arXiv research

Study of knots with generalized Mazur patterns and their invariants.

Study shows knot Floer homology inequalities for specific knots.

New infinite-rank summand found in knot concordance group.

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

Partial proof of a conjecture about knot concordance maps.

Formulas for tau and epsilon concordance invariants of braided satellite knots

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…

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…

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…

New knots not slice in rational 4-balls found.

Given a class of objects, a pattern theorem is a powerful result describing their structure. We show that alternating knots exhibit a pattern theorem, and use this result to prove a long-standing conjecture that alternating knots grow rare. This is currently the best possible analogue of a pair of theorems on alternati…

New method computes knot Floer homology for satellite knots.

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

Study on bounds of knot untangling for specific types of knots.

Satellite knots have a higher trunk number than their base 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 complexity measure for shake-slice knots established.

New knot polynomials reveal patterns and mutations.

Given a knot $K$ in $S^3$, a question raised by Cappell and Shaneson asks if the meridional rank of $K$ equals the bridge number of $K$. Using augmentations in knot contact homology we consider the persistence of equality between these two invariants under satellite operations on $K$ with a braid pattern. In particular…

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 …

The paper explores methods to construct knots with diffeomorphic $0$-traces.

Study knot groups from disc patterns in 3D space.

New game defined on origami patterns, linking number introduced.

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.

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

New method connects knot Floer homology with bordered Floer homology.

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

The paper develops finite knot theory using ropelength-filtered Reidemeister graphs.

Let K' be a knot that admits no cosmetic crossing changes and let C be a non-trivial, prime, non-cable knot. Then any knot that is a satellite of C with winding number zero and pattern K' admits no cosmetic crossing changes. As a consequence we prove the nugatory crossing conjecture for Whitehead doubles of prime, non-…

Deep learning uncovers patterns between knot types.

The paper constructs contractible manifolds with knotted spheres.

New satellite knots counter a conjecture about Lorenz knots.

New knots with specific properties have identical polynomial values.

Improved bounds for knot crossings in different mosaic patterns.

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 prove that for any winding number $m>0$ pattern $P$ and winding number $-m$ pattern $Q$, there exist knots $K$ such that the minimal genus of a cobordism between $P(K)$ and $Q(K)$ is arbitrarily large. This answers a question posed by Cochran-Harvey [CH17] and generalizes a result of Kim-Livingston [KL05].

Researchers compute Khovanov polynomials for satellite knots.

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…

Let $P(K)$ be a satellite knot where the pattern, $P$, is a Berge-Gabai knot (i.e., a knot in the solid torus with a non-trivial solid torus Dehn surgery), and the companion, $K$, is a non-trivial knot in $S^3$. We prove that $P(K)$ is an L-space knot if and only if $K$ is an L-space knot and $P$ is sufficiently positi…

We exhibit a knot $P$ in the solid torus, representing a generator of first homology, such that for any knot $K$ in the 3-sphere, the satellite knot with pattern $P$ and companion $K$ is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-…

Lower bounds on unknotting number for cabled knots.

Formula derived for cabled knots' concordance invariants.

A formula for bordered Floer homology of concordances and satellites

We show that there exists a $\mathbb{Z}^\infty$-summand in the subgroup of the knot concordance group generated by knots with trivial Alexander polynomial. To this end we use the invariant Upsilon $Υ$ recently introduced by Ozsváth, Stipsicz and Szabó using knot Floer homology. We partially compute $Υ$ of $(n,1)$-cable…

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…

Proof of Knot Entropy Conjecture for tube lattice polygons.

New torsion patterns found in Khovanov homology of link diagrams.

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…