arXiv research

Classifies Legendrian Hopf links in lens spaces.

Classifies exceptional Legendrian realizations of Hopf link connected sums.

The paper computes lens spaces resulting from rational surgeries on Hopf links.

Study representation varieties of twisted Hopf links using combinatorial and Hodge theory.

It was shown by Jim Davis that a 2-component link with Alexander polynomial one is topologically concordant to the Hopf link. In this paper, we show that there is a 2-component link with Alexander polynomial one that has unknotted components and is not smoothly concordant to the Hopf link, answering a question of Jim D…

We give new Casson-Gordon style obstructions for a two-component link to be topologically concordant to the Hopf link.

We prove that any link in $S^3$ whose Khovanov homology is the same as that of a Hopf link must be isotopic to that Hopf link. This holds for both reduced and unreduced Khovanov homology, and with coefficients in either $\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z}$.

Classifies Legendrian Hopf links in lens spaces.

Authors show that genus defects of Hopf arborescent links are decidable.

Study the embedding space of a Hopf link in 3D and 3-manifolds.

Following the recent work by T.-H. Chan in [HOMFLY polynomial of some generalized Hopf links, J. Knot Theory Ramif. 9 (2000) 865--883] on reverse string parallels of the Hopf link we give an alternative approach to finding the Homfly polynomials of these links, based on the Homfly skein of the annulus. We establish tha…

Elliptic curve governs Hopf linking in symmetric tensegrity.

We classify transverse Hopf links in the standard contact 3-space up to transverse isotopy in terms of their components' self-linking number.

Following the recent work by Chan, and by Morton and Hadji on the Homflypt polynomials of some generalized Hopf links, we investigate the Kauffman polynomials of generalized Hopf links. By studying the Kauffman skein module of the solid torus S^1\times D^2, we establish a similar skein map on the Kauffman skein module …

We give new examples of 2-component links with linking number one and unknotted components that are topologically concordant to the positive Hopf link, but not smoothly so - in fact they are not smoothly concordant to the positive Hopf link with a knot tied in the first component. Such examples were previously construc…

Four-dimensional surgery is used to show that a two component link with Alexander polynomial one is topologically concordant to the Hopf link.

We completely classify Legendrian realisations of the Hopf link, up to coarse equivalence, in the 3-sphere with any contact structure.

Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.

We give infinitely many $2$-component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any $2$-component link with trivial Alexander polynomial. Our examples are pairwise non-concordant.

We compute the $SU(2)$ Casson-Lin invariant for the Hopf link and determine the sign in the formula of Harper and Saveliev relating this invariant to the linking number.

Study extends knot polynomials to links, identifying them with known invariants.

Stratifies representation varieties of twisted Hopf links.

Link groups can only have certain SU(2) representations.

Optimizes the conformal capacity of linked curves in $S^3$.

The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements Q_λ, depending on partitions λ. We show how to find the 2-variable Homfly invariant <λ…

A link in the 3-sphere is called (smoothly) slice if its components bound disjoint smoothly embedded disks in the 4-ball. More generally, given a 4-manifold M with a distinguished circle in its boundary, a link in the 3-sphere is called M-slice if its components bound in the 4-ball disjoint embedded copies of M. A 4-ma…

Quantum invariants for fibered links determined by genus and Hopf invariant.

Oleg Viro studied in arXiv:math/0204290 two interpretations of the (multivariable) Alexander polynomial as a quantum link invariant: either by considering the quasi triangular Hopf algebra associated to $U_q sl(2)$ at fourth roots of unity, or by considering the super Hopf algebra $U_q gl(1|1)$. In this paper, we show …

We construct and prove a diagrammatic version of the Duflo isomorphism between the invariant subalgebra of the symmetric algebra of a Lie algebra and the center of the universal enveloping algebra. This version implies the original for metrized Lie algebras (Lie algebras with an invariant non-degenerate bilinear form).…

The recently suggested tangle calculus for knot polynomials is intimately related to topological string considerations and can help to build the HOMFLY-PT invariants from the topological vertices. We discuss this interplay in the simplest example of the Hopf link and link $L_{8n8}$. It turns out that the resolved conif…

New upper bound for non-trivial links in a cube, geometric approach.

Using a power sum (boson) realization for the Macdonald operators, we investigate the Gukov, Iqbal, Kozcaz and Vafa (GIKV) proposal for the homological invariants of the colored Hopf link, which include Khovanov-Rozansky homology as a special case. We prove the polynomiality of the invariants obtained by GIKV's proposa…

Study shows rank of knot Floer homology detects Hopf links and classifies second smallest links.

Method detects intersections between ellipses for Borromean linking.

Alexander quandles fail to distinguish certain links, thus not detecting causality.

Researchers describe character varieties for Hopf links, proving geometric properties.

Invariants for 3-manifolds with embedded links using Hopf algebras.

Let $L$ be an oriented link with an alternating diagram $D$. It is known that $L$ is a fibered link if and only if the surface $R$ obtained by applying Seifert's algorithm to $D$ is a Hopf plumbing. Here, we call $R$ a Hopf plumbing if $R$ is obtained by successively plumbing finite number of Hopf bands to a disk. In t…

The Reshetikhin-Turaev invariant, Turaev's TQFT, and many related constructions rely on the encoding of certain tangles (n-string links, or ribbon n-handles) as n-forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hop…

Researchers compute invariants for knots and links in lens spaces using large N and k limits.

Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.

The Hopf invariant is linked to null-homotopy properties of maps.

New tile types for knots and links reduce complexity.

Investigates BNSR invariants of link and knot groups, proving specific properties.

Study links Hopf differentials to curvature line flows on time-like CMC surfaces.

Empirical evidence suggests link polynomials can detect causality in spacetimes.

New method produces positive colored superpolynomials from four-point functions.

We use the terms, knot product and local move, as defined in the text of the paper. Let $n$ be an integer$\geqq3$. Let $\mathcal S_n$ be the set of simple spherical $n$-knots in $S^{n+2}$. Let $m$ be an integer$\geqq4$. We prove that the map $j:\mathcal S_{2m}\to\mathcal S_{2m+4}$ is bijective, where $j(K)=K\otimes$Hop…