Related papers: Every knot has characterising slopes

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…

For a knot $K$ with $\Delta_K(t)\doteq t^2-3t+1$ in a homology $3$-sphere, let $M$ be the result of $2/q$-surgery on $K$. We show that an appropriate assumption on the Reidemeister torsion of the universal abelian covering of $M$ implies…

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point $p$ of the cylinder is called {\em coherent} if all three branches intersect at $p$ pairwise with the same index. A {\em triple unknotting} of a classical knot…

Let K be a tame knot with irreducible exterior M(K) in a closed, connected, orientable 3--manifold Sigma such that pi_1(Sigma) is cyclic. If infinity is not a strict boundary slope, then the diameter of the set of strict boundary slopes of…

Given a tame knot K presented in the form of a knot diagram, we show that the problem of determining whether K is knotted is in the complexity class NP, assuming the generalized Riemann hypothesis (GRH). In other words, there exists a…

We say that two knots are friends if they share the same 0-surgery. Two friends with different sliceness status would provide a counterexample to the 4-dimensional smooth Poincar\'e conjecture. Here we create a census of all friends with…

It has been an open question whether all boundary slopes of hyperbolic knots are strongly detected by the character variety. The main result of this paper produces an infinite family of hyperbolic knots each of which has at least one strict…

Let $LHT$ be a left handed trefoil knot and $K$ be any knot. We define $M_n(K)$ to be the homology $3$-sphere which is represented by a simple link of $LHT$ and $LHT \sharp K$ with framings $0$ and $n$ respectively. Starting with this link,…

Let n be any integer greater than two. We prove that there exists a projection P having the following properties. (1) P is not the projection of any unknotted knot. (2) The singular point set of P consists of double points. (3) P is the…

This paper proves that every oriented non-disk Seifert surface $F$ for a knot $K$ in $S^3$ is smoothly concordant to a Seifert surface $F^{\prime}$ for a hyperbolic knot $K^{\prime}$ of arbitrarily large volume. This gives a new and simpler…

We show that every nonzero integer occurs in the denominator of a boundary slope for infinitely many (1,1)-knots and that infinitely many (1,1)-knots have boundary slopes of arbitrarily small difference. Specifically, we prove that for any…

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 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 establish a characterization of adequate knots in terms of the degree of their colored Jones polynomial. We show that, assuming the Strong Slope conjecture, our characterization can be reformulated in terms of "Jones slopes" of knots and…

We exhibit infinitely many ribbon knots, each of which bounds infinitely many pairwise non-isotopic ribbon disks whose exteriors are diffeomorphic. This family provides a positive answer to a stronger version of an old question of Hitt and…

We reconsider topological string realization of SU(N) Chern-Simons theory on S^3. At large N, for every knot K in S^3, we obtain a polynomial A_K(x,p;Q) in two variables x,p depending on the t'Hooft coupling parameter Q=e^{Ng_s}. Its…

We show that there exist hyperbolic knots in the 3-sphere such that the set of points of large injectivity radius in the complement take up the bulk of the volume. More precisely, given a finite volume hyperbolic manifold, for any bound R>0…

In this paper, we construct infinitely many non-isotopic 3-knots in the 5-sphere, each of which has four critical points with respect to the standard height function of the 5-sphere. This contrasts with a theorem of Scharlemann which says…

For the purposes of this paper, Dehn surgery along a curve K in a 3-manifold M with slope r is `exceptional' if the resulting 3-manifold M_K(r) is reducible or a solid torus, or the core of the surgery solid torus has finite order in the…

We show that if $p/q$-surgery on a nontrivial knot $K$ yields the branched double cover of an alternating knot or link, then $|p/q|\leq 4g(K)+3$. This generalises a bound for lens space surgeries first established by Rasmussen. We also show…