Related papers: Uniqueness of bridge surfaces for 2-bridge knots
In this paper, we show that any unknotting tunnel for a two bridge knot is isotopic to either one of known ones. This together with Morimoto-Sakuma's result gives the complete classification of unknotting tunnels for two bridge knots up to…
It is conjectured that for each knot $K$ in $S^3$, the fundamental group of its complement surjects onto only finitely many distinct knot groups. Applying character variety theory we obtain an affirmative solution of the conjecture for a…
For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot examples were obtained…
We have simulated the motion of a single vertical, two-dimensional liquid bridge spanning the gap between two flat, horizontal solid substrates of given wettabilities, using a multicomponent pseudopotential lattice Boltzmann method. For…
For any n\ge 2, we give infinitely many unsplittable links of n components in the 3-sphere which admit non-trivial surgery yielding the 3-sphere again and whose components are mutually distinct hyperbolic knots. Berge and Kawauchi gave…
We prove a simple necessary and sufficient condition for a two-bridge knot K(p,q) to be quasipositive, based on the continued fraction expansion of p/q. As an application, coupled with some classification results in contact and symplectic…
We show that there are hyperbolic tunnel-number one knots with arbitrarily high bridge number and that "most" tunnel-number one knots are not one-bridge with respect to an unknotted torus. The proof relies on a connection between bridge…
We study the equivariant concordance classes of two-bridge knots, providing an easy formula to compute their butterfly polynomial, and we give two different proofs that no two-bridge knot is equivariantly slice. Finally, we introduce a new…
We determine the set of all genus g bridge numbers of many iterated torus knots, listing these numbers in a sequence called the bridge spectrum. In addition, we prove a structural lemma about the decomposition of a strongly irreducible…
The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic…
We give a complete characterization of those essential simple loops on 2-bridge spheres of 2-bridge links which are null-homotopic in the link complements. By using this result, we describe all upper-meridian-pair-preserving epimorphisms…
Understanding ideal points in the character varieties of knot complements has led to a number of important invariants for 3-manifolds. Ohtsuki (1994) counted the ideal points for character varieties of 2-bridge knot complements, and he made…
We show that a small tree-decomposition of a knot diagram induces a small sphere-decomposition of the corresponding knot. This, in turn, implies that the knot admits a small essential planar meridional surface or a small bridge sphere. We…
It is a consequence of theorems of Gordon-Reid [Tangle decompositions of tunnel number one knots and links, J. Knot Theory and its Ramifications, 4 (1995) 389-409] and Thompson [Thin position and bridge number for knots in the 3-sphere,…
A 1-bridge torus knot in a 3-manifold of genus $\le 1$ is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form…
An $n$-plat 1-knot is one isotopic to the plat closure of some $2n$-braid, which is also called an $n$-bridge 1-knot. Schubert classified 2-bridge 1-knots by considering their double branched covers which are homeomorphic to lens spaces. A…
We prove that all 2-bridge ribbon knots are symmetric unions.
Given a knot in $S^3$, one can associate to it a surface diffeomorphism in two different ways. First, an arbitrary knot in $S^{3}$ can be represented by braids, which can be thought of as diffeomorphisms of punctured disks. Second, if the…
In a 3-manifold M, let K be a knot and R be an annulus which meets K transversely. We define the notion of the pair (R,K) being caught by a surface Q in the exterior of the link given by K and the boundary curves of R. For a caught pair…
Ng constructed an invariant of knots in ${\mathbb{R}}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${\mathbb{R}}^4$ using marked graph diagrams.