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In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for 1 9 3 knots (Tables A.1). [ABSTRACT FROM AUTHOR]

The Jones polynomial and Khovanov homology of a classical link are invariants that depend upon an initial choice of orientation for the link. In this paper, we give a Khovanov homology theory for unoriented virtual links. The graded Euler characteristic of this homology is proportional to a similarly-defined unoriented Jones polynomial for virtual links, which is a new invariant in the category of non-classical virtual links. The unoriented Jones polynomial continues to satisfy an important property of the usual one: for classical or even virtual links, the unoriented Jones polynomial evaluated at one is two to the power of the number of components of the link. As part of extending the main results of this paper to non-classical virtual links, a new framework for computing integral Khovanov homology based upon arc labeled diagrams is described. This framework can be efficiently and effectively implemented on a computer. We define an unoriented Lee homology theory for virtual links based upon the unoriented version of Khovanov homology. [ABSTRACT FROM AUTHOR]

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we examine the properties of the Jones polynomial using dimensionality reduction learning techniques combined with ideas from topological data analysis. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce and describe a method for using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. In particular, this method provides a new approach for analyzing knot invariants using Principal Component Analysis. Using this approach on the Jones polynomial data, we find that it can be viewed as an approximately three-dimensional subspace, that this description is surprisingly stable with respect to the filtration by the crossing number, and that the results suggest further structures to be examined and understood. [ABSTRACT FROM AUTHOR]

We show that Khovanov link homology is trivial in a range of gradings and utilize relations between Khovanov and chromatic graph homology to determine extreme Khovanov groups and corresponding coefficients of the Jones polynomial. The extent to which chromatic homology and the chromatic polynomial can be used to compute integral Khovanov homology of a link depends on the maximal girth of its all-positive graphs. In this paper, we define the girth of a link, discuss relations to other knot invariants, and describe possible values for girth. Analyzing girth leads to a description of possible all-A state graphs of any given link; e.g., if a link has a diagram such that the girth of the corresponding all-A graph is equal to ℓ > 2 , then the girth of the link is equal to ℓ. [ABSTRACT FROM AUTHOR]

Champanerkar and Kofman [Twisting quasi-alternating links, Proc. Amer. Math. Soc.137(7) (2009) 2451–2458] introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as c. As an application, we give new examples of quasi-alternating knots of 13 and 14 crossings. Moreover, we prove that the Jones polynomial of a quasi-alternating link that is obtained in this way has no gap if the original link has no gap in its Jones polynomial. This supports a conjecture introduced in [N. Chbili and K. Qazaqzeh, On the Jones polynomial of quasi-alternating links, Topology Appl.264 (2019) 1–11], which states that the Jones polynomial of any prime quasi-alternating link except (2 , p) -torus links has no gap. [ABSTRACT FROM AUTHOR]

This paper resolves the problem of comparing the skein modules defined using the skein relations discovered by Melvin and Kirby that underlie the quantum group-based Reshetikhin–Turaev model for SU (2) Chern–Simons theory to the Kauffman bracket skein modules. Several applications and examples are presented. [ABSTRACT FROM AUTHOR]

In this paper we prove that the family of colored Jones polynomials of a knot in S3 determines the family of ADO polynomials of this knot. More precisely, we construct a two variables knot invariant unifying both the ADO and the colored Jones polynomials. On the one hand, the first variable q can be evaluated at 2r roots of unity with r 2 N and we obtain the ADO polynomial over the Alexander polynomial. On the other hand, the second variable A evaluated at A D qn gives the colored Jones polynomials. From this, we exhibit a map sending, for any knot, the family of colored Jones polynomials to the family of ADO polynomials. As a direct application of this fact, we will prove that every ADO polynomial is holonomic and is annihilated by the same polynomial as of the colored Jones function. The construction of the unified invariant will use completions of rings and algebra. We will also show how to recover our invariant from Habiro's quantum sl2 completion studied by Habiro in [J. Pure Appl. Algebra 211 (2007), 265-292], showing that it corresponds in fact to the two-variable colored Jones invariant defined by Habiro in [Invent. Math. 171 (2008), 1-81]. [ABSTRACT FROM AUTHOR]

In this paper, we adjust the Yang–Baxter operators constructed by Jones for the HOMFLYPT polynomial. Then we compute the second homology for this family of Yang–Baxter operators. It has the potential to yield 2-cocycle invariants for links. [ABSTRACT FROM AUTHOR]

