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In this study, we introduce new criteria for a given link to detect the non- p -periodicity for a prime p ≥ 5. For this we use a congruence of the quantum N -invariant of periodic links. Our result is naturally consistent with Traczyk's criterion and Przytycki's criterion, and can be regarded as a generalization of these classical criteria. We shall give computational results of all the (alternating and non-alternating) knots of crossing number ≤ 1 6 for their non- p -periodicity (p ≥ 5). The computational results also show that our criteria has somewhat different nature with the classical criterion of Murasugi. [ABSTRACT FROM AUTHOR]

A spiral knot or link diagram (introduced in [C. Adams, R. Hudson, R. Morrison, W. George, L. Starkston, S. Taylor and O. Turanova, The spiral index of knots, Math. Proc. Cambridge Philos. Soc. 149(2) (2010) 297–315]) is an oriented knot or link diagram where, when traversing through the planar diagram, the curvature does not change sign. An oriented knot or link type is called curly if it admits a spiral diagram with fewer maxima than the braid index of that knot or link type. In this paper, we exhibit families of curly knots and links. [ABSTRACT FROM AUTHOR]

For a positive braid link, a link represented as a closed positive braid, we determine the first few coefficients of its HOMFLY polynomial in terms of geometric invariants such as, the maximum Euler characteristics, the number of split factors, and the number of prime factors. Our results give improvements of known results for the Conway and the Jones polynomial of positive braid links. In Appendix, we present a simpler proof of theorem of Cromwell, a positive braid diagram represents a composite link if and only if the diagram is composite. [ABSTRACT FROM AUTHOR]

Even trigonal bipyramid links (ETB links) are the mathematical models of DNA trigonal bipyramids such that each edge is composed of a twist tangle with 2n-crossing number (Lin et al. in J Mol Graph Model 95: 107492, 2020). In this paper, a basis of ETB links has been introduced as a set of trivial links obtained from any ETB link by respectively smoothing and switching each crossing, which is used to calculate the HOMFLY polynomial of each ETB link. Hence through establishing the algorithm for computing the HOMFLY polynomials of ETB links, an ETBL program has been developed in the Python programming language. As a result, the HOMFLY polynomials of 104 families of ETB links are given in terms of the parameter n. Based on these polynomials, we show that all ETB links are all chiral. Also, the structural property of 30 pairs of link families is further explored due to each pair of links having the same HOMFLY polynomial. These results provide a necessary theoretical support for predicting and controlling the topological structures of DNA trigonal bipyramids. [ABSTRACT FROM AUTHOR]

We give an obstruction for genus one knots K, K' to have Gordian distance one by using the 0th coefficient of the HOMFLY polynomials. As an application, we give a new constraint for a genus one knot to admit a (generalized) cosmetic crossing. Combining known results, we prove the (generalized) cosmetic crossing conjecture for genus one pretzel knots. [ABSTRACT FROM AUTHOR]

The interior polynomial is a Tutte-type invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to the maximal z -degree part of the HOMFLY polynomial of a naturally associated link. Note that the latter can be any oriented link. This result fits into a program aimed at deriving the HOMFLY polynomial from Floer homology. We also establish some other, more basic properties of the signed interior polynomial. For example, the HOMFLY polynomial of the mirror image of L is given by P L (− v − 1 , z). This implies a mirroring formula for the signed interior polynomial in the planar case. We prove that the same property holds for any bipartite graph and the same graph with all signs reversed. The proof relies on Ehrhart reciprocity applied to the so-called root polytope. We also establish formulas for the signed interior polynomial inspired by the knot theoretical notions of flyping and mutation. This leads to new identities for the original unsigned interior polynomial. [ABSTRACT FROM AUTHOR]

The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups F → and 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 F → and 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. [ABSTRACT FROM AUTHOR]

Abstract We provide a new proof of Jaeger's formula expressing the HOMFLY polynomial of a link presented in closed braid form, replacing the original representation theoretic proof with an easy combinatorial and geometric argument. Using new variants of Jaeger's result 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 now 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. [ABSTRACT FROM AUTHOR]

We give characterizations of the skein polynomial for links (as well as Jones and Alexander-Conway polynomials derivable from it), avoiding the usual 'smoothing of a crossing' move. As by-products, we have characterizations of these polynomials for knots and for links with any given number of components. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the properties of a knot K to be 2-adjacent to another knot W by analyzing their Conway polynomials, Jones polynomials and Homfly polynomials and give some very useful conditions. We discuss whether each pair of knots can be 2-adjacent to each other, i.e. whether 2-adjacency is a symmetric relation. We discuss also whether the trivial knot, the trefoil knot and the figure-eight knot can be 2-adjacent to any knot in Rolfsen's table and the opposite cases, except for 934 it is not decided whether it is 2-adjacent to 41. Finally, we give some examples to answer I. Torisu's problem partly and etc. [ABSTRACT FROM AUTHOR]

We give, using an explicit expression obtained in (Jones V, Ann Math 126:335, ), 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. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Brunnian links and braids are those that become trivial upon removing any of the components. It is well known that any link is the closure of some braid. However, a Brunnian link might not be the closure of any Brunnian braid. In this paper, we present two methods of constructing Brunnian links from Brunnian braids and show that our methods do result in Brunnian links that cannot be obtained as the closure of a Brunnian braid. [ABSTRACT FROM AUTHOR]

