相关论文: On the Upsilon invariant and satellite knots
By using grid homology theory, we give an explicit algorithm for computing Ozsv\'ath-Stipsicz-Szab\'o's $\Upsilon$-invariant and the $d$-invariant of Dehn surgeries along knots in $S^3$. As its application, we compute the two invariants for…
Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsvath-Szabo contact invariant we obtain an invariant of knots…
We describe an error in the proof of a key proposition, which was necessary for the proof of the main result. Alternate proofs of the main result are given by Ozsvath-Stipsicz-Szabo and Dai-Hom-Stoffregen-Truong.
Let \nu be any integer-valued additive knot invariant that bounds the smooth 4-genus of a knot K, |\nu(K)| <= g_4(K), and determines the 4-ball genus of positive torus knots, \nu(T_{p,q}) = (p-1)(q-1)/2. Either of the knot concordance…
A slice-torus invariant is an $\mathbb{R}$-valued homomorphism on the knot concordance group whose value gives a lower bound for the 4-genus such that the equality holds for any positive torus knot. Such invariants have been discovered in…
We prove that for particular infinite families of $L$-spaces, arising as branched double covers, the $d$-invariants defined by Ozsv\'ath and Szab\'o are arbitrarily large and small. As a consequence, we generalise a result by Greene and…
We compute explicitly the Khovanov polynomials (using the computer program from katlas.org) for the two simplest families of the satellite knots, which are the twisted Whitehead doubles and the two-strand cables. We find that a quantum…
We continue our study of the knot Floer homology invariants of cable knots. For large |n|, we prove that many of the filtered subcomplexes in the knot Floer homology filtration associated to the (p,pn+1) cable of a knot, K, are isomorphic…
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…
In this article, we apply slope detection techniques to study properties of toroidal $3$-manifolds obtained by performing Dehn surgeries on satellite knots in the context of the $L$-space conjecture. We show that if $K$ is an $L$-space knot…
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…
Given an L-space knot we show that its Upsilon function is the Legendre transform of a counting function equivalent to the d-invariants of its large surgeries. The unknotting obstruction obtained for the Upsilon function is, in the case of…
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…
We propose and analyze a structure with which to organize the difference between a knot in the 3-sphere bounding a topologically embedded 2-disk in the 4-ball and it bounding a smoothly embedded disk. The n-solvable filtration of the…
The fundamental quandle is a complete invariant for unoriented tame knots \cite{JO, Ma} and non-split links \cite{FR}. The proof involves proving a relationship between the components of the fundamental quandle and the cosets of the…
We discuss different invariants of knots and links that depend on a primitive root of unity. We clarify the definitions of existing invariants with the Reshetikhin-Turaev method, present the generalization of ADO invariants to…
We establish a satellite formula for the real Seiberg-Witten Floer homotopy types of knots with odd patterns. Using this, we derive several applications to knot concordance theory. The satellite formula follows from a version of the…
Let $\mathcal{T}$ be the group of smooth concordance classes of topologically slice knots, and $\{0\}\subset\cdots\subset \mathcal{T}_{n+1}\subset\mathcal{T}_{n}\subset \cdots\subset \mathcal{T}_{0}\subset \mathcal{T}$ be the bipolar…
We address primary decomposition conjectures for knot concordance groups, which predict direct sum decompositions into primary parts. We show that the smooth concordance group of topologically slice knots has a large subgroup for which the…
We generalize the Manolescu-Owens smooth concordance invariant delta(K) of knots K in the 3-sphere to invariants delta_{p^n}(K) obtained by considering covers of order p^n, with p prime. Our main result shows that for any odd prime p, the…