Related papers: Non-slice linear combinations of iterated torus kn…
We introduce a new technique for showing classical knots and links are not slice. As one application we resolve a long-standing question as to whether certain natural families of knots contain topologically slice knots. We also present a…
We give examples of a linear combination of algebraic knots and their mirrors that are algebraically slice, but whose topological and smooth four-genus is two. Our examples generalize an example of non-slice algebraically slice linear…
Torus knots are an important family of knots about which much is understood; invariants of torus knots often exhibit nice formulas, making them convenient and fundamental building blocks for examples in knot theory. Spiral knots, defined…
Algebraic knots are known to be iterated torus knots and to admit L-space surgeries. However, Hedden proved that there are iterated torus knots that admit L-space surgeries but are not algebraic. We present an infinite family of such…
We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance…
We show that the subgroup of the knot concordance group generated by links of isolated complex singularities intersects the subgroup of algebraically slice knots in an infinite rank subgroup.
We produce infinite families of knots $\{K^i\}_{i\geq 1}$ for which the set of cables $\{K^i_{p,1}\}_{i,p\geq 1}$ is linearly independent in the knot concordance group. We arrange that these examples lie arbitrarily deep in the solvable and…
As proved by Hedden and Ording, there exist knots for which the Ozsvath-Szabo and Rasmussen smooth concordance invariants, tau and s, differ. The Hedden-Ording examples have nontrivial Alexander polynomials and are not topologically slice.…
A knot in $S^3$ is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all…
As a corollary of work of Ozsvath and Szabo [math.GT/0301149], it is shown that the classical concordance group of algebraically slice knots has an infinite cyclic summand and in particular is not a divisible group.
The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman's theorem that Alexander polynomial one knots are topologically slice. This paper develops…
Continuing the work of Zemke, Livingston and Allen, we consider when linear combinations of torus knots are concordant to $L$-space knots. We begin by proving Allen's conjecture for alternating torus knots. That is, we prove that a linear…
In this paper we will associate a family $\{K_1,\dots,K_l\}\subset \mathbb{S}^3$ of iterated torus knots to a given free numerical semigroup. We will describe the fundamental group of the knot complement of each knot of the family. Finally,…
In 1997, T. Cochran, K. Orr, and P. Teichner defined a filtration {F_n} of the classical knot concordance group C. The filtration is important because of its strong connection to the classification of topological 4-manifolds. Here we…
We use a link invariant defined by Cimasoni-Florens to compute \rho-invariants. This generalizes results of Cochran-Teichner and Friedl on knots to the setting of links. As an application, we prove with only twelve possible exceptions that…
We consider knot invariants in the context of large $N$ transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicity constructed…
The existence of topologically slice knots that are of infinite order in the knot concordance group followed from Freedman's work on topological surgery and Donaldson's gauge theoretic approach to 4-manifolds. Here, as an application of…
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…
We compute rho-invariant for iterated torus knots $K$ for the standard representation of the knot group given by abelianisation. For algebraic knots, this invariant turns out to be very closely related to an invariant of a plane curve…
Many well studied knots can be realized as positive braid knots where the braid word contains a positive full twist; we say that such knots are twist positive. Some important families of knots are twist positive, including torus knots,…