arXiv research

Maps from rational homology solid tori yield rank inequalities in Heegaard Floer homology.

Classifies links with small Khovanov homology ranks.

The paper classifies links with low rank knot Floer and Khovanov homologies.

We show that if K is a non-trivial knot inside a homology sphere Y, then the rank of knot Floer homology associated with K is strictly bigger than the rank of Heegaard Floer homology of Y.

Study shows rank of knot Floer homology detects Hopf links and classifies second smallest links.

A well-known conjecture of Rasmussen states that for any knot $K$ in $S^{3}$, the rank of the reduced Khovanov homology of $K$ is greater than or equal to the rank of the reduced knot Floer homology of $K$. This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Flo…

New evidence refutes old conjectures about knot homology ranks, suggesting new congruences.

We establish three rank inequalities for the reduced flavor of Heegaard Floer homology of Seifert fibered integral homology spheres. Combining these inequalities with the known classifications of non-zero degree maps between Seifert fibered spaces, we prove that a map f from Y' to Y between Seifert homology spheres yie…

This paper studies knots in sutured manifolds achieving minimum instanton homology rank.

Study Brieskorn spheres using Floer homology, generating infinite rank summands in homology cobordism.

Classifies links with minimal Khovanov homology rank.

Improved homological dimension for certain subgroups in Lie groups.

Study shows infinite-rank summand in homology concordance group of knots.

We show that if K is a non-trivial knot inside a homology sphere X, the rank of the knot Floer homology group associated with K is strictly bigger than the rank of the Heegaard Floer homology group associated with X.

Paper stabilizes persistent homology rank functions for statistical inference.

Khovanov homology ranks 2 for certain knots in a specific bundle.

Knots without 2-torsion have minimal Khovanov homology rank.

Let $K$ denote a knot inside the homology sphere $Y$ and $K'$ denote a knot inside a homology sphere $L$-space. Let $X=Y(K,K')$ denote the 3-manifold obtained by splicing the complements of $K$ and $K'$. We show that $\text{rank}(\widehat{HF}(X)) \ge \text{rank}(\widehat{HF}(Y))$.

We establish a relationship between Heegaard Floer homology and the fractional Dehn twist coefficient of surface automorphisms. Specifically, we show that the rank of the Heegaard Floer homology of a 3-manifold bounds the absolute value of the fractional Dehn twist coefficient of the monodromy of any of its open book d…

We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil. We classify links whose branched double cover gives rise to this manifold. Together with a spectral sequence from Khovanov homology to the Floe…

Study shows knot grading properties in specific spaces.

In this paper we construct families of homology spheres which bound 4-manifolds with intersection forms isomorphic to $-E_8$. We show that these families have arbitrary large correction terms. This result says that among homology spheres, the difference of the maximal rank of minimal sub-lattice of definite filling and…

A well-known conjecture states that for any $l$-component link $L$ in $S^3$, the rank of the knot Floer homology of $L$ (over any field) is less than or equal to $2^{l-1}$ times the rank of the reduced Khovanov homology of $L$. In this paper, we describe a framework that might be used to prove this conjecture. We const…

The study shows a new upper bound on the homology rank of hyperbolic 3-manifolds.

The homology cobordism group of homology cylinders is a generalization of the mapping class group and the string link concordance group. We study this group and its filtrations by subgroups by developing new homomorphisms. First, we define extended Milnor invariants by combining the ideas of Milnor's link invariants an…

Proves homological inequality for cycles in Hadamard spaces of asymptotic rank 2.

Let $E_{k}^{F}(D)$ be the spectral sequence induced by the oriented cube of resolutions on knot Floer homology. We prove that $E_{2}^{F}(D)$ is a triply graded link invariant whose graded Euler characteristic is the HOMFLY-PT polynomial and that the higher pages are link invariants. By construction, the spectral sequen…

Let $\widetilde{K}$ be a 2-periodic knot in $S^3$ with quotient $K$. We prove a rank inequality between the knot Floer homology of $\widetilde{K}$ and the knot Floer homology of $K$ using a spectral sequence of Hendricks, Lipshitz and Sarkar. We also conjecture a filtered refinement of this inequality, for which we giv…

We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains a $\mathbb{Z}^\infty$-subgroup and a $\mathbb{Z}$-summand. Our proof proceeds by introducing an algebraic variant of the involutive Heegaard Floer package of He…

Study satellite operators expanding concordance groups and their applications.

Characterizes $(1,1)$ knots with large Dehn surgeries.

The aim of this paper is to define a homology theory for racks with finite rank N and use it to define invariants of knots generalizing the CJKLS 2-cocycle invariants related to the invariants defined in [15]. For this purpose, we prove that N -degenerate chains form a sub-complex of the classical complex defining rack…

We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Flo…

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 members of the family and list the resulting Heegaard--Floer homologies. We also show …

Study finds lower bounds on flat cycles in congruence covers of symmetric spaces.

Paper classifies $CFK^\infty$ type of almost L-space knots.

We use group homology to define invariants in algebraic K-theory and in an analogue of the Bloch group for Q-rank one lattices and for some other geometric structures. We also show that the Bloch invariants of CR structures and of flag structures can be recovered by a fundamental class construction.

We show that the set of even positive definite lattices that arise from smooth, simply-connected 4-manifolds bounded by a fixed homology 3-sphere can depend on more than the ranks of the lattices. We provide two homology 3-spheres with distinct sets of such lattices, each containing a distinct nonempty subset of the ra…

In this paper we compute the signature for a family of knots $W(k,n)$, the weaving knots of type $(k,n)$. By work of E.~S.~Lee the signature calculation implies a vanishing theorem for the Khovanov homology of weaving knots. Specializing to knots $W(3,n)$, we develop recursion relations that enable us to compute the Jo…

We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of righ…

Examples of knots and links distinguished by the total rank of their Khovanov homology but sharing the same two-fold branched cover are given. As a result, Khovanov homology does not yield an invariant of two-fold branched covers.

Positive braid knots have simple knot Floer homology.

We prove a basic inequality for the d-invariants of a splice of knots in homology spheres. As a result, we are able to prove a new relation on the rank of reduced Floer homology under maps between Seifert fibered homology spheres, improving results of the first and second authors. As a corollary, a degree one map betwe…

A deformation of the authors' instanton homology for webs is constructed by introducing a local system of coefficients. In the case that the web is planar, the rank of the deformed instanton homology is equal to the number of Tait colorings of the web.

Study shows inequality in Floer homologies for 3-manifold covers.

Study bounds topological entropy of maps on surfaces with punctures based on mapping torus homology.

We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant.

We prove a rank inequality on the instanton knot homology and the Khovanov homology of a link in $S^3$. The key step of the proof is to construct a spectral sequence relating Baldwin-Levine-Sarkar's pointed Khovanov homology to a singular instanton invariant for pointed links.