arXiv research

Study shows some knots can't be concordant to L-space knots.

Let $ν$ be either the Ozsváth-Szabó $τ$-invariant or the Rasmussen $s$-invariant, suitably normalized. For a knot $K$, Livingston and Naik defined the invariant $t_ν(K)$ to be the minimum of $k$ for which $ν$ of the $k$-twisted positive Whitehead double of $K$ vanishes. They proved that $t_ν(K)$ is bounded above by $-T…

New obstructions show links with vanishing Milnor invariants may not be concordant to homology boundary links.

A closer look at an example introduced by Livingston & Melvin and later studied by Miyazaki shows that a plumbing of two fibered ribbon knots (along their fiber surfaces) may be algebraically slice yet not ribbon.

Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain homomorphic images of link groups.

The slicing number of a knot, $u_s(K)$, is the minimum number of crossing changes required to convert $K$ to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus $g_s(K)$. We show that for many knots, previous bounds on unknotting number obtained by Ozsvath and Szabo and b…

We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.

Let $D_+(K,t)$ denote the positive $t$-twisted double of $K$. For a fixed integer-valued additive concordance invariant $ν$ that bounds the smooth four genus of a knot and determines the smooth four genus of positive torus knots, Livingston and Naik defined $t_ν(K)$ to be the greatest integer $t$ such that $ν(D_+(K,t))…

We prove that many pretzel knots of the form $P(2n,m,-2n\pm1,-m)$ are not topologically slice, even though their positive mutants $P(2n, -2n\pm1, m, -m)$ are ribbon. We use the sliceness obstruction of Kirk and Livingston related to the twisted Alexander polynomials associated to prime power cyclic covers of knots.

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].

A knot K in the 3-sphere is superslice if there is a slice disk D in the 4-ball such that the double of D along K is the unknotted 2-sphere S in $S^4$. Answering a question of Livingston-Meier, we find smoothly slice (in fact doubly slice) knots in the 3-sphere with Alexander polynomial equal to 1 that are not smoothly…

We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer theory and is a generalization of a result by Livingston and the first author. …

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 filtration. In this paper, we show that a proper collection of the knots employed by H…

We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus, generalising recent work of Borodzik and Livingston. We apply these inequalities to unicuspidal curves whose singularity has one Puiseux pair: we prove two identities tying the parameters of the singularity, the genus…

Study shows knots can have large genus difference from concordance.

We use Kirk's invariant of link maps $S^2\sqcup S^2\to S^4$ and its variations due to Koschorke and Kirk-Livingston to deduce results about classical links. Namely, we give a new proof of the Nakanishi-Ohyama classification of two-component links in $S^3$ up to $Δ$-link homotopy. We also prove its version for string li…

We give simple homological conditions for a rational homology 3-sphere Y to have infinite order in the rational homology cobordism group, and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when Y is the branched double cover of …

The set of homology cobordisms from a surface to itself with markings of their boundaries has a natural monoid structure. To investigate the structure of this monoid, we define and study its Magnus representation and Reidemeister torsion invariants by generalizing Kirk-Livingston-Wang's argument over the Gassner repres…

New examples show algebraically slice knots with specific genus bounds.

We study rational cuspidal curves in projective surfaces. We specify two criteria obstructing possible configurations of singular points that may occur on such curves. One criterion generalizes the result of Fernandez de Bobadilla, Luengo, Melle--Hernandez and Nemethi and is based on the Bezout theorem. The other one i…

Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. Invariants of stable equivalence include the concordance invari…

To a region $C$ of the plane satisfying a suitable convexity condition we associate a knot concordance invariant $Υ^C$. For appropriate choices of the domain this construction gives back some known knot Floer concordance invariants like Rasmussen's $h_i$ invariants, and the Ozsv\' ath-Stipsicz-Szab\' o upsilon invarian…

An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston showed that if L is achiral then the linking number of L is not congruent to 2 modulo 4. In this paper we study orientat…

We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A x I, has the structure of a bifiltered complex whose filtered chain homotopy type is an invariant of the isotopy class of L in A x I. Using ideas of Ozsvath-Stipsicz-Szabo as reinterpreted by Livingston, we use this structure to de…

Study concordance of alternating torus knots to L-space knots.

The paper studies twisted signature invariants of fibered knots and 3-manifolds.

New Seifert surfaces in 4-ball differ even when pushed in.

By a recent result of Livingston, it is known that if a knot has a prime power branched cyclic cover that is not a homology sphere, then there is an infinite family of non-concordant knots having the same Seifert form as the knot. In this paper, we extend this result to the full extent. We show that if the knot has non…

New method uses rational Witt span to bound concordance crosscap number of knots.

Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual homological notion of linking by taking into account the fundamental group of the ambient …

Let $D(K)$ be the positively-clasped untwisted Whitehead double of a knot $K$, and $T_{p,q}$ be the $(p,q)$ torus knot. We show that $D(T_{2,2m+1})$ and $D^2(T_{2,2m+1})$ are linearly independent in the smooth knot concordance group $\mathcal{C}$ for each $m\geq 2$. Further, $D(T_{2,5})$ and $D^2(T_{2,5})$ generate a $…

This paper shows hyperbolic knots can have arbitrarily large torsion in knot Floer homology.

Geography problem for nonorientable surfaces bounded by knots.

We give a new geometric obstruction to the iterated Bing double of a knot being a slice link: for n>1 the (n+1)-st iterated Bing double of a knot is rationally slice if and only if the n-th iterated Bing double of the knot is rationally slice. The main technique of the proof is a covering link construction simplifying …

New invariant defined for unoriented knots, proving no factorization through topological concordance.

Study knot invariants to answer questions about slice genus and clasp numbers.

A knot \widetilde{K} \subset S^3 is q-periodic if there is a \mathbb Z_q-action preserving \widetilde{K} whose fixed set is an unknot U. The quotient of \widetilde{K} under the action is a second knot K. We construct equivariant Heegaard diagrams for q-periodic knots, and show that Murasugi's classical condition on the…

We establish certain "non-triviality" results for several filtrations of the smooth and topological knot concordance groups. First, as regards the n-solvable filtration of the topological knot concordance group defined by K. Orr, P. Teichner and the first author, we refine the recent non-triviality results of Cochran a…

We note that the Conway potential function $Ω_L$ of an $m$-component link $L$, $m>1$, can be expressed as $Ω_L(x_1,\dots,x_m)=Θ_L(\nabla_L(x_1-x_1^{-1},\dots,x_m-x_m^{-1}))$ for a unique $\nabla_L\in\mathbb Z[z_1,\dots,z_m]$, where $Θ_L$ is a certain endomorphism of the additive group of $\mathbb Z[x_1^{\pm1},\dots,x_m…

Optimizes Metropolis-Hastings algorithms for efficient sampling in high dimensions.

Every knot is concordant to a prime hyperbolic knot.

This thesis develops some general calculational techniques for finding the orders of knots in the topological concordance group C. The techniques currently available in the literature are either too theoretical, applying to only a small number of knots, or are designed to only deal with a specific knot. The thesis buil…

Bayesian Neural Networks detect gravitational wave events with high accuracy and real-time potential.

We prove the following results (1) (2) (3) on relations between $n$-links and their components. (1) Let L=(L_1, L_2) be a (4k+1)-link (4k+1\geq 5). Then we have Arf L=Arf L_1+Arf L_2. (2) Let L=(L_1, L_2) be a (4k+3)-link (4k+3\geq3). Then we have σL=σL_1+σL_2. (3) Let n\geq1. Then there is a nonribbon n-link L=(L_1, L…