Related papers: Knots and 4-manifolds
We give uniform, explicit, and simple face-pairing descriptions of all the branched cyclic covers of the 3-sphere, branched over two-bridge knots. Our method is to use the bi-twisted face-pairing constructions of Cannon, Floyd, and Parry;…
We prove that an odd pretzel knot is doubly slice if it has $2n+1$ twist parameters consisting of $n+1$ copies of $a$ and $n$ copies of $-a$ for some odd integer $a$. Combined with the work of Issa and McCoy, it follows that these are the…
A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The decision problem whether a graph has a disconnected cut is called Disconnected Cut. This problem is closely related to several…
In this paper we show how generalized quaternions, including 2X2 matrices, can be used to find solutions of a non-commuting equation intimately connected with braid groups. These solutions can then be used to find polynomial invariants of…
A companion paper to "On knot Floer homology in branched double covers" applied to braided branched loci. We reprove the main result of that paper concerning alternating branched loci when projected to an annulus, without using Khovanov…
We use the combinatorial techniques of graphs of intersection to study reducible Dehn surgeries on knots in the three-sphere. In particular, in the event that a reducible surgery on a knot K in the three-sphere of slope r produces a…
This is a review article on the Bennequin-Birman-Menasco machinery for studying embedded incompressible surfaces in 3-space via their `braid foliations'. Two cases are investigated: case (1) The surface has non-empty boundary; the boundary…
It is known that the Alexander polynomial detects fibered knots and 3-manifolds that fiber over the circle. In this note, we show that when the Alexander polynomial becomes inconclusive, the notion of "knot adjacency", studied in the paper…
We establish the slice-ribbon conjecture for a large family of Montesinos' knots by means of Donaldson's theorem on the intersection forms of definite 4-manifolds.
We introduce and study knotoids. Knotoids are represented by diagrams in a surface which differ from the usual knot diagrams in that the underlying curve is a segment rather than a circle. Knotoid diagrams are considered up to Reidemeister…
The unknotting number of a knot is bounded from below by its slice genus. It is a well-known fact that the genera and unknotting numbers of torus knots coincide. In this note we characterize quasipositive knots for which the genus bound is…
We investigate the properties of knots in S^3 which bound Klein bottles, such that a pushoff of the knot has zero linking number with the knot, i.e. has zero framing. This is motivated by the many results in the literature regarding slice…
Techniques are introduced which determine the geometric structure of non-simple two-generator $3$-manifolds from purely algebraic data. As an application, the satellite knots in the $3$-sphere with a two-generator presentation in which at…
These introductory notes on Whitney towers in 4-manifolds, as developed in collaboration with Jim Conant and Peter Teichner, are an expansion of three expository lectures given at the Winter Braids X conference February 2020 in Pisa, Italy.…
In recent work the author investigates perfect matchings of a bipartite graph obtained from a knot diagram and demonstrates that these correspond to discrete Morse functions on a 2-complex for the 2-sphere. This relationship is expounded…
This manuscript introduces a new framework for the study of knots by exploring the neighborhood of knot embeddings in the space of simple open and closed curves in 3-space. The latter gives rise to a knotoid spectrum, which determines the…
These are lecture notes from a mini-course taught at Winterbraids XIII (Montpellier, 2024). The main character of these notes are curves in the complex projective plane, viewed from a topological perspective.
These notes summarize and expand on a mini-course given at CIRM in February 2018 as part of Winter Braids VIII. We somewhat obsessively develop the slogan `Trisections are to 4-manifolds as Heegaard splittings are to 3-manifolds', focusing…
We study the geometry of interacting knotted solitons. The interaction is local and advances either as a three-body or as a four-body process, depending on the relative orientation and a degeneracy of the solitons involved. The splitting…
We study spectral gaps of cellular differentials for finite cyclic coverings of knot complements. Their asymptotics can be expressed in terms of irrationality exponents associated with ratios of logarithms of algebraic numbers determined by…