It has been known that the coefficients of the series expansion of the Jones polynomial evaluated at ex are rational-valued Vassiliev invariants. In this paper, we calculate minimal multiplying factor, λ, needed for these rational-valued invariants to become integer-valued Vassiliev invariants. By doing that, we obtain a set of integer-valued Vassiliev invariants. [ABSTRACT FROM AUTHOR]

We investigate coincidences of the (one-variable) Jones polynomial amongst rational knots, what we call “Jones rational coincidences”. We provide moves on the continued fraction expansion of the associated rational which we prove do not change the Jones polynomial and conjecture (based on experimental evidence from all rational knots with determinant <900) that these moves are sufficient to generate all Jones rational coincidences. In the process we give a new formula for the Jones polynomial of a rational knot based on a continued fraction expansion of the associated rational, which has significantly fewer terms than other formulae known to us. The paper is based on the second author’s Ph.D. thesis and gives an essentially self-contained account. [ABSTRACT FROM AUTHOR]

We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 crossings. Following an approach of Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed their Jones polynomials by a divide-and-conquer method, and tested those with trivial Jones polynomials for unknottedness with the computer program SnapPy. We employed numerous novel strategies for reducing the computation time per knot diagram and the number of knot diagrams to be considered. That made computations up to 21 crossings possible on a single processor desktop computer. We explain these strategies in this paper. We also provide total numbers of algebraic tangles up to 18 crossings and of Conway polyhedra up to 22 vertices. We encountered new unknot diagrams with no crossing-reducing pass moves in our search. We report one such diagram in this paper. [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]

In (Compos. Math. 152(7): 1333–1384, 2016), Berest and Samuelson proposed a conjecture that the Kauffman bracket skein module of any knot in S 3 carries a natural action of a rank 1 double-affine Hecke algebra S H q , t 1 , t 2 depending on 3 parameters q , t 1 , t 2 . As a consequence, for a knot K satisfying this conjecture, we defined a three-variable polynomial invariant J n K (q , t 1 , t 2) generalizing the classical coloured Jones polynomials J n K (q) . In this paper, we give explicit formulas and provide a quantum group interpretation for the polynomials J n K (q , t 1 , t 2) . Our formulas generalize the so-called cyclotomic expansion of the classical Jones polynomials constructed by Habiro (Invent. Math. 171(1): 1–81, 2008) : as in the classical case, they imply the integrality of J n K (q , t 1 , t 2) and, in fact, make sense for an arbitrary knot K independent of whether or not it satisfies the conjecture of Berest and Samuelson (Compos. Math. 152(7): 1333–1384, 2016). When one of the Hecke deformation parameters is set to be 1, we show that the coefficients of the (generalized) cyclotomic expansion of J n K (q , t 1) are expressed in terms of Macdonald orthogonal polynomials. [ABSTRACT FROM AUTHOR]

This paper studies equivariant Jones polynomials of periodic links. Namely, to every -periodic link and any divisor of , we associate a polynomial that is a graded Euler characteristic of -graded equivariant Khovanov homology. The first main result shows that certain linear combinations of these polynomials, called the difference Jones polynomials, satisfy an appropriate version of the skein relation. This relation is used to generalize Przytycki's periodicity criterion. We also provide an example showing that the new criterion is indeed stronger. The second main result gives a state-sum formula for the difference Jones polynomials. This formula is used to give an alternative proof of the periodicity criterion of Murasugi. [ABSTRACT FROM AUTHOR]

It is well known that Kauffman constructed a state model of the Jones polynomial based on unoriented link diagrams. In his approach, in order to obtain Jones polynomial one needs to calculate both the writhe and the Kauffman bracket. Stimulated by a paper of Altintas (An oriented state model for the Jones polynomial and its applications to alternating links, Appl. Math. Comput.194 (2007) 168–178), in this paper we present a state sum model based on oriented link diagrams. In our approach, we succeed in adding the writhe to the state sum model and need not to compute the writher any more. We further show that, via our state sum model, Jones polynomial of any link (alternating or not) is a special parametrization of the dichromatic polynomial of a weighted graph with two different edge weights. [ABSTRACT FROM AUTHOR]

Let t i , j be the coefficient of x i y j in the Tutte polynomial T(G; x, y) of a connected bridgeless and loopless graph G with order v and size e. It is trivial that t 0 , e - v + 1 = 1 and t v - 1 , 0 = 1 . In this paper, we obtain expressions for another six extreme coefficients t i , j 's with (i , j) = (0 , e - v) , (0 , e - v - 1) , (v - 2 , 0) , (v - 3 , 0) , (1 , e - v) and (v - 2 , 1) in terms of small substructures of G. We also discuss their duality properties and their specializations to extreme coefficients of the Jones polynomial. [ABSTRACT FROM AUTHOR]

