Knot Polynomials Research Articles

An Analog of the Kauffman Bracket Polynomial for Knots in the Non-orientable Thickening of a Non-orientable Surface

An Analog of the Kauffman Bracket Polynomial for Knots in the Non-orientable Thickening of a Non-orientable Surface

Planar decomposition of the HOMFLY polynomial for bipartite knots and links

AbstractThe theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from $$\mathfrak {sl}(2)$$ sl ( 2 ) to $$\mathfrak {sl}(N)$$ sl ( N ) at arbitrary N – but for a special class of bipartite diagrams made entirely from the antiparallel lock tangle. Many amusing and important knots and links can be described in this way, from twist and double braid knots to the celebrated Kanenobu knots for even parameters – and for all of them the entire HOMFLY polynomials possess planar decomposition. This provides an approach to evaluation of HOMFLY polynomials, which is complementary to the arborescent calculus, and this opens a new direction to homological techniques, parallel to Khovanov–Rozansky generalizations of the Kauffman calculus. Moreover, this planar calculus is also applicable to other symmetric representations beyond the fundamental one, and to links which are not fully bipartite what is illustrated by examples of Kanenobu-like links.

Clasp pass moves and arrow polynomials of virtual knots

For classical knots, clasp pass moves are closely related to Vassiliev invariants of degree [Formula: see text]. Tsukamoto showed that the values of the Vassiliev invariant of degree [Formula: see text] induced from the Jones polynomial for two knots differ by [Formula: see text] or [Formula: see text], if they are related by a single clasp pass move. For virtual knots, the arrow polynomial is a generalization of the Jones polynomial and induces a Vassiliev invariant of degree [Formula: see text]. We show that the values of the Vassiliev invariant of degree [Formula: see text] induced from the arrow polynomial of two virtual knots differ by [Formula: see text] or [Formula: see text], if they are related by a single clasp pass move. We also obtain a lower bound of the distance between virtual knots by clasp pass moves.

Factors of HOMFLY polynomials

We study factorizations of HOMFLY polynomials of certain prime knots and oriented links. We begin with a computer analysis of prime knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for HOMFLY polynomials of oriented links associated to 2-connected plane graphs.

Quantum solution of the relationship between the 19-vertex model and the Jones polynomial

The challenge is to create an efficient quantum algorithm for the bosonic model capable of calculating the Jones polynomials for a knot resulting from interweaving or interlacing n-vertices. This weave is the construction of braid group representations from nineteen-vertex model. We present eigenbases and eigenvalues for lattice generators and their usefulness for the direct computation of Jones polynomials. The calculation shows that the Temperley-Lieb operators can be used for any braid word. Therefore, we propose a quantum sequence using these singular operators as quantum gates operating on the state of n qubits. We show that quantum calculations give the Jones polynomial for achiral knots and links.

N股Montesinos纽结的着色Jones多项式

N股Montesinos纽结的着色Jones多项式

A two-variable polynomial invariant of virtual knots

In this paper, we introduce a polynomial invariant of virtual knots by using weights of crossings. It is motivated from the writhe polynomial and the affine index polynomial for virtual knots. The new polynomial invariant is distinct from these polynomials and is a Vassiliev invariant of degree 1. By using the polynomial invariant we give a lower bound for the Gordian distance of two homotopic virtual knots.

Characterizations of pretzel knots which are simple-ribbon

We give a recursive formula for the Alexander polynomials of pretzel knots with a pair of integer parameters with opposite signs. Using the formula, we characterize certain pretzel knots which are simple-ribbon.

The 3-loop polynomial of knots obtained by plumbing the doubles of two knots

The 3-loop invariant (or, the 3-loop polynomial) of a knot is a rational form (or, a polynomial) presenting the 3-loop part of the Kontsevich invariant of knots. In this paper, we calculate the 3-loop polynomial of knots obtained by plumbing the doubles of two knots; this class of knots includes untwisted Whitehead doubles. We construct the 3-loop invariant by calculating the rational version of the Aarhus integral of a surgery presentation. As a consequence, we obtain an explicit presentation of the 3-loop polynomial for the knots.

The intersection polynomials of a virtual knot II: Connected sums

Although the connected sum of trivial knots is trivial in classical knot theory, there is a nontrivial connected sum of trivial knots in virtual knot theory. In this paper, we give connected sum formulae of the intersection polynomials of virtual knots introduced in the preceding paper [6]. As an application, we show that there are infinitely many connected sums of any pair of virtual knots.

On the Jones polynomial modulo primes

AbstractWe derive an upper bound on the density of Jones polynomials of knots modulo a prime number $p$ , within a sufficiently large degree range: $4/p^7$ . As an application, we classify knot Jones polynomials modulo two of span up to eight.

