Related papers: Null-homotopic knots have Property R

We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y(K_1,K_2)$ obtained by splicing the complements of the knots $K_i\subset Y_i$, $i=1,2$, in terms of the knot Floer homology of $K_1$ and $K_2$. We also…

This paper gives a proof that the fundamental group of a class of closed orientable 3-manifolds constructed from three injective handlebodies has a solvable word problem. This is done by giving an algorithm to decide if a closed curve in…

We construct two homology 3-spheres for which the (unperturbed) $SU(2)$ Chern-Simons function is not Morse-Bott. In one case, there is a degenerate isolated critical point. In the other, a path component of the critical set is not…

Let $K$ denote a knot inside the homology sphere $Y$. The zero-framed longitude of $K$ gives the complement of $K$ in $Y$ the structure of a bordered three-manifold, which may be denoted by $Y(K)$. We compute the quasi-isomorphism type of…

Let K be a fibered knot in the 3-sphere. We show that if the monodromy of K is sufficiently complicated, then Dehn surgery on K cannot yield a lens space. Work of Yi Ni shows that if K has a lens space surgery then it is fibered. Combining…

We prove that for any integer $n$ there exist infinitely many different knots in $S^3$ such that $n$-surgery on those knots yields the same 3-manifold. In particular, when $|n|=1$ homology spheres arise from these surgeries. This answers…

A conjecture of Shumakovitch states that every nontrivial knot has 2-torsion in its Khovanov homology. We show that if a knot $K$ has no 2-torsion in its Khovanov homology, then the rank of its reduced Khovanov homology is minimal among all…

A principal bundle over the connected sum of two manifolds need not be diffeomorphic or even homotopy equivalent to a non-trivial connected sum of manifolds. We show however that the homology of the total space of a bundle formed a pullback…

A knot K is called n-adjacent to another knot K', if K admits a projection containing n generalized crossings such that changing any 0 < m \leq n of them yields a projection of K'. We apply techniques from the theory of sutured 3-manifolds,…

In this paper we define and investigate Z/2-homology cobordism invariants of Z/2-homology 3-spheres which turn out to be related to classical invariants of knots. As an application we show that many lens spaces have infinite order in the…

In this paper, we generalize a result of Satoh to show that for any odd natural $n$, the connected sum of the $n$-twist spun sphere of a knot $K$ and an unknotted projective plane in the 4-sphere is equivalent to the same unknotted…

We prove that the moment-angle complex $\mathcal Z_K$ corresponding to a 3-dimensional simplicial sphere $K$ has the cohomology ring isomorphic to the cohomology ring of a connected sum of products of spheres if and only if either (a) $K$…

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 short note, we prove that if a knot in the Poincare homology sphere is homotopically essential, then it does not admit any purely cosmetic surgeries.

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…

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the configuration space of ordered pairs of…

In this paper, we establish a "pseudo-effective" version of the holonomy principle for compact K\"{a}hler manifolds with nonnegative holomorphic sectional curvature. As applications, we prove that if a compact complex manifold $M$ admits a…

In this paper, we conjecture a connection between the $A$-polynomial of a knot in $\mathbb{S}^{3}$ and the hyperbolic volume of its exterior $\mathcal{M}_{K}$ : the knots with zero hyperbolic volume are exactly the knots with an…

Band surgery is an operation relating pairs of knots or links in the three-sphere. We prove that if two quasi-alternating knots $K$ and $K'$ of the same square-free determinant are related by a band surgery, then the absolute value of the…

We show that in a prime, closed, oriented 3-manifold M, equivalent knots are isotopic if and only if the orientation preserving mapping class group is trivial. In the case of irreducible, closed, oriented $3$-manifolds we show the more…