We frequently encounter knots in the flow of our daily life. Either we knot a tie or we tie a knot on our shoes. We can even see a fisherman knotting the rope of his boat. Of course, the knot as a mathematical model is not that simple. These are the reflections of knots embedded in three-dimensional space in our daily lives. In fact, the studies on knots are meant to create a complete classification of them. This has been achieved for a large number of knots today. But we cannot say that it has been terminated yet. There are various effective instruments while carrying out all these studies. One of these effective tools is graphs. Graphs are have made a great contribution to the development of algebraic topology. Along with this support, knot theory has taken an important place in low dimensional manifold topology. In 1984, Jones introduced a new polynomial for knots. The discovery of that polynomial opened a new era in knot theory. In a short time, this polynomial was defined by algebraic arguments and its combinatorial definition was made. The Jones polynomials of knot graphs and their applications were introduced by Murasugi. T. UÄŸur and A. Kopuzlu found an algorithm for the Jones polynomials of torus knots K(2, q) in 2006. In this paper, first of all, it has been obtained signed graphs of the twist knots which are a special family of knots. We subsequently compute the Jones polynomials for graphs of twist knots. We will consider signed graphs associated with each twist knot diagrams. [ABSTRACT FROM AUTHOR]

This paper explores the idea that within the framework of three-dimensional quantum gravity one can extend the notion of Feynman diagram to include the coupling of the particles in the diagram with quantum gravity. The paper concentrates on the non-trivial part of the gravitational response, which is to the large momenta propagating around a closed loop. By taking a limiting case one can give a simple geometric description of this gravitational response. This is calculated in detail for the example of a closed Feynman loop in the form of a trefoil knot. The results show that when the magnitude of the momentum passes a certain threshold value, non-trivial gravitational configurations of the knot play an important role. [ABSTRACT FROM AUTHOR]

We show that every alternating link of two components and crossings can be reduced by -moves to the trivial link or the Hopf link. It answers the question asked in one of the last papers by Slavik Jablan. [ABSTRACT FROM AUTHOR]

DAHA-Jones polynomials of torus knots T( r, s) are studied systematically for reduced root systems and in the case of $$C^\vee C_1$$ . We prove the polynomiality and evaluation conjectures from the author's previous paper on torus knots and extend the theory by the color exchange and further symmetries. The DAHA-Jones polynomials for $$C^\vee C_1$$ depend on five parameters. Their surprising connection to the DAHA-superpolynomials (type A) for the knots $$T(2p+1,2)$$ is obtained, a remarkable combination of the color exchange conditions and the author's duality conjecture (justified by Gorsky and Negut). The uncolored DAHA-superpolynomials of torus knots are expected to coincide with the Khovanov-Rozansky stable polynomials and the superpolynomials defined via rational DAHA and/or in terms of certain Hilbert schemes. We end the paper with certain arithmetic counterparts of DAHA-Jones polynomials for the absolute Galois group in the case of $$C^\vee C_1$$ , developing the author's previous results for $$A_1$$ . [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]

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]

The Tait conjecture states that alternating reduced diagrams of links in have the minimal number of crossings. It has been proved in 1987 by Thistlethwaite, Kauffman and Murasugi studying the Jones polynomial. In [A. Carrega, The Tait conjecture in , J. Knot Theory Ramifications 25(11) (2016) 1650063], the author proved an analogous result for alternating links in giving a complete answer to this problem. In this paper, we extend the result to alternating links in the connected sum of copies of . In and , the appropriate version of the statement is true for -homologically trivial links, and the proof also uses the Jones polynomial. Unfortunately, in the general case, the method provides just a partial result and we are not able to say if the appropriate statement is true. [ABSTRACT FROM AUTHOR]

