arXiv research

Satellite links with many twists have simpler companions.

The virtual genus of a virtual satellite link is equal to that of its companion.

In this paper we prove a Markov Theorem for virtual braids and for some analogs of this structure. The virtual braid group is the natural companion in the category of virtual knots, just as the Artin braid group is the natural companion to classical knots and links. In this paper we follow the L--move methods to prove …

New proof for minimizing tunnel systems in satellite chain links.

Links can be transformed into many others using a specific operation.

Essential surfaces in link diagrams on surfaces are crucial for understanding link properties.

Study satellite knots and their quandles related to incompressible tori.

Bing doubling is an operation which gives a satellite of a knot. It is also applied to a link by specifying a component of the link. We give a formula to compute the reduced colored Jones polynomial of a Bing double by using that of the companion. This formula enables us to compute a lot of examples of the reduced colo…

This paper defines a theory of cobordism for virtual knots and studies this theory for standard and rotational virtual knots and links. Non-trivial examples of virtual slice knots are given. Determinations of the four-ball genus of positive virtual knots are given using the results of a companion paper by the author an…

The study improves genus 1 bridge number bounds for satellite knots.

New satellite knots counter a conjecture about Lorenz knots.

Interpretable companion model for black-box classifiers.

Study geometrically measures to decide if modular companions are conformally equivalent.

In this article we complete the proof---for a broad class of four-manifolds---of Witten's conjecture that the Donaldson and Seiberg-Witten series coincide, at least through terms of degree less than or equal to c-2, where c is a linear combination of the Euler characteristic and signature of the four-manifold. This art…

Given a hyperelliptic Klein surface, we construct companion Klein bottles, extending our technique of companion tori already exploited by the authors in the genus 2 case. Bavard's short loops on such companion surfaces are studied in relation to the original surface so to improve a systolic inequality of Gromov's. A ba…

Given a hyperelliptic Klein surface, we construct companion Klein bottles. Bavard's short loops on companion bottles are studied in relation to the surface to improve an inequality of Gromov's in systolic geometry.

The study computes invariants of satellite knots using bordered Floer homology.

Thin position for knots in the 3-sphere was introduced by Gabai and has been used in a variety of contexts. We conjecture an analogue to a theorem of Schubert and Schultens concerning the bridge number of satellite knots. For a satellite knot K, we use the companion torus T to provide a lower bound for w(K), proving th…

In this work we establish the tightest lower bound up-to-date for the minimal crossing number of a satellite knot based on the minimal crossing number of the companion used to build the satellite. If $M$ is the wrapping number of the pattern knot, we essentially show that $c(Sat(P,C))>\frac{M^2}{2}c(C)$. The existence …

In the paper we prove the conjecture by Alexander Zupan that $w(K) \geqslant n^2w(J)$ where w denote the width and $K$ and $J$ are satellite knot and its companion with winding number $n$. Also we proved that for satellite knot with braid pattern, the equality holds.

We exploit the symmetry concepts developed in the companion review of this article to introduce a stochastic version of link reversal symmetry, which leads to an improved understanding of the reciprocity of directed networks. We apply our formalism to the international trade network and show that a strong embedding in …

We study the knot invariant called trunk, as defined by Ozawa, and the relation of the trunk of a satellite knot with the trunk of its companion knot. Our first result is ${\rm trunk}(K) \geq n \cdot {\rm trunk}(J)$ where ${\rm trunk}(\cdot)$ denotes the trunk of a knot, $K$ is a satellite knot with companion $J$, and …

The paper explores identifiability and stability in drifting fields using companion-elliptic kernels.

Study tunnel numbers of cable knots and their companions, proving new bounds and constructing examples.

Detect slopes in toroidal 3-manifolds to prove properties of fundamental groups.

We study D3-brane theories that are dually described as deformations of two different $\mathcal{N}=2$ superconformal theories with massless monopoles and dyons. These arise at the self-intersection of a seven-brane in F-theory, which cuts out a link on a small three-sphere surrounding the self-intersection. The spectru…

Investigates ropelength of complex knots and links.

Study satellite operators expanding concordance groups and their applications.

New satellite knots found that can't be represented by positive braids with full twists.

Researchers compute Khovanov polynomials for satellite knots.

We show that the crossing number of a satellite knot is at least 10^{-13} times the crossing number of its companion knot.

This article is the second of a pair of articles about the Goldman symplectic form on the PGL(V)-Hitchin component of a closed, connected, oriented, hyperbolic surface S. We show that any ideal triangulation on S and any compatible bridge system determine a symplectic trivialization of the tangent bundle to the PGL(V)-…

Let $k$ and $k'$ be two knots in 3-sphere. Say $k$ 1--dominates $k'$, if there is a proper degree 1 map $f\co E(k)\to E(k')$, between knot exterior of $k_i$. Theorem: Suppose that any companion of $k$ is prime. If $k$ 1--dominates $k'$ with the same Gromov volume, then $k'$ can be obtained from $k$ by finitely many de-…

A companion result of the the Tits alternative for $Out(F_n)$ is proved: Every solvable subgroup of $Out(F_n)$ is finitely generated and virtually abelian.

Lower bounds on rational slice genus using Heegaard Floer invariants.

New procedures connect braid charts, triplane diagrams, and braid movies for knotted surfaces.

Study on flat singular points of area-minimizing currents, defining a singularity degree.

Novel probabilistic solver speeds up solving related linear systems.

From a pseudo-triangulation with $n$ tetrahedra $T$ of an arbitrary closed orientable connected 3-manifold (for short, {\em a 3D-space}) $M^3$, we present a gem $J '$, inducing $\IS^3$, with the following characteristics: (a) its number of vertices is O(n); (b) it has a set of $p$ pairwise disjoint couples of vertices …

This paper treats the theory of Mukai duality on K3 surfaces from the differential geometric perspective, taylored to the need of the author's companion paper about Mukai duality of adiabatic coassociative K3 fibrations.

In this paper we construct quasiconformal embeddings from Y-pieces that contain a short boundary geodesic into degenerate ones. These results are used in a companion paper to study the Jacobian tori of Riemann surfaces that contain small simple closed geodesics.

These notes are a companion to the article "Lorentz spacetimes of constant curvature" by Geoffrey Mess, which was first written in 1990 but never published. Mess' paper will appear together with these notes in a forthcoming issue of Geometriae Dedicata.

In the present paper and the companion paper [9] a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on a complex algebraic varieties X is introduced, by sampling "temperature deformed" determinantal point processes. The main new ingredient is a large deviation principle for Gibbs…

Paper studies identifiability and stability of drifting fields in generative modeling.

We derive Verlinde's formula from the fixed point formula for loop groups proved in the companion paper "A fixed point formula for loop group actions", and extend it to compact, connected groups that are not necessarily simply-connected.

In this note, we explain that Ross-Thomas' result on the weighted Bergman kernels on orbifolds can be directly deduced from our previous result. This result plays an important role in the companion paper to prove an orbifold version of Donaldson Theorem.

This is the third installment in our series of articles (dg-ga/9712005, dg-ga/9710032) on the application of the PU(2) monopole equations to prove Witten's conjecture (hep-th/9411102) concerning the relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. The moduli space of solutions to t…

A holomorphic triple over a compact Riemann surface consists of two holomorphic vector bundles and a holomorphic map between them. After fixing the topological types of the bundles and a real parameter, there exist moduli spaces of stable holomorphic triples. In this paper we study non-emptiness, irreducibility, smooth…