Related papers: Knots without cosmetic crossings

Let $T$ be a satellite knot, link, or spatial graph in a 3-manifold $M$ that is either $S^3$ or a lens space. Let $\mathfrak{b}_0$ and $\mathfrak{b}_1$ denote genus 0 and genus 1 bridge number, respectively. Suppose that $T$ has a companion…

We develop purely algebraic methods for proving that a knot is prime. Our approach uses the Heegaard Floer polynomial in conjunction with classical knot-theoretic methods: cyclic, dihedral, and metacyclic covering spaces. The theory of…

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…

Kearton observed that mutation can change the concordance class of a knot. A close examination of his example reveals that it is of 4-genus 1 and has a mutant of 4-genus 0. The first goal of this paper is to construct examples to show that…

We study symmetric crossing change operations for strongly invertible knots. Our main theorem is that the most natural notion of equivariant unknotting number is not additive under connected sum, in contrast with the longstanding conjecture…

We study positive braid knots (the knots in the three-sphere realized as positive braid closures) through the lens of the L-space conjecture. This conjecture predicts that if $K$ is a non-trivial positive braid knot, then for all $r <…

In this paper, we prove that $w(K) =4w(J)$, where $w(.)$ is the width of a knot and $K$ is the Whitehead double of a nontrivial knot $J$.

A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J…

A pair of surgeries on a knot is chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. We find new obstructions to the existence of such surgeries coming from Heegaard Floer homology; in particular, we make…

We give a topological characterisation of alternating knot exteriors based on the presence of special spanning surfaces. This shows that alternating is a topological property of the knot exterior and not just a property of diagrams,…

This paper contains linear systems of equations which can distinguish knots without knot invariants. Let $M_n$ be the topological moduli space of all n-component string links and such that a fixed projection into the plane is an immersion.…

A knot k in a closed orientable 3-manifold is called nonsimple if the exterior of k possesses a properly embedded essential surface of nonnegative Euler characteristic. We show that if k is a nonsimple prime tunnel number one knot in a lens…

We show the existence of infinitely many prime knots each of which having in their complements meridional essential surfaces with two boundary components and arbitrarily high genus.

A (weak chord) index is a function on the crossings of knot diagrams such that: 1) the index of a crossing does not change under Reidemeister moves; 2) crossings which can be paired by a second Reidemeister move have the same index. We show…

In Theorem 1.2 of the paper math.GT/0002110 the author claimed to have proved that all transversal knots whose topological knot type is that of an iterated torus knot (we call them cable knots) are transversally simple. That theorem is…

Let $K$ be a nontrivial knot. For each $n\in \mathbb{N}$, we prove that the rank of its $n$th iterated Whitehead doubled knot group $\pi_1(S^3 \setminus \operatorname{WD}^n(K))$ is bounded below by $n+1$. As an application, we show that…

We show that if a classical knot diagram satisfies a certain combinatorial condition then it is minimal with respect to the number of classical crossings. This statement is proved by using the Kauffman bracket and the construction of atoms…

We define the property (D) for nontrivial knots. We show that the fundamental group of the manifold obtained by Dehn surgery on a knot $K$ with property (D) with slope $\frac{p}{q}\ge 2g(K)-1$ is not left orderable. By making full use of…

Positive permutation braids on n strings, which are defined to be positive n-braids where each pair of strings crosses at most once, form the elementary but non-trivial building blocks in many studies of conjugacy in the braid groups. We…

M. Scharlemann has recently proved that any genus one tunnel number one knot is either a satellite or 2-bridge knot, as conjectured by H. Goda and M. Teragaito; all such knots admit a (1,1) decomposition. In this paper we give a…