Motivated by double crossover DNA polyhedra (He et al. in Nature 452:198, ; Lin et al. in Biochemistry 48:1663, ; Zhang et al. in J Am Chem Soc 131:1413, ; Zhang et al. in Proc Natl Acad Sci USA 105:10665, ; He et al. in Angew Chem Int Ed 49:748, ), 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. [ABSTRACT FROM AUTHOR]

In this paper, an expression for the Homfly polynomial of a classical pretzel link with a fixed orientation was obtained. As an application, we proved that a (p, q, r)-pretzel knot is achiral if and only if {p,q,r} = {2,1,1}, {--2, --1, --1}, {2, --1,3}, {--2, 1, --3}, {1, --1, k}, {1, k, --2 -- k} or { -- 1, -- k, 2 + k}, where k is an odd integer. [ABSTRACT FROM AUTHOR]

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 $$G.$$ [ABSTRACT FROM AUTHOR]

In this paper, we first associate a plane graph with an oriented link via replacing each edge of the graph by an alternatingly oriented 2-tangle. Then we establish a relation between the Homfly polynomial of the associated oriented link and the dichromatic polynomial of the plane graph by assigning suitable edge weights. This result extends work of F. Jaeger and L. Traldi. [ABSTRACT FROM AUTHOR]

Some evaluations of the Gordian distance from a knot to another knot are given by using three polynomial invariants called the HOMFLY polynomial, the Jones polynomial and the Q-polynomial. Furthermore, Gordian distances between a lot of pairs of knots are determined. [ABSTRACT FROM AUTHOR]

Abstract: We claim that the Homfly polynomial (that is to say, Ocneanu''s trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin''s braid group. These bear a close connection to the Morton–Franks–Williams inequality. With respect to these structures, the set of positive, respectively negative permutation braids becomes an orthonormal basis. In the second case, many inner products can be geometrically interpreted through Legendrian fronts and rulings. [Copyright &y& Elsevier]

A general approach is proposed to elucidate the topological characteristics of molecules with the shape of polyhedral links. For an arbitrary polyhedral graph, four classes of polyhedral links can be obtained by applying the operation of ‘ X-tangle covering’ to the related reduced sets. The relationships between the W-polynomial of a polyhedral graph and the HOMFLY polynomial of each kind of polyhedral links are established. These relationships not only simplify the computation but also provide a method of constructing a general formula for the HOMFLY polynomial of polyhedral links. [ABSTRACT FROM AUTHOR]

We explain progress in computing the cabled Jones, HOMFLY and Kauffman polynomial. This is applied, first, in combination with some group theoretic considerations, to the tabulation of low-crossing mutants. Then we study the distinction of mutants, with particular regard to the symmetric mutants. We discuss the determination of braid index as another application of our computational methods. [ABSTRACT FROM AUTHOR]

Abstract: Recently S. Chmutov introduced a generalization of the dual of a ribbon graph (or equivalently an embedded graph) and proved a relation between Bollobás and Riordan’s ribbon graph polynomial of a ribbon graph and of its generalized duals. Here I show that the duality relation satisfied by the ribbon graph polynomial can be understood in terms of knot theory and I give a simple proof of the relation which used the homfly polynomial of a knot. [Copyright &y& Elsevier]

By using a tangle decomposition of a knot, we give a method for the construction of a knot with the lowest trivial HOMFLY coefficient polynomial. Applying this, we show that there exist infinitely many 2-bridge knots with such a coefficient polynomial. [ABSTRACT FROM AUTHOR]

Pairs of genus 2 mutant knots can have different Homfly polynomials, for example some 3-string satellites of Conway mutant pairs. We give examples which have different Kauffman 2-variable polynomials, answering a question raised by Dunfield et al. in their study of genus 2 mutants. While pairs of genus 2 mutant knots have the same Jones polynomial, given from the Homfly polynomial by setting v = s2, we give examples whose Homfly polynomials differ when v = s3. We also give examples which differ in a Vassiliev invariant of degree 7, in contrast to satellites of Conway mutant knots. [ABSTRACT FROM AUTHOR]

We exhibit an infinite family of knots that are detected chiral by the Kauffman polynomial but not by the HOMFLY polynomial. [ABSTRACT FROM AUTHOR]

The HOMFLY polynomial, which is an invariant of an oriented knot or link L, is the two-variable polynomial P(L; t, z) ∈ ℤ[t±1, z±1]. If we arrange the HOMFLY polynomial by the variable z, then the coefficient of zk is the polynomial of t, which is called the Pk polynomial. Let $P_{k}^{(n)} (L;1)$ be the nth derivative of the Pk polynomial of a link L evaluated at 1. It is a Vassiliev invariant of order less than or equal to max {n + k, 0}. Let K and K′ be oriented knots such that K′ is obtained from K by a Cn-move (n ≥ 3). We show that the value $P_{0}^{(n)} (K;1) - P_{0}^{(n)} (K';1)$ is 0 or ±n! · 2n by using Jacobi diagrams and moreover it is determined by the planarity of the Jacobi diagram corresponding to the Cn-move. By the result, we can decide the above value only by the knot diagrams of K and K′. [ABSTRACT FROM AUTHOR]

The Vassiliev conjecture states that the Vassiliev invariants are dense in the space of all numerical link invariants in the sense that any link invariant is a pointwise limit of Vassiliev invariants. In this article, we prove that the Vassiliev conjecture holds in the case of the coefficients of the HOMFLY and the Kauffman polynomials. [ABSTRACT FROM AUTHOR]