Negative amphichiral knots and the half-Conway polynomial

In 1979, Hartley and Kawauchi proved that the Conway polynomial of a strongly negative amphichiral knot factors as f(z)f(-z) . In this paper, we normalize the factor f(z) to define the half-Conway polynomial. First, we prove that the half-Conway polynomial satisfies an equivariant skein relation, giving the first feasible computational method, which we use to compute the half-Conway polynomial for knots with 12 or fewer crossings. This skein relation also leads to a diagrammatic interpretation of the degree-one coefficient, from which we obtain a lower bound on the equivariant unknotting number. Second, we completely characterize polynomials arising as half-Conway polynomials of knots in S^3 , answering a problem of Hartley–Kawauchi. As a special case, we construct the first examples of non-slice strongly negative amphichiral knots with determinant one, answering a question of Manolescu. The double branched covers of these knots provide potentially non-trivial torsion elements in the homology cobordism group.

A new framework for the Jones polynomial of fluid knots

A new framework for the Jones polynomial of fluid knots

Knot Theory in Linear Algebra: An Example with Trip Matrices

Summary The trip matrix, introduced by Louis Zulli [6], gives a method for computing the Jones polynomial of a knot using basic Linear Algebra. We use this method to provide an alternative proof of the formula for the Jones polynomial of torus knots. In doing so, we provide a partial solution to an open question related to finding an elementary proof of the formula for general torus knots.

The writhe and first intersection polynomials of virtual knots of supporting genus one

The supporting genus of a virtual knot is the minimal genus of all thickened surfaces where the virtual knot lies. In this paper, we study several polynomial invariants of a virtual knot of supporting genus one. We show that the first intersection polynomial of any virtual knot of supporting genus one is reciprocal. We also give a characterization of the writhe polynomial of a virtual knot of supporting genus one.

The mixed mock modularity of certain duals of generalized quantum modular forms of Hikami and Lovejoy

We resolve a question of Hikami and Lovejoy that sits at the intersection of quantum modular forms, mock modular forms, and knot theory. We express a family of Hecke–Appell-type sums of Hikami and Lovejoy in terms of mixed mock modular forms; in particular, we express the sums in terms of Appell functions and theta functions. They obtained their family of Hecke–Appell-type sums by considering certain duals of generalized quantum modular forms, where the quantum modular forms are motivated by the colored Jones polynomial for special torus knots.

On a generalization of Jones polynomial and its categorification for Legendrian knots

In this article, we explore a polynomial invariant for Legendrian knots which is a natural extension of Jones polynomial for (topological) knots. To this end, a new type of skein relation is introduced for the front projections of Legendrian knots. Further, we give a categorification of the polynomial invariant for Legendrian knots which is a natural extension of Khovanov homology. The Thurston-Bennequin invariant of Legendrian knot appears naturally in the construction of the homology as the grade shift.The constructions of the polynomial invariant and its categorification are natural in the sense that if we treat Legendrian knots as only knots (that is, we forget the geometry on the knots), then we recover the Jones polynomial and Khovanov homology respectively. In the end, we discuss strengths and limitations of these invariants.

K-theory of Jones polynomials

We recover the Jones polynomials of knots and links from the K-theory of a cluster C⁎-algebra of the sphere with two cusps. In particular, an interplay between the Chebyshev and Jones polynomials is studied.

Twist moves and the affine index polynomials of virtual knots

In this paper, we give a necessary condition for two virtual knots to be related by a finite sequence of twist moves by using the affine index polynomial, which is a Vassiliev invariant of degree [Formula: see text]. Trapp showed that a numerical Vassiliev invariant of degree [Formula: see text] has a polynomial growth of degree [Formula: see text] on a twist sequence of knots, which can be extended to a twist sequence of virtual knots. We calculate the growth of the affine index polynomial for a twist sequence of virtual knots and find the difference of the affine index polynomials of two virtual knots, which are related by a twist move. Moreover, we give a lower bound for the number of twist moves needed to transform [Formula: see text] to [Formula: see text] if [Formula: see text] and [Formula: see text] are virtual knots related by a finite sequence of twist moves.

Colored HOMFLY-PT polynomials of quasi-alternating 3-braid knots

Obtaining a closed form expression for the colored HOMFLY-PT polynomials of knots from 3-strand braids carrying arbitrary SU(N) representation is a challenging problem. In this paper, we confine our interest to twisted generalized hybrid weaving knots which we denote hereafter by Qˆ3(m1,−m2,n,ℓ). This family of knots not only generalizes the well-known class of weaving knots but also contains an infinite family of quasi-alternating knots. Interestingly, we obtain a closed form expression for the HOMFLY-PT polynomial of Qˆ3(m1,−m2,n,ℓ) using a modified version of the Reshitikhin-Turaev method. In addition, we compute the exact coefficients of the Jones polynomials and the Alexander polynomials of quasi-alternating knots Qˆ3(1,−1,n,±1). For these homologically-thin knots, such coefficients are known to be the ranks of their Khovanov and link Floer homologies, respectively. We also show that the asymptotic behavior of the coefficients of the Alexander polynomial is trapezoidal. On the other hand, we compute the [r]-colored HOMFLY-PT polynomials of quasi alternating knots for small values of r. Remarkably, the study of the determinants of certain twisted weaving knots leads to establish a connection with enumerative geometry related to mth Lucas numbers, denoted hereafter as Lm,2n. At the end, we verify that the reformulated invariants satisfy Ooguri-Vafa conjecture and we express certain BPS integers in terms of hyper-geometric functions F12[a,b,c;z].