arXiv research

In this paper, we construct two families of satellite constructions for Brunnian links, called the satellite sum and the satellite tie. An interesting fact is that by applying the satellite sum and the satellite tie constructions, we can build infinitely many new Brunnian links from any given Brunnian links. With the h…

Satellite links with many twists have simpler companions.

Satellite links of fully positive braids are characterized.

New proof for minimizing tunnel systems in satellite chain links.

New links identified with Alexander polynomial signs to detect satellite links.

Formula for satellite operators using knot Floer homology.

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

Starfield optimizes satellite links for LEO mega-constellations by aligning ISLs with regional traffic patterns.

The paper classifies hyperbolic and satellite T-links formed by twisting.

In a recent work "Arc-presentation of links: Monotonic simplification" Ivan Dynnikov showed that each rectangular diagram of the unknot, composite link, or split link can be monotonically simplified into a trivial, composite, or split diagram, respectively. The following natural question arises: Is it always possible t…

Formula calculates linking numbers in knot theory.

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

Given an $n$-component link $L$ in any 3-manifold $M$, the space $\mathcal{L} \subset (\mathbb{Q}\cup \mkern-1.5mu\{\infty\})^n$ of rational surgery slopes yielding L-spaces is already fully characterized (in joint work by the author) when $n\!=\!1$ and $\mathcal{L}$ is nontrivial. For $n\mkern-2mu>\mkern-3mu1$, howeve…

Satellite constructions on a knot can be thought of as taking some strands of a knot and then tying in another knot. Using satellite constructions one can construct many distinct isotopy classes of knots. Pushing this further one can construct distinct concordance classes of knots which preserve some algebraic invarian…

This study analyzes satellite communication latency using a stochastic geometry model.

Satellite operations with winding number ≠ 1 are not homomorphisms.

Study satellite knots and their quandles related to incompressible tori.

New satellite knots counter a conjecture about Lorenz knots.

Let P be a knot in an unknotted solid torus (i.e. a satellite operator or pattern), K a knot in S^3 and P(K) the satellite of K with pattern P. For any satellite operator P, this correspondence gives a function P : C -> C on the set of smooth concordance classes of knots. We give examples of winding number one satellit…

The paper constructs exotic surface links in 4-ball, proving their Brunnian nature.

New Milnor's invariant condition for topologically slice links.

We give a new proof that the Levine-Tristram signatures of a link give lower bounds for the minimal sum of the genera of a collection of oriented, locally flat, disjointly embedded surfaces that the link can bound in the 4-ball. We call this minimal sum the 4-genus of the link. We also extend a theorem of Cochran, Frie…

We provide a diagrammatic criterion for semi-adequate links to be hyperbolic. We also give a conjectural description of the satellite structures of semi-adequate links. One application of our result is that the closures of sufficiently complicated positive braids are hyperbolic links.

Study shows links with unbounded annular Khovanov gradings but bounded Floer gradings.

We examine the Legendrian analogue of the topological satellite construction for knots, and deduce some results for specific Legendrian knots and links in standard contact three-space and the solid torus. In particular, we show that the Chekanov-Eliashberg contact homology invariants of Legendrian Whitehead doubles of …

Links with minimum tunnel number have one less component than their number of parts.

Formula for Heegaard Floer multicurves of double tangles from knot complements.

For each three-bridge link of a certain form, we construct a taut Seifert surface for the link and establish whether the link is fibred. Using this, we also give the genus and fibredness of satellite knots whose pattern is constructed from a two-component two-bridge link in the case not addressed by work of Hirasawa an…

For an oriented surface link $S$, we can take a satellite construction called a 2-dimensional braid over $S$, which is a surface link in the form of a covering over $S$. We demonstrate that 2-dimensional braids over surface links are useful for showing the distinctness of surface links. We investigate non-trivial examp…

For an oriented surface link $F$ in $\mathbb{R}^4$, we consider a satellite construction of a surface link, called a 2-dimensional braid over $F$, which is in the form of a covering over $F$. We introduce the notion of an $m$-chart on a surface diagram $π(F)\subset \mathbb{R}^3$ of $F$, which is a finite graph on $π(F)…

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

We prove that links with meridional rank 3 whose 2-fold branched covers are graph manifolds are 3-bridge links. This gives a partial answer to a question by S. Cappell and J. Shaneson on the relation between the bridge numbers and meridional ranks of links. To prove this, we also show that the meridional rank of any sa…

We show that if $K$ is a nontrivial knot then the proportion of satellites of $K$ among all of the prime non-split links of $n$ or fewer crossings does not converge to $0$ as $n$ approaches infinity. This implies in particular that the proportion of hyperbolic links among all of the prime non-split links of $n$ or fewe…

We show that a link in an open book can be realized as a strongly quasipositive braid if and only if it bounds a Legendrian ribbon with respect to the associated contact structure. This generalizes a result due to Baader and Ishikawa for links in the three-sphere. We highlight some related techniques for determining wh…

Investigates ropelength of complex knots and links.

Introduces slice knots and concordance, linking to exotic smooth structures.

New findings on T-links derived from torus links.

We show that the fundamental quandle defines a functor from the oriented tangle category to a suitably defined quandle category. Given a tangle decomposition of a link $L$, the fundamental quandle of $L$ may be obtained from the fundamental quandles of tangles. We apply this result to derive a presentation of the funda…

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…

We show that the Kakimizu complex of a knot may be locally infinite, answering a question of Przytycki--Schultens. We then prove that if a link $L$ only has connected Seifert surfaces and has a locally infinite Kakimizu complex then $L$ is a satellite of either a torus knot, a cable knot or a connected sum, with windin…

Call a smooth knot (or smooth link) in the unit sphere in $\mathbb{C}^2$ analytic (respectively, smoothly analytic) if it bounds a complex curve (respectively, a smooth complex curve) in the complex ball. Let $K$ be a smoothly analytic knot. For a small tubular neighbourhood of $K$ we give a sharp lower bound for the 4…

For pattern knots admitting genus-one bordered Heegaard diagrams, we show the knot Floer chain complexes of the corresponding satellite knots can be computed using immersed curves. This, in particular, gives a convenient way to compute the $τ$-invariant. For patterns $P$ obtained from two-bridge links $b(p,q)$, we deri…

Study of Lorenz links and T-links, showing equivalence and unique presentations.

Let P be a knot in a solid torus, K a knot in 3-space and P(K) the satellite knot of K with pattern P. This defines an operator on the set of knot types and induces a satellite operator P:C--> C on the set of smooth concordance classes of knots. There has been considerable interest in whether certain such functions are…

Differential graded algebra invariants are constructed for Legendrian links in the 1-jet space of the circle. In parallel to the theory for R^3, Poincare-Chekanov polynomials and characteristic algebras can be associated to such links. The theory is applied to distinguish various knots, as well as links that are closur…

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

Formula for τ-invariant of satellite knots derived from L-space satellite operators.

The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p,p+1)-torus knots, and show how composing and taking satellites can be used to obtain t…