相关论文: Berge-Gabai knots and L-space satellite operations

We prove that the (p,q)-cable of a knot K in S^3 admits a positive L-space surgery if and only if K admits a positive L-space surgery and q/p \geq 2g(K)-1, where g(K) is the Seifert genus of K. The "if" direction is due to Hedden.

We give a formula for the $\tau$-invariant of a satellite knot $P(K,n)$ when $P$ is an L-space satellite operator. Our formula holds for general L-space satellite operators $P$ when the companion $K$ satisfies $\epsilon(K)=1$. When…

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 give sufficient conditions for a satellite knot to admit an L-space surgery, and use this result to give new infinite families of patterns which produce satellite L-space knots.

Let P be a knot in an unknotted solid torus (i.e. a satellite operator or pattern), K a knot in S^3 and P(K) the satellite of K with pattern P. For any satellite operator P, this correspondence gives a function P : C -> C on the set of…

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…

A slope $p/q$ is said to be characterizing for a knot $K$ if the homeomorphism type of the $p/q$-Dehn surgery along $K$ determines the knot up to isotopy. Extending previous work of Lackenby and McCoy on hyperbolic and torus knots…

A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, i.e. a rational homology 3-sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c…

Let $K$ be a nontrivial knot in $S^{3}$ and $t(K)$ its tunnel number. For any $(p\geq 2,q)$-slope in the torus boundary of a closed regular neighborhood of $ K$ in $S^{3}$, denoted by $K^{\star}$, it is a nontrivial cable knot in $S^{3}$.…

We exhibit a knot $P$ in the solid torus, representing a generator of first homology, such that for any knot $K$ in the 3-sphere, the satellite knot with pattern $P$ and companion $K$ is not smoothly slice in any homology 4-ball. As a…

Let $D$ be a diagram of an alternating knot with unknotting number one. The branched double cover of $S^3$ branched over $D$ is an L-space obtained by half integral surgery on a knot $K_D$. We denote the set of all such knots $K_D$ by…

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…

An L-space link is a link in $S^3$ on which all sufficiently large integral surgeries are L-spaces. We prove that for m, n relatively prime, the r-component cable link $K_{rm,rn}$ is an L-space link if and only if K is an L-space knot and…

Given an $n$-component link $L$ in any 3-manifold $M$, the space $\mathcal{L} \subset (\mathbb{Q}\cup \mkern-1.5mu\{\infty\})^n$ of rational surgery slopes yielding L-spaces is already fully characterized (in joint work by the author) when…

For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r<R(K) define satellite structures, or local knotting. We explore…

Necessary and sufficient conditions are given for a satellite knot to be fibered. Any knot $\tilde k$ embeds in an unknotted solid torus $\tilde V$ with arbitrary winding number in such a way that no satellite knot with pattern $(\tilde V,…

Thin position for knots in the 3-sphere was introduced by Gabai and has been used in a variety of contexts. We conjecture an analogue to a theorem of Schubert and Schultens concerning the bridge number of satellite knots. For a satellite…

A knot K is called n-adjacent to the unknot, if K admits a projection containing n generalized crossings such that changing any m (no larger than n) of them yields a projection of the unknot. We show that a non-trivial satellite knot K is…

A link in $S^{3}$ is a fully positive braid link if it is the closure of a positive braid that contains at least one full-twist. We show that a fully positive braid link is a satellite link if and only if it is the satellite of a fully…

An $L$-space knot is a knot that admits a positive Dehn surgery yielding an $L$-space. Many known hyperbolic $L$-space knots are braid positive, meaning they can be represented as the closure of a positive braid. Recently, Baker and Kegel…