Related papers: Exceptional and cosmetic surgeries on knots

Band surgery is an operation relating pairs of knots or links in the three-sphere. We prove that if two quasi-alternating knots $K$ and $K'$ of the same square-free determinant are related by a band surgery, then the absolute value of the…

Let K be a non-trivial knot in the 3-sphere with a lens space surgery and L(p,q) a lens space obtained by a Dehn surgery on K. We study a relationship between the order of the fundamental group of L(p,q) and the Seifert genus of K.

Let M be a 3-manifold admitting a strongly irreducible Heegaard surface S and f:M \to M an involution. We construct an invariant Heegaard surface for M of genus at most 8 g(S) - 7. As a consequence, given a (possibly branched) double cover…

We give a necessary and sufficient condition for a simple closed curve on the boundary of a genus two handlebody to decompose the handlebody into (torus with one boundary component times [0,1]. We use this condition to decide whether a…

Let $K\subseteq S^3$ be a knot with exterior $E_K$, and denote by $\rho\colon \pi_1(E_K)\twoheadrightarrow G$ a quotient of its group. We give a sharp obstruction to the existence of a connected, oriented, smooth surface $F\subseteq B^4$…

Two Dehn surgeries on a knot are called cosmetic if they yield homeomorphic manifolds. For a null-homologous knot with certain conditions on the Thurston norm of the ambient manifold, if the knot admits cosmetic surgeries, then the surgery…

We present new lower bounds on the complexity of Dehn surgery manifolds of knots, using our recent result on the Cheeger-Gromov rho invariants and triangulations. As an application, we give explicit examples of closed hyperbolic 3-manifolds…

Given a knot $K$ in a closed orientable manifold $M$ we define the growth rate of the tunnel number of $K$ to be $gr_t(K) = \limsup_{n \to \infty} \frac{t(nK) - n t(K)}{n-1}$. As our main result we prove that the Heegaard genus of $M$ is…

Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of…

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…

We prove that a positive two-bridge knot other than the $(2,k)$ torus knot does not admit chirally cosmetic surgeries, a pair of Dehn surgeries along distinct slopes yielding orientation-reversingly homeomorphic 3-manifolds.

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…

We extend techniques due to Pardon to show that there is a lower bound on the distortion of a knot in $\mathbb{R}^3$ proportional to the minimum of the bridge distance and the bridge number of the knot. We also exhibit an infinite family of…

In this paper, we study reducible surgeries on knots in $S^3$. We develop thickness bounds for L-space knots that admit reducible surgeries, and lower bounds on the slice genus for general knots that admit reducible surgeries. The L-space…

We use the methods of Hedden, Juhasz, and Sarkar to exhibit a set of arborescent knots that bound large numbers of non-isotopic minimal genus spanning surfaces. In particular, we describe a sequence of prime knots K_{n} which will bound at…

We show that a special alternating knot with sufficiently large number (more than $63$) of twist regions has no chirally cosmetic surgeries, a pair of Dehn surgeries producing orientation-reversingly homeomorphic $3$-manifolds. In the…

We show that the problem of deciding whether a knot in a fixed closed orientable 3-dimensional manifold bounds a surface of genus at most $g$ is in co-NP. This answers a question of Agol, Hass, and Thurston in 2002. Previously, this was…

Let $M_1$ and $M_2$ be orientable irreducible 3--manifolds with connected boundary and suppose $\partial M_1\cong\partial M_2$. Let $M$ be a closed 3--manifold obtained by gluing $M_1$ to $M_2$ along the boundary. We show that if the gluing…

We give a diagrammatic characterization of the $(1,1)$ knots in the three-sphere and lens spaces which admit large Dehn surgeries to manifolds with Heegaard Floer homology of next-to-minimal rank. This is inspired by a corresponding result…

By examining the homology groups of a 4-manifold associated to an integral surgery on a knot $K$ in a rational homology 3-sphere $Y$ yielding a rational homology 3-sphere $Y^*$ with surgery dual knot $K^*$, we show that the subgroups…