arXiv research

The paper classifies knot Floer complexes of low width, simplifying knot bases.

To each knot $K\subset S^3$ one can associated its knot Floer homology $\hat{HFK}(K)$, a finitely generated bigraded abelian group. In general, the nonzero ranks of these homology groups lie on a finite number of slope one lines with respect to the bigrading. The width of the homology is, in essence, the largest horizo…

Knot Floer homology matches fixed point Floer for fibred knots.

Study links with annuli using sutured Floer homology.

New spectral sequences define knot invariants.

Proves properties of instanton knot Floer homology and connected sum formula.

New colored knot Floer homology defined using infinite full twists.

Algorithm calculates knot Floer homology for a specific knot type.

Formula for Heegaard Floer multicurves of double tangles from knot complements.

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

New link detection results using knot and link Floer homology.

Study of knot Floer homology and its relation to Heegaard Floer homology via equivariant surgery.

Knot Floer homology is a knot invariant defined using holomorphic curves. In more recent work, taking cues from bordered Floer homology,the authors described another knot invariant, called "bordered knot Floer homology", which has an explicit algebraic and combinatorial construction. In the present paper, we extend the…

Knot Floer homology stabilizes with twists.

Lecture notes on Heegaard Floer homology for 3-manifolds and knots.

Positive braid knots have simple knot Floer homology.

Study shows how knot Floer homology and bordered Floer theory are linked.

This note explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology whic…

We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus two mutant which shares the same total dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its genus two mutant by both knot Floer hom…

In this paper, we introduce a sequence of invariants of a knot K in S^3: the knot Floer homology groups of the preimage of K in the m-fold cyclic branched cover over K. We exhibit the knot Floer homology in the m-fold branched cover as the categorification of a multiple of the Turaev torsion in the case where the m-fol…

New method connects knot Floer homology with bordered Floer homology.

We define the Wirtinger width of a knot. Then we prove the Wirtinger width of a knot equals its Gabai width. The algorithmic nature of the Wirtinger width leads to an efficient technique for establishing upper bounds on Gabai width. As an application, we use this technique to calculate the Gabai width of approximately …

Algorithm computes knot Floer complex for knots of thickness one.

We derive symmetries and adjunction inequalities of the knot Floer homology groups which appear to be especially interesting for homologically essential knots. Furthermore, we obtain an adjunction inequality for cobordism maps in knot Floer homologies. We demonstrate the adjunction inequalities and symmetries in explic…

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

We develop a skein exact sequence for knot Floer homology, involving singular knots. This leads to an explicit, algebraic description of knot Floer homology in terms of a braid projection of the knot.

Study shows non-trivial knot Floer homology for specific knots.

Study on knot concordance and homology cobordism using Heegaard Floer homology.

Floer homology detects right-veering monodromy in fibered knots.

Paper classifies Heegaard Floer minimal knots in sutured manifolds.

In "Width complexes for knots and 3-manifolds," Jennifer Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the 3-sphere and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the…

Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.

Study corrects previous work on knot Floer homology of certain pretzel knots.

Formula for satellite operators using knot Floer homology.

A formula for bordered Floer homology of concordances and satellites

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, changes as the three-manifold undergoes Dehn surgery along a knot. Since its origin…

New Heegaard Floer homology findings block chirally cosmetic surgeries.

We extend the classical definition of {\it width} to higher dimensional, smooth codimension 2 knots and show in each dimension there are knots of arbitrarily large width.

Study contact invariants using Floer homology to understand knots.

In this paper, we compute the symplectic Floer homology of the figure eight knot. This provides first nontrivial knot with trivial symplectic Floer homology.

The paper studies knot Floer homology under Murasugi sum and establishes graded isomorphisms.

Study calculates instanton Floer homology for surgeries on L-space knots.

Study satellite knots using bordered Floer theory, proving non-thinness and calculating genus.

It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of $K$ detects more structure of minimal genus Seifert surfaces for $K$. We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homol…

Study computes SL(2,C) Floer cohomology for surgeries on knots.

Study knot Floer homology to create concordance invariants and slice genus bounds.

Algebraic methods prove knot primality using Floer homology.

Tests using knot Floer homology detect prime knots with high accuracy.