arXiv research

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

Stratifies representation varieties of twisted Hopf links.

It follows from earlier work of Silver-Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexand…

A virtual link can be understood as a link in a trivial I-bundle over an orientable compact surface with genus. A twisted virtual link is a link in a trivial I-bundle over a not-necessarily orientable compact surface. A twisted virtual birack is an algebraic structure with axioms derived from the twisted virtual Reidem…

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

New algebraic structure derived from Hopf algebra and Drinfel'd twist.

New tile types for knots and links reduce complexity.

Proves Poincaré duality for Hopf algebroids with bijective antipode.

This paper classifies fibered links in 3-sphere using open book decompositions.

The full 6d Hopf-Wess-Zumino term in the action functional for the M5-brane is anomalous as traditionally defined. What has been missing is a condition implying the higher analogue of level quantization familiar from the 2d Wess-Zumino term. We prove that the anomaly cancellation condition is implied by the hypothesis …

Classifies Legendrian Hopf links in lens spaces.

The paper generalizes Kuperberg invariants using twisted Drinfeld doubles.

Classifies exceptional Legendrian realizations of Hopf link connected sums.

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

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.

Theory of packing diabolic domains in liquid crystals.

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.

The paper extends the Hopf differential concept to associative submanifolds in G2-manifolds.

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 …

The paper sets genus bounds for twisted quantum invariants.

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.

Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.

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

Quantum invariant derived from ternary cohomology of self-distributive structures.

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.

The Hopf fibration has inspired any number of geometric structures in physical systems, in particular in chiral liquid crystalline materials. Because the Hopf fibration lives on the three sphere, $\mathbb{S}^3$, some method of projection or distortion must be employed to realize textures in flat space. Here, we explore…

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

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

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

The study finds hyperbolic twisted torus links for certain twists.

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…

The paper introduces colorings and invariants for twisted links and shows how double coverings can be equivalent.

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.

New deformation of link homology for colored diagrams.

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 …

The paper classifies twisted torus links that are unlinks.

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).…

Satellite links with many twists have simpler companions.