Related papers: A note on the concordance invariants epsilon and u…

We compute the knot Floer filtration induced by a cable of the meridian of a knot in the manifold obtained by large integer surgery along the knot. We give a formula in terms of the original knot Floer complex of the knot in the…

We compute the effect of concordance surgery, a generalization of knot surgery defined using a self-concordance of a knot, on the Ozsv\'ath-Szab\'o 4-manifold invariant. The formula involves the graded Lefschetz number of the concordance…

We give a purely combinatorial proof of a K\"{u}nneth formula for the minus version of knot Floer homology of connected sums by constructing a quasi-isomorphism of grid chain complexes. The quasi-isomorphism naturally deduces that the…

Knot Floer homology is an invariant for knots discovered by the authors and, independently, Jacob Rasmussen. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology,…

We use bordered Floer homology to give a formula for the knot Floer homology of any (p, pn+1)-cable of a thin knot K in terms of Delta_K(t), tau(K), p, and n. We also give a formula for the Ozsvath-Szabo concordance invariant tau(K_{p,…

Based on the data of 12-17-crossing knots, we establish three new conjectures about the hyperbolic volume and knot cohomology: (1) There exists a constant $a \in R_{>0}$ such that the percentage of knots for which the following inequality…

We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds…

In this paper, we give a combinatorial description of the concordance invariant $\varepsilon$ defined by Hom in \cite{hom2011knot}, prove some properties of this invariant using grid homology techniques. We also compute $\varepsilon$ of…

We define band maps in unoriented link Floer homology and show that they form an unoriented skein exact triangle. These band maps are similar to the band maps in equivariant Khovanov homology given by the Lee deformation. As a key tool, we…

This note corrects the mistakes in the splicing formulas of the paper "Floer homology and splicing knot complements". The mistakes are the result of the incorrect assumption that for a knot $K$ inside a homology sphere $Y$, the involution…

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…

We show that an infinite family of contractible 4-manifolds have the same boundary as a special type of plumbing. Consequently their Ozsvath--Szabo invariants can be calculated algorithmically. We run this algorithm for the first few…

Ozsvath and Szabo proved that knot Floer homology determines the genera of knots in S^3. We will generalize this deep result to links in homology 3-spheres, by adapting their method. Our proof relies on a result of Gabai and some…

We use unoriented versions of instanton and knot Floer homology to prove inequalities involving the Euler characteristic and the number of local maxima appearing in unorientable cobordisms, which mirror results of a recent paper by Juhasz,…

Inspired by the $S^n$ colored version of Khovanov and Khovanov-Rozansky homology, we define a colored version of knot Floer homology by studying the colimit of a directed system of link Floer homology with infinite full twists.…

We prove the equivalence of the invariants EH(L) and LOSS-(L) for oriented Legendrian knots L in the 3-sphere equipped with the standard contact structure, partially extending a previous result by Stipsicz and Vertesi. In the course of the…

The Heegaard Floer correction term ($d$-invariant) is an invariant of rational homology 3-spheres equipped with a Spin$^c$ structure. In particular, the correction term of 1-surgeries along knots in $S^3$ is a ($2\mathbb{Z}$-valued) knot…

We equip the basic local crossing bimodules in Ozsv\'ath-Szab\'o's theory of bordered knot Floer homology with the structure of 1-morphisms of 2-representations, categorifying the $U_q(\mathfrak{gl}(1|1)^+)$-intertwining property of the…

We automate the process of machine learning correlations between knot invariants. For nearly 200,000 distinct sets of input knot invariants together with an output invariant, we attempt to learn the output invariant by training a neural…

In a previous work, we defined an unoriented skein exact triangle in unoriented link Floer homology. In this paper, we iterate a modified version of this exact triangle and obtain a spectral sequence from various versions of Khovanov…