Related papers: Non-integer characterizing slopes and knot Floer h…
This is a companion paper to earlier work of the authors, which proved an integral surgery formula for framed instanton homology. First, we present an enhancement of the large surgery formula, a rational surgery formula for null-homologous…
The nonorientable four-ball genus of a knot $K$ in $S^3$ is the minimal first Betti number of nonorientable surfaces in $B^4$ bounded by $K$. By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we…
In this paper we extend the idea of bordered Floer homology to knots and links in $S^3$: Using a specific Heegaard diagram, we construct gluable combinatorial invariants of tangles in $S^3$, $D^3$ and $I\times S^2$. The special case of…
We study the classification of slice disks of knots up to isotopy and diffeomorphism using an invariant in knot Floer homology. We compute the invariant of a slice disk obtained by deform-spinning, and show that it can be effectively used…
It has been an open question whether all boundary slopes of hyperbolic knots are strongly detected by the character variety. The main result of this paper produces an infinite family of hyperbolic knots each of which has at least one strict…
We classify all finite group actions on knots in the 3-sphere. By geometrization, all such actions are conjugate to actions by isometries, and so we may use orthogonal representation theory to describe three cyclic and seven dihedral…
We use bordered Floer homology, specifically the immersed curve interpretation of the bordered pairing theorem, to compute various three- and four-dimensional invariants of satellite knots with arbitrary companions and patterns from a…
We study the representation spaces $R(K;\bf{i})$ as appearing in Kronheimer and Mrowka's framed instanton knot Floer homology, for a class of pretzel knots. In particular, for pretzel knots $P(p,q,r)$ with $p, q, r$ pairwise coprime, these…
We provide infinitely many rational homology 3-spheres with weight-one fundamental groups which do not arise from Dehn surgery on knots in $S^3$. In contrast with previously known examples, our proofs do not require any gauge theory or…
We define an invariant of transverse links in the standard contact 3-sphere as a distinguished element of the Khovanov homology of the link. The quantum grading of this invariant is the self-linking number of the link. For knots, this gives…
We prove the existence of an exact triangle for the Pin(2)-monopole Floer homology groups of three manifolds related by specific Dehn surgeries on a given knot. Unlike the counterpart in usual monopole Floer homology, only two of the three…
Real Heegaard Floer homology is an invariant associated to a three-manifold equipped with an involution with nonempty fixed set of codimension two. We show that when the image of the fixed point set is nullhomologous in the quotient, the…
We construct the first explicit example of a simplicial 3-ball B_{15,66} that is not collapsible. It has only 15 vertices. We exhibit a second 3-ball B_{12,38} with 12 vertices that is collapsible and evasive, but not shellable. Finally, we…
We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It carries information about the Floer homology…
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.…
The $L$-space conjecture asserts the equivalence, for prime $3$-manifolds, of three properties: not being an $L$-space ($NLS$), having a left-orderable fundamental group ($LO$), and admitting a co-orientable taut foliation ($CTF$). In this…
We study a theory of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries. It is an analogue in the setting of the rational homology of the Goussarov-Rozansky…
We explore certain restrictions on knots in the three-sphere which admit non-trivial Seifert fibered surgeries. These restrictions stem from the Heegaard Floer homology for Seifert fibered spaces, and hence they have consequences for both…
We define the longitude Floer homology of a knot K in S^3 and show that it is a topological invariant of K. Some basic properties of these homology groups are derived. In particular, we show that they distinguish the genus of K. We also…
Consider an unknot $c$ in $S^3$ and a knot $K$ in ${S^3-N(c)}$. Twisting the knot $K$ along $c$, or equivalently applying $\frac{1}{m}$-surgery on $c$, produces a family of knots $\{K_m\}_{m \in \mathbb{Z}}$. We use bordered Floer homology…