Related papers: Uniqueness of bridge surfaces for 2-bridge knots

In this paper, we characterize all links in the 3-sphere with bridge number at least three that have a bridge sphere of distance two. We show that a link L has a bridge sphere of distance at most two then it falls into at least one of three…

We show that a two-bridge ribbon knot $K(m^2 , m k \pm 1)$ with $m > k >0$ and $(m,k)=1$ admits a symmetric union presentation with partial knot which is a two-bridge knot $K(m,k)$. Similar descriptions for all the other two-bridge ribbon…

We define a 2-normal surface to be one which intersects every 3-simplex of a triangulated 3-manifold in normal triangles and quadrilaterals, with one or two exceptions. The possible exceptions are a pair of octagons, a pair of unknotted…

We determine the Dehn surgeries on 2-bridge links, which yield reducible 3-manifolds. Further, we show the conditions that we obtain a torus or cable knot from one component of a 2-bridge link by a surgery on another component.

A knot in the 3-sphere in genus-1 1-bridge position (called a (1,1)-position) can be described by an element of the braid group of two points in the torus. Our main results tell how to translate between a braid group element and the…

In this paper, we give a complete classification of exceptional Dehn surgeries on a component of a hyperbolic two-bridge link in the 3-sphere.

We generalize the results of [AS], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each the lift of a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a…

We prove that for any knot $K$, there exists a one-vertex triangulation of the $3$-sphere containing an edge forming $K$. The proof is constructive, and based on fully augmented links. We use our method to produce ``complicated'' simplicial…

In this paper, we consider two properties on the braid index of a two-bridge knot. We prove an inequality on the braid indices of two-bridge knots if there exists an epimorphism between their knot groups. Moreover, we provide the average…

J. Hempel's definition of the distance of a Heegaard surface generalizes to a complexity for a knot which is in bridge position with respect to a Heegaard surface. Our main result is that the distance of a knot in bridge position is bounded…

We consider closed acylindrical surfaces in 3-manifolds and in knot and link complements, and show that the genus of these surfaces is bounded linearly by the number of tetrahedra in the triangulation of the manifold and by the number of…

We show that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, i.e., no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds. Our argument, based on…

We will classify all exceptional Dehn surgeries on 2-bridge knots according to whether they produce reducible, toroidal, or small Seifert fibered manifolds.

In this paper, we define the rectangle condition on the bridge sphere for a $n$-bridge decomposition of a knot whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of $3$-manifolds. We show that…

We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot.…

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…

It is known that there are only finitely many knots with super bridge index 3. Jin and Jeon have provided a list of possible such candidates. However, they conjectured that the only knots with super bridge index 3 are trefoil and the figure…

We prove that hyperbolic 2-bridge knots are determined amongst all compact 3-manifolds by the profinite completions of their knot groups.

A knotted surface in the 4-sphere may be described by means of a hyperbolic diagram that captures the 0-section of a special Morse function, called a hyperbolic decomposition. We show that every hyperbolic decomposition of a knotted surface…

Let K' be a hyperbolic knot in S^3 and suppose that some Dehn surgery on K' with distance at least 3 from the meridian yields a 3-manifold M of Heegaard genus 2. We show that if M does not contain an embedded Dyck's surface (the closed…