Related papers: A general Heegaard Floer surgery formula

In this paper we study the knot Floer homology of a subfamily of twisted $(p, q)$ torus knots where $q \equiv\pm1$ (mod $p$). Specifically, we classify the knots in this subfamily that admit L-space surgeries. To do calculations, we use the…

Fix a 3-manifold $Y$ with boundary $F\amalg F$ and an orientation-preserving involution $\tau: Y\to Y$ exchanging the boundary components, with nonempty fixed set. To an appropriate kind of Heegaard diagram for $Y$, we describe how to…

In 2003, Ozsv\'ath and Szab\'o defined the concordance invariant $\tau$ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of $\tau$ for knots in $S^3$ and a…

Let Y(r) be the closed, oriented three-manifold obtained by performing rational r-surgery on the right-handed trefoil knot in the three-sphere. Using contact surgery and the Heegaard Floer contact invariants we construct positive, tight…

We explore the Fourier transform of the Heegaard Floer $d$-invariants, which is particularly well-behaved with respect to connected sum. As corollaries, we show that lens spaces are cancellable in the monoid of 3-manifolds up to integer…

We establish some general relations between Heegaard Floer based contact invariants. In particular, we observe that if the contact invariant of large negative, respectively positive, contact surgeries along a Legendrian knot does not…

In previous works, the author described an associative algebra whose $A_\infty$-module categories encode the Heegaard Floer Dehn surgery formulas. In this article, we describe the Koszul dual of this algebra. We construct dualizing…

We first present three graphic surgery formulae for the degree $n$ part $Z_n$ of the Kontsevich-Kuperberg-Thurston universal finite type invariant of rational homology spheres. Each of these three formulae determines an alternate sum of the…

Turaev showed that there is a well-defined map assigning to an oriented link L in the three-sphere a Spin structure t_0 on Sigma(L), the 2-fold cover of S^3 branched along L. We prove, generalizing results of Manolescu-Owens and…

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 relate decategorifications of Ozsv\'ath-Szab\'o's new bordered theory for knot Floer homology to representations of $\mathcal{U}_q(\mathfrak{gl}(1|1))$. Specifically, we consider two subalgebras $\mathcal{C}_r(n,\mathcal{S})$ and…

We continue our study of the integer-valued knot invariants $\nu^\sharp(K)$ and $r_0(K)$, which together determine the dimensions of the framed instanton homologies of all nonzero Dehn surgeries on $K$. We first establish a "conjugation"…

If a 3--manifold $Y$ contains a non-separating sphere, then some twisted Heegaard Floer homology of $Y$ is zero. This simple fact allows us to prove several results about Dehn surgery on knots in such manifolds. Similar results have been…

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…

We prove that the cut-system complex of a sutured compression body, with vertices representing cut-systems and edges corresponding to handleslides, becomes simply connected when six kinds of 2-cells are attached. Moreover, we define tight…

In this paper we prove another pairing theorem for bordered Floer homology. Unlike the original pairing theorem, this one is stated in terms of homomorphisms, not tensor products. The present formulation is closer in spirit to the usual…

We show that Ozsv\'ath-Szab\'o's bordered algebra used to efficiently compute knot Floer homology is a graded flat deformation of the regular block of a $\mathfrak{q}$-presentable quotient of parabolic category $\mathcal{O}$. We identify…

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…

This paper studies the existence of $2$-torsion in instanton Floer homology with $\mathbb{Z}$ coefficients for closed $3$-manifolds and singular knots. First, we show that the non-existence of $2$-torsion in the framed instanton Floer…

The cosmetic surgery conjecture predicts that for a non-trivial knot in the three-sphere, performing two different Dehn surgeries results in distinct oriented three-manifolds. Hanselman reduced the problem to $\pm 2$ or $\pm 1/n$ surgeries…