On the Jones polynomial of quasi-alternating links

Mediterranean Journal of Mathematics, 2022

We prove that the length of any gap in the differential grading of the Khovanov homology of any quasi-alternating link is one. As a consequence, we obtain that the length of any gap in the Jones polynomial of any such link is one. This establishes a weaker version of Conjecture 2.3 in [5]. Moreover, we obtain a lower bound for the determinant of any such link in terms of the breadth of its Jones polynomial. This establishes a weaker version of Conjecture 3.8 in [16]. The main tool in obtaining this result is proving the Knight Move Conjecture [2] for the class of quasi-alternating links.

Journal of Mathematical Chemistry, 2011

In this paper, we first recall some known architectures of polyhedral links . 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 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 . 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 the present authors in 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 of polyhedral links based on edge covering. Finally, we give some remarks and pose some problems for further study. We introduce the notions of polyhedral links based on edge covering, vertex covering, and mixed edge and vertex covering. These new architectures include polyhedral links in [10-16] as special cases. We hope that these new architectures will become the potential synthetical objects of chemists and biologists. Then we review a general method on the computation of the Jones polynomial of polyhedral links based on the edge covering developed by the present authors in , at the same time, by using this method to 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 open problems for further study.

Kodai Mathematical Journal, 2016

We give an explicit formula for the Jones polynomial of any rational link in terms of the denominators of the canonical continued fraction of the slope of the given rational link.

Topology and its Applications, 1992

Almost alternating links, Topology and its Applications 46 (1992) 151-165. We introduce the category of almost alternating links: nonalternating links which have a projection for which one crossing change yields an alternating projection. We extend this category to m-almost alternating links which require m crossing changes to yield an alternating projection. We show that all but five of the nonalternating knots up through eleven crossings and links up through ten crossings are almost alternating. We also prove that a prime almost alternating knot is either a hyperbolic knot or a torus knot. We then obtain a bound on the span of the bracket polynomial for m-almost alternating links and discuss applications.

2010

Garoufalidis conjectured a relation between the boundary slopes of a knot and its colored Jones polynomials. According to the conjecture, certain boundary slopes are detected by the sequence of degrees of the colored Jones polynomials. We verify this conjecture for adequate knots, a class that vastly generalizes that of alternating knots.

arXiv (Cornell University), 2024

We prove that the Jones diameter of a link is twice its crossing number whenever the breadth of its Jones polynomial equals the difference between the crossing number and the Turaev genus. This implies that such link is adequate, as per the characterization provided in [5, Theorem 1.1]. By combining this with the result in [1, Theorem 3.2], we obtain a characterization of adequate links using these numerical link invariants. As an application, we provide a criterion to obstruct a link from being quasi-alternating. Furthermore, we establish a lower bound for the crossing number of certain classes of links, aiding in determining the crossing number of the link in specific cases.

Topology, 2003

For each k ¿ 2, we exhibit inÿnite families of prime k-component links with Jones polynomial equal to that of the k-component unlink. ?

Algebraic & Geometric Topology, 2006

We show that if {L n } is any infinite sequence of links with twist number τ (L n ) and with cyclotomic Jones polynomials of increasing span, then lim sup τ (L n ) = ∞. This implies that any infinite sequence of prime alternating links with cyclotomic Jones polynomials must have unbounded hyperbolic volume. The main tool is the multivariable twist-bracket polynomial, which generalizes the Kauffman bracket to link diagrams with open twist sites.

2010

This article contains general formulas for the Tutte and Jones polynomials of families of knots and links given in Conway notation and the corresponding plots of zeroes for the Jones polynomials.

Proceedings of the American Mathematical Society, 2016

We show that the head and tail functions of the colored Jones polynomial of adequate links are the product of head and tail functions of the colored Jones polynomial of alternating links that can be read-off an adequate diagram of the link. We apply this to strengthen a theorem of Kalfagianni, Futer and Purcell on the fiberedness of adequate links.