Let D be a general connected reduced alternating link diagram, C be the set of crossings of D and C′ be the nonempty subset of C. In this paper we first define a multiple crossing-twisted link family { D( C′)| n=1,2,...} based on D and C′, which produces (2,2 n+1)-torus knot family, the link family A defined in Chang and Shrock (Physica A 301:196–218, ) and the pretzel link family P( n, n, n) as special cases. Then by applying Beraha-Kahane-Weiss’s Theorem we prove that limits of zeros of Jones polynomials of { D( C′)| n=1,2,...} are the unit circle | z|=1 (It is independent of the selections of D and C′) and several isolated limits, which can be determined by computing flow polynomials of subgraphs of G corresponding to D. Furthermore, we use the method of Brown and Hickman (Discrete Math. 242:17–30, ) to prove that, for any ε>0, all zeros of Jones polynomial of the link D( C) lie inside the circle | z|=1+ ε, provided that n is large enough. Our results extend results of F.Y. Wu, J. Wang, S.-C. Chang, R. Shrock and the present authors and refine partial result of A. Champanerkar and L. Kofman. [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]

We give an elementary construction of the Fibonacci model, a unitary braid group representation that is universal for quantum computation. This paper is dedicated to Professor C. N. Yang, on his 85th birthday. [ABSTRACT FROM AUTHOR]

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

We revisit the issue of the existence of infinitely many distinct prime knots with the same Alexander invariant. We present infinitely many distinct families, each family made up of infinitely many distinct knots. Within each family, the Alexander invariant is the same. Unlike, other examples in the literature, ours are elementary and based on a sub-collection of pretzel knots with three tassels. [ABSTRACT FROM AUTHOR]

The Tait conjecture states that reduced alternating diagrams of links in have the minimal number of crossings. It has been proved in 1987 by M. Thistlethwaite, L. H. Kauffman and K. Murasugi studying the Jones polynomial. In this paper, we prove an analogous result for alternating links in giving a complete answer to this problem. In we find a dichotomy: the appropriate version of the statement is true for -homologically trivial links, and our proof also uses the Jones polynomial. On the other hand, the statement is false for -homologically non-trivial links, for which the Jones polynomial vanishes. [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]

In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation. [ABSTRACT FROM PUBLISHER]

The slope conjecture [S. Garoufalidis, The degree of a q-holonomic sequence is a quadratic quasi-polynomial, Electron. J. Combin. 18 (2011) 4-27] gives a precise relation between the degree of the colored Jones polynomial of a knot and the boundary slopes of essential surfaces in the knot complement. In this paper, we propose a generalization of the slope conjecture to links. We prove the conjecture for all alternating or more generally adequate links. We also verify the conjecture for torus links. [ABSTRACT FROM AUTHOR]

Extending upon our previous work, we verify the Jones Unknot Conjecture for all knots up to 24 crossings. We describe the method of our approach and analyze the growth of the computational complexity of its different components. [ABSTRACT FROM AUTHOR]

In this paper, a method is given to calculate the Jones polynomial of the 6-plat presentations of knots by using a representation of the braid group 픹6 into a group of 5 × 5 matrices. We also can calculate the Jones polynomial of the 2n-plat presentations of knots by generalizing the method for the 6-plat presentations of knots. Also, it helps us to detect 3-bridge knots in 3-plat presentations. [ABSTRACT FROM AUTHOR]

The clasp number c(K) of a knot K is the minimum number of clasp singularities among all clasp disks bounded by K. It is known that the genus g(K) and the unknotting number u(K) are lower bounds of the clasp number, that is, {g(K), u(K)} ≤ c(K). Then it is natural to ask whether there exists a knot K such that {g(K), u(K)} < c(K). In this paper, we prove that there exists an infinite family of prime knots such that the question above is affirmative. [ABSTRACT FROM AUTHOR]

This essay is a remembrance of Vaughan Jones and a diagrammatic exposition of the remarkable breakthroughs in knot theory and low-dimensional topology that were catalyzed by his work. [ABSTRACT FROM AUTHOR]

This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with g boundary points and n crossings in the Kauffman bracket skein module is a linear combination of O(29) basis elements, with each coefficient a polynomial with at most n non-zero terms, each with integer coefficients, and that the link can be built one crossing at a time as a sequence of tangles with maximum number of boundary points bounded by C√n for some C. From this it follows that the computation of the Kauffman bracket of the link takes time and memory a polynomial in n times (2C√n. [ABSTRACT FROM AUTHOR]

A state of a virtual knot diagram D is a disjoint union of circles obtained from D by splicing all real crossings. For each integer n > 0, we denote by sn(D) the number of states of D consisting of n circles. The state number of a virtual knot K is the minimal number of sn(D) for all diagrams D of K and denoted by sn(K). The aim of this paper is to study sn(D) and sn(K). [ABSTRACT FROM AUTHOR]

