Link Polynomials Research Articles

U(N) torus link invariants in the large N limit from the matrix model approach

In this paper we study U(N) colored HOMFLY-PT polynomials of torus links in the double scaling limit (polynomial variable q→1, N→∞ keeping qN fixed). We show that, in this limit, the colored HOMFLY-PT polynomial of any (Lα,Lβ) torus link can be expressed in terms of the colored HOMFLY-PT polynomial of (L,L) torus link. Using the connection between matrix models and the Chern-Simons field theoretic invariants, we show that the colored torus link invariants are uniquely expressed in terms of connected correlation functions of operators in U(N) matrix model. We determine the leading and subleading contribution to some of the correlators at large N from the matrix model approach and find that they match exactly with those obtained from the corresponding colored HOMFLY-PT polynomials. Published by the American Physical Society 2024

A formula to evaluate type-A webs and link polynomials

A formula to evaluate type-A webs and link polynomials

On twist positive braids with the 2-variable link invariant: 3-braids of width 2

The possible [Formula: see text]-variable link polynomial [Formula: see text] for an oriented link [Formula: see text], which has width [Formula: see text] in the variable [Formula: see text] is studied, where width of [Formula: see text] is the minimal number of strands allowed by the index bound. It is shown that if the 2-variable link invariant [Formula: see text] for an oriented link [Formula: see text] has width [Formula: see text] in the variable [Formula: see text] then it is the same as the polynomial of a closed [Formula: see text]-braid, [Formula: see text] Also a complete list of [Formula: see text]- braids of width [Formula: see text] which close to knots, are given. Consequently it is shown that [Formula: see text] determines [Formula: see text] for the full [Formula: see text]-braid [Formula: see text] Finally, the 2-variable polynomial for the non-full 3-braid is calculated.

Evaluations of Link Polynomials and Recent Constructions in Heegaard Floer Theory

Using a definition of Euler characteristic for fractionally-graded complexes based on roots of unity, we show that the Euler characteristics of Dowlin's "$\mathfrak{sl}(n)$-like" Heegaard Floer knot invariants $HFK_n$ recover both Alexander polynomial evaluations and $\mathfrak{sl}(n)$ polynomial evaluations at certain roots of unity for links in $S^3$. We show that the equality of these evaluations can be viewed as the decategorified content of the conjectured spectral sequences relating $\mathfrak{sl}(n)$ homology and $HFK_n$.

HOMFLY-PT polynomials of open links

We numerically estimate the superposition of the HOMFLY-PT polynomial of an open two-component link, define its spread, and describe how this quantity may be employed to quantify the degree of entanglement of confined two component open links.

The one-row-colored 𝔰𝔩3 Jones polynomials for pretzel links

The colored [Formula: see text] Jones polynomials [Formula: see text] are given by a link and an [Formula: see text]-irreducible representation of [Formula: see text]. In general, it is hard to calculate [Formula: see text] for an oriented link [Formula: see text]. However, we calculate the one-row-colored [Formula: see text] Jones polynomials [Formula: see text] for three-parameter families of oriented pretzel links [Formula: see text] by using Kuperberg’s linear skein theory by setting [Formula: see text]. Furthermore, we show the existence of the tails of [Formula: see text] for the alternating pretzel knots [Formula: see text].

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.

A Note on the Jones Polynomials of 3-Braid Links

A Note on the Jones Polynomials of 3-Braid Links

New structures for colored HOMFLY-PT invariants

In this paper, we present several new structures for the colored HOMFLY-PT invariants of framed links. First, we prove the strong integrality property for the normalized colored HOMFLY-PT invariants by purely using the HOMFLY-PT skein theory developed by Morton and his collaborators. By this strong integrality property, we immediately obtain several symmetric properties for the full colored HOMFLY-PT invariants of links. Then we apply our results to refine the mathematical structures appearing in the Labastida-Mariño-Ooguri-Vafa (LMOV) integrality conjecture for framed links. As another application of the strong integrality, we obtain that the q = 1 and a = 1 specializations of the normalized colored HOMFLY-PT invariant are well-defined link polynomials. We find that a conjectural formula for the colored Alexander polynomial which is the a = 1 specialization of the normalized colored HOMFLY-PT invariant implies that a special case of the LMOV conjecture for framed knots holds.

The Jones Polynomials of the Special Alternative Links

The Jones Polynomials of the Special Alternative Links

Independence polynomials and Alexander-Conway polynomials of plumbing links

We use the Chudnovsky-Seymour Real Root Theorem for independence polynomials to obtain some statements about the coefficients and roots of the Alexander and Conway polynomial of some types of plumbing links, addressing conjectures of Fox, Hoste and Liechti.

On arrow polynomials of checkerboard colorable virtual links

In this paper, we give two new criteria of detecting the checkerboard colorability of virtual links by using the odd writhe and the arrow polynomial of virtual links, respectively. As a result, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings.

Application of braiding sequences IV: link polynomials and geometric invariants

Nous appliquons le concept de séquences de tressage aux polynômes de skein et de Conway, mais aussi à quelques invariants géométriques des entrelacs positifs. On donne des estimations pour l’indice des tresses et pour des invariants legendriens, en utilisant le degré et des propriétés de croissance des coefficients du polynôme de Conway. Nous énumérons asymptotiquement à une constante près les entrelacs alternants (et quelques autres) de genre donné par leur indice de tresses.

Local invariants of braiding quantum gates—associated link polynomials and entangling power

For a generic n-qubit system, local invariants under the action of characterize non-local properties of entanglement. In general, such properties are not immediately apparent and hard to construct. Here we consider two-qubit Yang–Baxter operators and show that their eigenvalues completely determine the non-local properties of the system. Moreover, we apply the Turaev procedure to these operators and obtain their associated link/knot polynomials. We also compute their entangling power and compare it with that of a generic two-qubit operator.

Do link polynomials detect causality in globally hyperbolic spacetimes?

Let X be a (2 + 1)-dimensional globally hyperbolic spacetime with a Cauchy surface Σ whose universal cover is homeomorphic to R2. We provide empirical evidence suggesting that the Jones polynomial detects causality in X. We introduce a new invariant of certain tangles related to the Conway polynomial and prove that the Conway polynomial does not detect the connected sum of two Hopf links among relevant three-component links, which suggests that the Conway polynomial does not detect causality in the scenario described.

2-Verma modules and the Khovanov–Rozansky link homologies

We explain how Queffelec–Sartori’s construction of the HOMFLY-PT link polynomial can be interpreted in terms of parabolic Verma modules for \({\mathfrak {gl}_{2n}}\). Lifting the construction to the world of categorification, we use parabolic 2-Verma modules to give a higher representation theory construction of Khovanov–Rozansky’s triply graded link homology.

Span of the Jones polynomials of certain v-adequate virtual links

It is known that the Kauffman–Murasugi–Thislethwaite type inequality becomes an equality for any (possibly virtual) adequate link diagram. We refine this condition. As an application we obtain a criterion for virtual link diagram with exactly one virtual crossing to represent a properly virtual link.

A closed formula for the evaluation of foams

We give a purely combinatorial formula for evaluating closed, decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral, equivariant version of colored Khovanov–Rozansky link homology categorifying the \mathfrak {sl}_N link polynomial. We also provide connections to the equivariant cohomology rings of partial flag varieties.

Knot polynomials of open and closed curves.

In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.

Twisted Alexander polynomials of torus links

In this paper, we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the [Formula: see text]-character variety. We also discuss similar things for the higher-dimensional twisted Alexander polynomial and the Reidemeister torsion.