相关论文: Knots in homology lens spaces determined by their …

Given a compact orientable 3-manifold M whose boundary is a hyperbolic surface and a simple closed curve C in its boundary, every knot in M is homotopic to one whose complement admits a complete hyperbolic structure with totally geodesic…

Let M be $S^3$, $S^1\times S^2$, or a lens space L(p,q), and let k be a (1,1)-knot in M, i.e., a knot which is of 1-bridge with respect to a Heegaard torus. We show that if there is a closed meridionally incompressible surface in the…

We prove that if $P$ is a $(1,1)$-pattern knot, the two inequalities $\dim \widehat{HFK} (P(K)) \geqslant \dim \widehat{HFK} (P(U))$ and $\dim \widehat{HFK} (P(K)) \geqslant \dim \widehat{HFK} (K)$ hold for the unknot $U\subset S^3$ and any…

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…

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…

We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of spaces of maps between configuration spaces and another which is cosimplicial.…

This paper introduces a new approach to finding knots and links with hidden symmetries using "hidden extensions", a class of hidden symmetries defined here. We exhibit a family of tangle complements in the ball whose boundaries have…

Given a one-dimensional homology class in a lens space, a question related to the Berge conjecture on lens space surgeries is to determine all knots realizing the minimal rational genus of all knots in this homology class. It is known that…

We show that if K is a non-trivial knot inside a homology sphere Y, then the rank of knot Floer homology associated with K is strictly bigger than the rank of Heegaard Floer homology of Y.

This paper gives a partial description of the homotopy type of K, the space of long knots in 3-dimensional Euclidean space. The primary result is the construction of a homotopy equivalence between K and the free little 2-cubes object over…

Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a…

We prove that every lens space contains a genus one homologically fibered knot, which is contrast to the fact that some lens spaces contain no genus one fibered knot. In the proof, the Chebotarev density theorem and binary quadratic forms…

Knot contact homology is an ambient isotopy invariant of knots and links in $\mathbb R^3$. The purpose of this paper is to extend this definition to an ambient isotopy invariant of tangles and prove that gluing of tangles gives a gluing…

Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…

In 1999, Khovanov showed that a link invariant known as the Jones polynomial is the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups, and to explain their…

We discuss the fundamental (relative) 3-classes of knots (or hyperbolic links), and provide diagrammatic descriptions of the push-forwards with respect to every link-group representation. The point is an observation of a bridge between the…

We describe necessary and sufficient conditions for a knot in an L-space to have an L-space homology sphere surgery. We use these conditions to reformulate a conjecture of Berge about which knots in S^3 admit lens space surgeries.

In view of the self-linking invariant, the number $|K|$ of framed knots in $S^3$ with given underlying knot $K$ is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that $|K|$ is…

For a closed 4-manifold X and closed 3-manifold M we investigate the smallest integer n (perhaps infinity) such that M embeds in the connected sum of n copies of X. It is proven that any lens space (or homology lens space) embeds…

We prove that there are compact submanifolds of the 3-sphere whose interiors are not homeomorphic to any geometric limit of hyperbolic knot complements.