Related papers: Alternating knots do not admit cosmetic crossings

We consider knots equipped with a representation of their knot groups onto a dihedral group D_{2n} (where n is odd). To each such knot there corresponds a closed 3-manifold, the (irregular) dihedral branched covering space, with the…

We present various examples of cosmetic bandings on knots and links, that is, bandings on knots and links leaving their types unchanged. As a byproduct, we give a hyperbolic knot which admits exotic chirally cosmetic surgeries yielding…

We give a new, conceptually simpler proof of the fact that knots in $S^3$ with positive L-space surgeries are fibered and strongly quasipositive. Our motivation for doing so is that this new proof uses comparatively little Heegaard…

This paper establishes a new technique that enables us to access some fundamental structural properties of instanton Floer homology. As an application, we establish, for the first time, a relation between the instanton Floer homology of a…

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…

A crown diagram of a smooth, closed oriented 4-manifold can be thought of as the projection of a link in the product of a closed surface and the circle, with chords in the circle direction connecting the strands of each crossing. This paper…

We prove that instanton L-space knots are fibered and strongly quasipositive. Our proof differs conceptually from proofs of the analogous result in Heegaard Floer homology, and includes a new decomposition theorem for cobordism maps in…

We study the sutured Floer homology invariants of the sutured manifold obtained by cutting a knot complement along a Seifert surface, R. We show that these invariants are finer than the "top term" of the knot Floer homology, which they…

The pair (K,r) consisting of a knot K and a surjective map r from the knot group onto a dihedral group is said to be a p-colored knot. D. Moskovich conjectured that for any odd prime p there are exactly p equivalence classes of p-colored…

Using a Heegaard diagram for the pullback of a knot $K \subset S^3$ in its cyclic double branched cover $\Sigma_2(K)$, we give a combinatorial proof for the invariance of knot Floer homology over $\mathbb{Z}$.

In an earlier paper, we introduced a collection of graded Abelian groups $\HFKa(Y,K)$ associated to knots in a three-manifold. The aim of the present paper is to investigate these groups for several specific families of knots, including the…

We study knots in $S^3$ with infinitely many $SU(2)$-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into $SU(2)$ has cyclic image. We show that for every such nontrivial knot…

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…

Using the covering involution on the double branched cover of the three-sphere branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot invariants and apply them to deduce novel linear…

Let $K$ be a prime knot in $S^3$ and $G(K)=\pi_1(S^3-K)$ the knot group. We write $K_1 \geq K_2$ if there exists a surjective homomorphism from $G(K_1)$ onto $G(K_2)$. In this paper, we determine this partial order on the set of prime knots…

Let $\widehat{\mathcal{C}}_{\mathbb{Z}}$ denote the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms. Answering a question of Matsumoto, the second author previously showed that…

For a nullhomologous Legendrian knot in a closed contact 3-manifold Y we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of Y is mapped by a…

We prove the nugatory crossing conjecture for fibered knots. We also show that if a knot $K$ is $n$-adjacent to a fibered knot $K'$, for some $n>1$, then either the genus of $K$ is larger than that of $K'$ or $K$ is isotopic to $K'$.

There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every…

We produce the first examples of closed, tight contact 3-manifolds which become overtwisted after performing admissible transverse surgeries. Along the way, we clarify the relationship between admissible transverse surgery and Legendrian…