Related papers: 1-domination of knots

Let $k$ and $k'$ be two knots in 3-sphere. Say $k$ 1--dominates $k'$, if there is a proper degree 1 map $f\co E(k)\to E(k')$, between knot exterior of $k_i$. Theorem: Suppose that any companion of $k$ is prime. If $k$ 1--dominates $k'$ with…

We discuss an "extrinsic" property of knots in a 3-subspace of the 3-sphere $S^3$ to characterize how the subspace is embedded in $S^3$. Specifically, we show that every knot in a subspace of the 3-sphere is transient if and only if the…

We show that, for any prime p, a knot K in the 3-sphere is determined by its p-fold cyclic unbranched covering. We also investigate when the m-fold cyclic unbranched covering of a knot coincides with the n-fold cyclic unbranched covering of…

We consider the space of all smooth knots in the 3-sphere isotopic to a given knot, with the aim of finding a small subspace onto which this large space deformation retracts. For torus knots and many hyperbolic knots we show the subspace…

By using double branched covers, we prove that there is a 1-1 correspondence between the set of knotoids in the 2-sphere, up to orientation reversion and rotation, and knots with a strong inversion, up to conjugacy. This correspondence…

In this paper, we prove than given two cubic knots $K_1$, $K_2$ in $\mathbb{R}^3$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister…

We review the polynomial parameterization of classical knots and prove the analogous results for long $2$ knots. We also construct polynomial parameterizations for certain classes of knotted spheres (such as spun and twist spun of the…

We introduce a geometric invariant of knots in the three-sphere, called the first-order genus, that is derived from certain 2-complexes called gropes, and we show it is computable for many examples. While computing this invariant, we draw…

We show that if each of $K_1$ and $K_2$ is a trefoil knot or figure eight knot, the homology 3-sphere defined by the Kirby diagram which is a simple link of $K_1$ and $K_2$ with framing (0, n) is represented by an n-twisted Whitehead double…

In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander…

Kirby and Lickorish showed that every knot in the 3-sphere is concordant to a prime knot, equivalently, every concordance class contains a prime knot. We prove here that their result can be strengthened: Every knot in the 3-sphere is…

Let K be a knot in the 3-sphere. A slope p/q is said to be characterising for K if whenever p/q surgery on K is homeomorphic, via an orientation-preserving homeomorphism, to p/q surgery on another knot K' in the 3-sphere, then K and K' are…

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…

We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot.…

A knot K in the 3-sphere is superslice if there is a slice disk D in the 4-ball such that the double of D along K is the unknotted 2-sphere S in $S^4$. Answering a question of Livingston-Meier, we find smoothly slice (in fact doubly slice)…

In this article we study a partial ordering on knots in the 3-sphere where K_1 is greater than or equal to K_2 if there is an epimorphism from the knot group of K_1 onto the knot group of K_2 which preserves peripheral structure. If K_1 is…

We show that, for any positive real number, there exists a knot in the 3-sphere admitting a pair of boundary slopes whose difference is at most the given number.

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 K be a knot in the 3-sphere with 2-fold branched covering space M. If for some prime p congruent to 3 mod 4 the p-torsion in the first homology of M is cyclic with odd exponent, then K is of infinite order in the knot concordance group.…

For n >1, if the Seifert form of a knotted 2n-1 sphere K in S^{2n+1} has a metabolizer, then the knot is slice. Casson and Gordon proved that this is false in dimension three (n = 1). However, in the three dimensional case it is true that…