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S3. In this paper, we define the Gordian complex of virtual knots by using forbidden moves. We show that for any virtual knot K0 and for any given natural number n, there exists a family of virtual knots {K0, K1, ..., Kn} such that for any pair (Ki, Kj) of distinct elements of the family, the Gordian distance of virtual knots by forbidden moves dF(Ki, Kj) = 1. [ABSTRACT FROM AUTHOR]

This paper contains general formulae for the reduced relative Tutte, Kauffman bracket and Jones polynomials of families of virtual knots and links given in Conway notation and discussion of a counterexample to the Z-move conjecture of Fenn, Kauffman and Manturov. [ABSTRACT FROM AUTHOR]

In the present paper we construct an equivalence between the set of graph-knots and the set of homotopy classes of looped graphs. Moreover, the graph-knot and the homotopy class constructed from a given knot are related by this equivalence. This equivalence is given by a simple formula. [ABSTRACT FROM AUTHOR]

In this paper, we first recall some known architectures of polyhedral links (Qiu and Zhai in J Mol Struct (THEOCHEM) 756:163-166, ; Yang and Qiu in MATCH Commun Math Comput Chem 58:635-646, ; Qiu et al. in Sci China Ser B Chem 51:13-18, ; Hu et al. in J Math Chem 46:592-603, ; Cheng et al. in MATCH Commun Math Comput Chem 62:115-130, ; Cheng et al. in MATCH Commun Math Comput Chem 63:115-130, ; Liu et al. in J Math Chem 48:439-456 ). Motivated by these architectures we introduce the notions of polyhedral links based on edge covering, vertex covering, and mixed edge and vertex covering, which include all polyhedral links in Qiu and Zhai (J Mol Struct (THEOCHEM) 756:163-166, ), Yang and Qiu (MATCH Commun Math Comput Chem 58:635-646, ), Qiu et al. (Sci China Ser B Chem 51:13-18, ), Hu et al. (J Math Chem 46:592-603, ), Cheng et al. (MATCH Commun Math Comput Chem 62:115-130, ), Cheng et al. (MATCH Commun Math Comput Chem 63:115-130, ), Liu et al. (J Math Chem 48:439-456, ) as special cases. The analysis of chirality of polyhedral links is very important in stereochemistry and the Jones polynomial is powerful in differentiating the chirality (Flapan in When topology meets chemistry. Cambridge Univ. Press, Cambridge, ). Then we give a detailed account of a result on the computation of the Jones polynomial of polyhedral links based on edge covering developed by Jin, Zhang, Dong and Tay (Electron. J. Comb. 17(1): R94, ) and, at the same time, by using this method we obtain some new computational results on polyhedral links of rational type and uniform polyhedral links with small edge covering units. These new computational results are helpful to judge the chirality of polyhedral links based on edge covering. Finally, we give some remarks and pose some problems for further study. [ABSTRACT FROM AUTHOR]

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form $J(n) =\sum_k c(n,k) \hat{J} (k)$ given a recursion relation for $(\hat{J} (n))$ and the hypergeometric kernel c(n, k). As an application of our method, we explicitly compute the non-commutative A-polynomial for twist knots with -15 and 15 crossings. The non-commutative A-polynomial of a knot encodes the monic, linear, minimal order q-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to q = 1 is conjectured to be the better-known A-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the A-polynomial is harder to compute and already unknown for some knots with 12 crossings. [ABSTRACT FROM AUTHOR]

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

This paper defines a new invariant of virtual knots and flat virtual knots. We study this invariant in two forms: the extended bracket invariant and the arrow polyomial. The extended bracket polynomial takes the form of a sum of virtual graphs with polynomial coefficients. The arrow polynomial is a polynomial with a finite number of variables for any given virtual knot or link. We show how the extended bracket polynomial can be used to detect non-classicality and to estimate virtual crossing number and genus for virtual knots and links. [ABSTRACT FROM AUTHOR]

The Jones polynomial of a knot in 3-space is a Laurent polynomial in q, with integer coefficients. Many people have pondered why this is so, and what a proper generalization of the Jones polynomial for knots in other closed 3-manifolds is. Our paper centers around this question. After reviewing several existing definitions of the Jones polynomial, we argue that the Jones polynomial is really an analytic function, in the sense of Habiro. Using this, we extend the holonomicity properties of the colored Jones function of a knot in 3-space to the case of a knot in an integer homology sphere, and we formulate an analogue of the AJ Conjecture. Our main tools are various integrality properties of topological quantum field theory invariants of links in 3-manifolds, manifested in Habiro's work on the colored Jones function. [ABSTRACT FROM AUTHOR]