Related papers: Unique Surgery Descriptions along Knots

Suppose that a hyperbolic knot in $S^3$ admits a finite surgery, Boyer and Zhang proved that the surgery slope must be either integral or half-integral, and they conjectured that the latter case does not happen. Using the correction terms…

We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there…

We provide infinitely many rational homology 3-spheres with weight-one fundamental groups which do not arise from Dehn surgery on knots in $S^3$. In contrast with previously known examples, our proofs do not require any gauge theory or…

We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is rule out knotted curves in the singular set of the orbit space. As an extension of this work we…

The untwisting number of a knot K is the minimum number of null-homologous twists required to convert K to the unknot. Such a twist can be viewed as a generalization of a crossing change, since a classical crossing change can be effected by…

In this note we show that $+1$-contact surgery on distinct Legendrian knots frequently produces contactomorphic manifolds. We also give examples where this happens for $-1$-contact surgery. As an amusing corollary we find overtwisted…

Kashaev and Reshetikhin previously described a way to define holonomy invariants of knots using quantum $\mathfrak{sl}_2$ at a root of unity. These are generalized quantum invariants depend both on a knot $K$ and a representation of the…

Associated to a hyperbolic knot complement in $S^3$ is a set of prime numbers corresponding to the residue characteristics of the ramified places of the quaternion algebras obtained by Dehn surgery on the knots. Previous work by…

A knot K in the 3-sphere is said to have Property nR if, whenever K is a component of an n-component link L and some integral surgery on L produces the connected sum of n copies of S^1 x S^2, there is a sequence of handle slides on L that…

A pair of surgeries on a knot is called purely cosmetic if the pair of resulting 3-manifolds are homeomorphic as oriented manifolds. Using recent work of Hanselman, we show that (nontrivial) knots which arise as the closure of a 3-stranded…

A knot in the 3-sphere is called an L--space knot if it admits a nontrivial Dehn surgery yielding an L--space. Like torus knots and Berge knots, many L--space knots admit also a Seifert fibered surgery. We give a concrete example of a…

A celebrated theorem of Kirby identifies the set of closed oriented connected 3-manifolds with the set of framed links in $S^3$ modulo two moves. We give a similar description for the set of knots (and more generally, boundary links) in…

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 cosmetic crossing conjecture (also known as the "nugatory crossing conjecture") asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery…

We prove that for particular infinite families of $L$-spaces, arising as branched double covers, the $d$-invariants defined by Ozsv\'ath and Szab\'o are arbitrarily large and small. As a consequence, we generalise a result by Greene and…

A Seifert surgery is a pair (K, m) of a knot K in the 3-sphere and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for…

The $m$-trace of a knot is the $4$-manifold obtained from $\mathbf{B}^4$ by attaching a $2$-handle along the knot with $m$-framing. In 2015, Abe, Jong, Luecke and Osoinach introduced a technique to construct infinitely many knots with the…

We prove that 0 is a characterizing slope for infinitely many knots, namely the genus-1 knots whose knot Floer homology is 2-dimensional in the top Alexander grading, which we classified in recent work and which include all $(-3,3,2n+1)$…

A fixed knot $K$ acts via Murasugi sum on the space $\mathcal{S}$ of isotopy classes of knots. This operation endows $\mathcal{S}$ with a directed graph structure denoted by $M\kern-1pt SG(K)$. We show that any given family of knots in…

We give a description of degree-one maps between closed, oriented 3-manifolds in terms of surgery. Namely, we show that there is a degree-one map from a closed, oriented 3-manifold $M$ to a closed, oriented 3-manifold $N$ if and only if $M$…