Related papers: Cabling Conjecture for Small Bridge Number

A consequence of the Cabling Conjecture of Gonzalez-Acu\~{n}a and Short is that Dehn surgery on a knot in $S^3$ cannot produce a manifold with more than two connected summands. In the event that some Dehn surgery produces a manifold with…

The Cabling Conjecture of Gonz\'alez-Acu\~na and Short holds that only cable knots admit Dehn surgery to a manifold containing an essential sphere. We approach this conjecture for thin knots using Heegaard Floer homology, primarily via…

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…

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.

Closed 3-string braids admit many bandings to two-bridge links. By way of the Montesinos Trick, this allows us to construct infinite families of knots in the connected sum of lens spaces L(r,1) # L(s,1) that admit a surgery to a lens space…

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus g…

We prove that if positive integer p-surgery along a knot K \subset S^3 produces an L-space and it bounds a sharp 4-manifold, then the knot genus obeys the bound 2g(K) -1 \leq p - \sqrt{3p+1}. Moreover, there exists an infinite family of…

The Cabling Conjecture states that surgery on hyperbolic knots in $S^3$ never produces reducible manifolds. In contrast, there do exist hyperbolic knots in some lens spaces with non-prime surgeries. Baker constructed a family of such…

Suppose F is a compact orientable surface, K is a knot in F x I, and N is the 3-manifold obtained by some non-trivial surgery on K. If F x {0} compresses in N, then there is an annulus in F x I with one end K and the other end an essential…

We show that the distance of a link $K$ with respect to a bridge surface of any genus determines a lower bound on the genus of essential surfaces and Heegaard surfaces in the manifolds that result from non-trivial Dehn surgeries on 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…

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we…

The construction of knots via annular twisting has been used to create families of knots yielding the same manifold via Dehn surgery. Prior examples have all involved Dehn surgery where the surgery slope is an integral multiple of 2. In…

We introduce a new numerical knot invariant, termed the \textit{segment number}, which is derived from partitioned knot diagrams subject to specific over/under-crossing constraints. We prove that a knot is non-trivial if and only if its…

The $AJ$-conjecture for a knot $K \subset S^3$ relates the $A$-polynomial and the colored Jones polynomial of $K$. If a two-bridge knot $K$ satisfies the $AJ$-conjecture, we give sufficient conditions on $K$ for the $(r,2)$-cable knot $C$…

We examine geometric properties of a knot J that are unchanged by taking a (p,q)-cable K of J. Specifically, we relate w(K) to w(J), where w(K) is the width of K in the sense of Gabai. We use this information to demonstrate that thin…

Suppose M is a closed irreducible orientable 3-manifold, K is a knot in M, P and Q are bridge surfaces for K and K is not removable with respect to Q. We show that either Q is equivalent to P or $d(K,P) \leq 2-\chi(Q-K)$. If K is not a two…

Let $M$ be a $3$--dimensional handlebody of genus $g$. This paper gives examples of hyperbolic knots in $M$ with arbitrarily large genus $g$ bridge number which admit Dehn surgeries which are boundary-reducible manifolds.

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

Let $K$ be a tunnel number one knot in $M$ with irreducible knot exterior, where $M$ is either $S^3$, or a connected sum of $S^2\times S^1$ with any lens space. (In particular, this includes $M = S^2\times S^1$.) We prove that if a…