostics of complex flows.

It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known Morton-Frank-Williams inequality reveals a close relationship between the HOMFLY polynomial of a link and its braid index. In the case that a link is already presented in a closed braid form, Jaeger derived a special formulation of the HOMFLY polynomial. In this paper, we prove a variant of Jaeger's result as well as a dual version of it. Unlike Jaeger's original reasoning, which relies on representation theory, our proof uses only elementary geometric and combinatorial observations. Using our variant and its dual version, we provide a direct and elementary proof of the fact that the braid index of a link that has an $n$-string closed braid diagram that is also reduced and alternating, is exactly $n$. Until know this fact was only known as a consequence of a result due to Murasugi on fibered links that are star products of elementary torus links and of the fact that alternating braids are fibered., Substantial revisions: abstract changed relation of main theorems to earlier results better explained

This paper gives a connection between well chosen reductions of the Links-Gould invariants of oriented links and powers of the Alexander-Conway polynomial. We prove these formulas by showing the representations of the braid groups we derive the specialized Links-Gould polynomials from can be seen as exterior powers of copies of Burau representations., 19 pages

We give a new interpretation of the Alexander polynomial $\Delta_0$ for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, $\Delta_0$ determines the writhe polynomial of Cheng and Gao (equivalently, Kauffman's affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications., Comment: 22 pages, 19 figures

This chapter gives an expository account of some unexpected connections which have arisen over the last few years between Macdonald polynomials, invariants of torus knots, and lattice path combinatorics. The study of polynomial knot invariants is a well-known branch of topology which originated in the 1920s with the one-parameter Alexander polynomial [1]. In the early 1980s Jones [41] introduced a different one-parameter polynomial invariant, with important connections to physics. Shortly thereafter a number of authors more or less simultaneously discovered the HOMFLY polynomial, a two-parameter invariant which includes both the Alexander and Jones polynomials as special cases. The HOMFLY polynomial can be calculated recursively through skein relations. In the late 1980s Witten showed that the Jones polynomial and related invariants have an interpretation in terms of Chern–Simons theory, which is central to string theory. In 2006 Dunfield, Gukov, and Rasmussen [14] hypothesized the existence of a three-parameter knot invariant, now known as the “superpolynomial knot invariant” of a knot K, denoted \(\mathcal{P}_{K}(a,q,t)\), which includes the HOMFLY polynomial as a special case. Since then various authors proposed different possible definitions of the superpolynomial, which are conjecturally all equivalent. These definitions typically involve homology though, and are difficult to compute. In the case of torus knots an accepted definition of the superpolynomial has recently emerged from work of Aganagic and Shakirov [6, 7] (using refined Chern–Simons theory) and Cherednik [12] (using the double affine Hecke algebra). Gorsky and Negut [24] showed that these two different constructions yield the same three-parameter knot invariant which is now accepted as the definition of the superpolynomial for torus knots. These constructions involve symmetric functions in a set of variables X known as Macdonald polynomials, which also depend on two extra parameters q, t. These symmetric functions are important in algebraic combinatorics and other areas, and play a central role in various character formulas for S n modules connected to the Hilbert scheme from algebraic geometry. In particular, Haiman’s formula for the bigraded character of the space DH n of diagonal harmonics under the diagonal action of the symmetric group is expressed in terms of Macdonald polynomials. E. Gorsky [22], [47, Appendix] noticed that the coefficient of a j in the superpolynomial of the (n + 1, n) torus knot equals the bigraded multiplicity of a certain hook shape in the character of DH n . This polynomial is known as the (q, t)-Schroder polynomial since the author showed it can be expressed as a weighted sum over Schroder lattice paths in the n × n + 1 rectangle. Gorsky and Negut have shown that the coefficient of a j in the superpolynomial of the (m, n) torus knot can be viewed as the coefficient of a certain hook Schur function in a symmetric function expression involving Macdonald polynomials, and they have derived many explicit identities for this object. In addition Oblomkov, Rasmussen, and Shende [47] have introduced a conjectured extension of the q, t-Schroder polynomial to general (m, n) giving a positive strictly combinatorial expression for the superpolynomial of the (m, n) torus knot. This conjecture connects nicely with an important conjecture in algebraic combinatorics called the rational shuffle conjecture. In the following pages we will describe these developments in more detail.

This paper shows how the Formal Knot Theory state model for the Alexander-Conway polynomial is related to Knot Floer Homology. In particular we prove a parity result about the states in this model that clarifies certain relationships of the model with Knot Floer Homology., 10 pages, 4 figures, LaTeX document