We resolve parts (A) and (B) of Problem 1.100 from Kirby’s list [Problems in low-dimensional topology, in Geometric topology, AMS/IP Studies in Advanced Mathematics, vol. 2 (American Mathematical Society, Providence, RI, 1997), 35–473] by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontrivial cross-sections.

We give a method for constructing a Legendrian representative of a knot in $S^{3}$ which realizes its maximal Thurston–Bennequin number under a certain condition. The method utilizes Stein handle decompositions of $D^{4}$ , and the resulting Legendrian representative is often very complicated (relative to the complexity of the topological knot type). As an application, we construct infinitely many knots in $S^{3}$ each of which yields a reducible 3-manifold by a Legendrian surgery in the standard tight contact structure. This disproves a conjecture of Lidman and Sivek.

This paper contains some applications of the description of knot diagrams by genus, and Gabai’s methods of disk decomposition. We show that there exists no genus one knot of canonical genus 2, and that canonical genus 2 fiber surfaces realize almost every Alexander polynomial only finitely many times (partially confirming a conjecture of Neuwirth).

This paper establishes a version of the Chebotarev density theorem in which number fields are replaced by 3-manifolds.

Given a diagram D of a knot K, we give easily computable bounds for Rasmussen’s concordance invariant s(K). The bounds are not independent of the diagram D chosen, but we show that for diagrams satisfying a given condition the bounds are tight. As a corollary we improve on previously known Bennequin-type bounds on the slice genus.

Scharlemann and Thompson introduced an invariant $\rho(K,\gamma)\in \mathbb{Q}/2\mathbb{Z}$ for the pair of a knot $K$ in the 3-sphere and an unknotting tunnel $\gamma$ for $K$. The paper studies the relationship between the invariant $\rho(K,\gamma)$ of a (1,1)-knot and the distance of its (1,1)-splitting introduced by the author.

Let $l$ be an oriented link of $d$ components in a homology $3$-sphere. For any nonnegative integer $q$, let $l(q)$ be the link of $d-1$ components obtained from $l$ by performing $1/q$ surgery on its $d$th component $l_d$. The Mahler measure of the multivariable Alexander polynomial $\Delta_{l(q)}$ converges to the Mahler measure of $\Delta_l$ as $q$ goes to infinity, provided that $l_d$ has nonzero linking number with some other component. If $l_d$ has zero linking number with each of the other components, then the Mahler measure of $\Delta_{l(q)}$ has a well defined but different limiting behavior. Examples are given of links $l$ such that the Mahler measure of $\Delta_l$ is small. Possible connections with hyperbolic volume are discussed.

A regular $n$-gon inscribing a knot is a sequence of $n$ points on a knot, such that the distances between adjacent points are all the same. It is shown that any smooth knot is inscribed by a regular $n$-gon for any $n$.The author was partially supported by NSF grant #DMS 9802558.

Using the Fiedler–Polyak–Viro Gauß diagram formulae we study the Vassiliev invariants of degree 2 and 3 on almost positive knots. As a consequence we show that the number of almost positive knots of a given genus or unknotting number grows polynomially in the crossing number, and also recover and extend, inter alia to their untwisted Whitehead doubles, previous results on the polynomials and signatures of such knots. In particular, we prove that there are no achiral almost positive knots and classify all almost positive diagrams of the unknot. We give an application to contact geometry (Legendrian knots) and property P.