Pacific Journal of Mathematics, 1992

Exchange moves were introduced in an earlier paper by the same authors. They take one closed n-braid representative of a link to another, and can lead to examples where there are infinitely many conjugacy classes of n-braids representing a single link type. THEOREM I. If a link type has infinitely many conjugacy classes of closed n-braid representatives, then n > 4 and the infinitely many classes divide into finitely many equivalence classes under the equivalence relation generated by exchange moves. This theorem is the last of the preliminary steps in the authors' program for the development of a calculus on links in S 3. THEOREM 2. Choose integers n, g > 1. Then there are at most finitely many link types with braid index n and genus g.

10TH INTERNATIONAL CONFERENCE ON APPLIED SCIENCE AND TECHNOLOGY

The Jones polynomial of an alternating link is known to have no gap of length greater than 1. This result extends to quasi-alternating links as well. Our purpose is to study the structure, in particular the number of gaps, of the Jones polynomial of an arbitrary link with the ultimate aim of characterizing Laurent polynomials which arise as the Jones polynomial of a link.

arXiv (Cornell University), 2022

We apply the twisting technique that was first introduced in [1] and later generalized in [9] to obtain an infinite family of adequate, homogeneous or alternative links from a given adequate, homogeneous or alternative link, respectively. Thus we conclude that these three classes of links are infinite. 1. introduction Many classes of links have been defined in trying to generalize the class of alternating links. In particular, Lickorish and Thistlewaite in [8] introduced the class of adequate links and Cromwell in [2] introduced the class of homogeneous links. Later, Kauffman in [5] introduced the class of alternative links. These classes of links are natural generalization of alternating links each in its own aspect. We recall the definition of these classes for the sake of making this paper more self-contained. First, we recall the definition of adequate links as in [8]. Let L be a link of a diagram D with crossings c 1 , c 2 ,. .. , c n. A state of the diagram D is a function s : {c 1 , c 2 ,. .. , c n } → {1, −1}, and the state diagram sD is the diagram obtained from the diagram D after smoothing all of its crossings according to the state s. We apply the A-or B-smoothing to the crossing c i if s(c i) = 1 or s(c i) = −1, respectively according to the scheme in Figure 1. Note that sD consists of simple closed curves and the number of components will be denoted by |sD|. There are two special state diagrams s A D and s B D. The first one is obtained when s A (c i) = 1 and the second one is obtained when s B (c i) = −1 for all i such that 1 ≤ i ≤ n. The link diagram is A-adequate if |s A D| > |sD| for every state diagram sD with n i=1 s(c i) = n − 2 and is Badequate if |s B D| > |sD| for every state diagram sD with n i=1 s(c i) = 2−n. This is equivalent to say that the two arcs after smoothing each crossing in s A D or s B D belong to two different components, respectively. The link diagram is adequate if it is A-and B-adequate at the same time and the link is adequate if it admits a diagram that is adequate. Figure 1 below shows a link diagram at the crossing c with sign(c) = 1 and both of its smoothings, A-and B-smoothings, respectively. Thistlewaite in [12] proved that any adequate diagram of an adequate link has the minimal crossing number among all diagrams that represent the same link using the Kauffman polynomial. The same result is obtained using the Jones polynomial [8]. Moreover, the result of [12] implies that the minimal crossing diagram of an adequate link is also adequate. Second, we recall the definition of homogeneous links as in [2]. Seifert in [10] defined an algorithm for constructing an orientable surface spanning the link L from its oriented diagram Date: 22/11/2022. 2020 Mathematics Subject Classification. 57K10, 57K14.

Applied Mathematics and Computation, 2007

In this paper, we define a polynomial invariant of regular isotopy, G L , for oriented knot and link diagrams L. From G L by multiplying it by a normalizing factor, we obtain an ambient isotopy invariant, N L , for oriented knots and links. We compare the polynomial N L with the original Jones polynomial and with the normalized bracket polynomial. We show that the polynomial N L yields the Jones polynomial and the normalized bracket polynomial. As examples, we give the polynomial G L of some knot and link diagrams and compute the polynomial G L for torus links of type (2, n), and applying computer algebra (MAPLE) techniques, we calculate the polynomial G L of torus links of type (2, n). Furthermore we give its applications to alternating links.

After reconsidering the Dasbach-Hougardy counterexample to the Kauffman conjecture on alternating knots, we reformulate the conjecture and prove this reformulated conjecture for alternating links as the Mirror Theorem.