arXiv research

Satellite knots can be trivialized by a single band move.

New satellite knots counter a conjecture about Lorenz knots.

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

The Gluck twist preserves the diffeomorphism type of certain satellite 2-knots.

Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set o…

Necessary and sufficient conditions are given for a satellite knot to be fibered. Any knot $\tilde k$ embeds in an unknotted solid torus $\tilde V$ with arbitrary winding number in such a way that no satellite knot with pattern $(\tilde V, \tilde k)$ is fibered. In particular, there exist nonfibered satellite knots wit…

A simplified proof for satellite knot signatures.

New findings on cosmetic surgeries for satellite knots.

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

Satellite knots have a higher trunk number than their base knots.

We give sufficient conditions for a satellite knot to admit an L-space surgery, and use this result to give new infinite families of patterns which produce satellite L-space knots.

Satellite formula connects knot concordance invariants to surgery.

Knot Floer homology is an invariant for knots in the three-sphere for which the Euler characteristic is the Alexander-Conway polynomial of the knot. The aim of this paper is to study this homology for a class of satellite knots, so as to see how a certain relation between the Alexander-Conway polynomials of the satelli…

New invariants help study satellite knots and their concordance.

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

Researchers compute Khovanov polynomials for satellite knots.

Formula for satellite operators using knot Floer homology.

Study satellite operations on knot invariant θ, proving additivity and distinguishing knots.

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…

Study satellite knots using bordered Floer theory, proving non-thinness and calculating genus.

New infinite-rank summand found in knot concordance group.

We investigate the ramifications of the Legendrian satellite construction on the relation of Lagrangian cobordism between Legendrian knots. Under a simple hypothesis, we construct a Lagrangian concordance between two Legendrian satellites by stacking up a sequence of elementary cobordisms. This construction narrows the…

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

New knots with specific properties have identical polynomial values.

We study the effect of satellite operations on the Upsilon invariant of Ozsvath-Stipsicz-Szabo. We obtain results concerning when a knot and its satellites are independent; for example, we show that the set $\{D_{2^i,1}\}_{i=1}^\infty$ is a basis for an infinite rank summand of the group of smooth concordance classes o…

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.

Study satellite knots and their quandles related to incompressible tori.

We use four dimensional techniques to derive general bounds on the $τ$ invariant of a satellite knot in $S^{3}$.

In this master thesis, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot $K$ with Alexander polynomial $Δ_K(t)$, I then use these lassos as patterns to construct families of satellite knots that have Alexander polynomial $Δ_K(t^d)$ where $d\in\mathbb{N}\cup \{0\}$. In …

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

New research shows that many slopes are characterizing for satellite knots.

We begin the systematic study of knot polynomials for the twist satellites of a knot, when its strand is substituted by a 2-strand twist knot. This is a generalization of cabling (torus satellites), when the substitute of the strand was a torus knot. We describe a general decomposition of satellite's colored HOMFLY in …

New knots found with non-trivial Steenrod operations on Khovanov homology.

In this paper we study Legendrian knots in the knot types of satellite knots. In particular, we classify Legendrian Whitehead patterns and learn a great deal about Legendrian braided patterns. We also show how the classification of Legendrian patterns can lead to a classification of the associated satellite knots if th…

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

Study of knots with generalized Mazur patterns and their invariants.

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…

New proof for minimizing tunnel systems in satellite chain links.

The genus of satellite tunnel number one knots and torti-rational knots is computed using the tools introduced by Floyd and Hatcher. An implementation of an algorithm is given to compute genus and slopes of minimal genus Seifert surfaces for such knots.

Mutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general their m-string satellites can have different Homfly polynomials for m>2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the…

Formulae for Rasmussen invariant of satellite knots with wrapping number 2 proved.

New method computes knot Floer homology for satellite knots.

We show that if K is a satellite knot which admits a generalized cosmetic crossing change of order q with |q| \geq 6, then K admits a pattern knot with a generalized cosmetic crossing change of the same order. As a consequence of this, we find that any prime satellite knot which admits a pattern knot that is fibered ca…

New examples show satellite operations can expand the concordance group in topological knot theory.

Formulas for tau and epsilon concordance invariants of braided satellite knots

Satellite operators generate infinite rank subgroups in knot concordance.

We generalize C. Van Cott's results on the $τ$ and $s$-invariants of cabled knots to apply to general satellite knots. This paper uses no Heegaard-Floer homology, relying on more geometric techniques.

We study satellites of Legendrian knots in R^3 and their relation to the Chekanov-Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in R^3 and augmentations of its DGA by showing that the DGA has finite-dimensional represe…