Detects figure-eight knot using Khovanov homology.
problem Detecting the figure-eight knot.
method Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology.
result Reduced Khovanov homology (over Q) detects the figure-eight knot.
Knot Floer homology matches fixed point Floer for fibred knots.
problem Understanding the algebraic structure of knot Floer homology.
method Proved isomorphism between Knot Floer and fixed point Floer homologies.
result Knot Floer homology of fibred knots matches fixed point Floer homology.
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…
Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable …
New knot homologies detect non-fibered knots, expanding on previous results.
problem Detecting non-fibered knots using knot homologies.
method Developed new knot homologies (knot Floer, Khovanov, HOMFLY) and proved a technical theorem.
result New homologies detect infinitely many non-fibered knots, expanding on previous results.
New colored knot Floer homology defined using infinite full twists.
problem Defining a new homology theory for knots.
method Defining colored knot Floer homology through colimit of link Floer homology with infinite full twists.
result Colored knot Floer homology is a module over the colored knot Floer homology of the unknot.
New spectral sequences define knot invariants.
problem Understanding strongly invertible knots.
method Two spectral sequences in knot Floer homology.
result Numerical invariant defined for strongly invertible knots.
Study shows infinite-rank summand in homology concordance group of knots.
problem Homology concordance of knots in integer homology three-spheres.
method Knot Floer homology to construct homology concordance homomorphisms.
result Homology concordance group modulo knots from S^3 contains an infinite-rank summand.
Spectral sequence connects knot homologies to quotient knots.
problem Distinguishing knots using homology.
method Construct spectral sequence relating Khovanov homology to quotient knots.
result Khovanov homology distinguishes certain slice disks.
Study on knot concordance and homology cobordism using Heegaard Floer homology.
problem Knot concordance and homology cobordism.
method Heegaard Floer homology.
result Recent results in knot concordance and homology cobordism.
Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.
problem Determine grid homology of diagonal knots and compare them to other knot types.
method Use grid diagrams and combinatorial knot Floer homology to analyze diagonal knots.
result Grid homology detects the number of prime factors and decompositions of the knot into non-integer tangles.
Algorithm calculates knot Floer homology for a specific knot type.
problem Computing knot Floer homology for (1,1) knots. method Algorithm based on fundamental group of (1,1) knots. result Algorithm successfully computes knot Floer homology.
Study links with annuli using sutured Floer homology.
problem Characterize links with specific cable structures.
method Apply sutured Floer homology techniques.
result Characterizations of links with (n,nm)-cables and (2,2m)-cables. Any knot in S3 may be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a homology sphere is nullhomotopic in a smooth homology ball if and only if that k…
New subgroup found in knot homology concordance group.
problem Understanding the structure of knot homology concordance groups.
method Applying filtered mapping cone formula to L-space knots and using connected knot complex.
result Contains a Z∞ subgroup. New cobordism maps for instanton knot homology help compute surgeries on knots.
problem Computing surgeries on knots using instanton knot homology.
method Construct cobordism maps for instanton knot homology.
result Compute the framed instanton Floer homology of certain knots.
New link detection results using knot and link Floer homology.
problem Detecting specific links and knots using Floer homology.
method Inspired by Khovanov homology, uses knot and link Floer homology.
result Detects specific links and knots with high precision.
Study shows rank of knot Floer homology detects Hopf links and classifies second smallest links.
problem Detecting and classifying links based on knot Floer homology ranks.
method Analyzes knot Floer homology ranks to identify and classify links.
result Classifies links of the second smallest knot Floer homology.
New example shows figure eight knot not smoothly concordant but homology cobordant.
problem Smooth concordance vs homology cobordism of knots.
method Construction of knots with specific properties.
result Figure eight knot not smoothly concordant but homology cobordant.
This paper defines a spectral sequence connecting knot homologies.
problem Link Floer homology and its connections to knot homologies.
method Iterating a modified skein exact triangle to create a spectral sequence.
result A spectral sequence from reduced Khovanov homology of the mirror of a knot to knot Floer homology of the knot.
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…
Study shows infinite hyperbolic knots with odd torsion in Khovanov homology.
problem Understanding torsion in Khovanov homology for hyperbolic knots.
method Analyzes knots in S3 with specific torsion in Khovanov homology. result Infinitely many hyperbolic knots with odd torsion in Khovanov homology.
New proofs of knot detection using instanton Floer homology.
problem Detecting specific knots using homology theories.
method Using instanton Floer homology instead of knot Floer homology.
result Proves detection of specific knots using instanton homology.
The paper calculates the slicing degree of knots using advanced homology theories.
problem Determining the minimum slicing degree of knots.
method Rasmussen's s-invariant, knot Floer homology, and singular instanton homology.
result Computed slicing degrees for many small knots and some families of torus knots.
Knot Floer homology stabilizes with twists.
problem Understanding knot behavior through homology.
method Analyzing m full twists on n parallel strands. result Knot Floer homologies stabilize as twists increase.
Positive braid knots have simple knot Floer homology.
problem Computing knot Floer homology for positive braids.
method Computed the next-to-top term of knot Floer homology.
result Rank is 1 for any prime positive braid knot.
Study of knot Floer homology and its relation to Heegaard Floer homology via equivariant surgery.
problem Understanding the action of symmetries on knot Floer homology.
method Relating knot Floer homology to Heegaard Floer homology via equivariant surgeries.
result Identify the action of the involution on Heegaard Floer homology with an action on knot Floer homology.
Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
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…
Computes upper bounds for instanton knot homology.
problem Computing the exact dimension of instanton knot homology.
method Developed an algorithm using knot diagrams and Heegaard diagrams.
result Algorithm provides sharp bounds for knots up to seven crossings.
The paper conjectures Khovanov homology can distinguish torus and twist knots.
problem Detecting and distinguishing knots using Khovanov homology.
method Examining all prime knots with up to 20 crossings, conjecturing Legendrian simplicity.
result Numerical evidence supports Khovanov homology distinguishing torus and twist knots.
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…
A well-known conjecture of Rasmussen states that for any knot K in S3, 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…
Knots without 2-torsion have minimal Khovanov homology rank.
problem Proving the minimality of Khovanov homology rank for knots without 2-torsion.
method Analyzing the effect of rational tangle replacements on Khovanov homology rank.
result Knots without 2-torsion have minimal Khovanov homology rank.
This paper computes the second quandle homology group of knot n-quandles.
problem Characterizing knots using quandle homology groups.
method Computation of the second quandle homology group for knot n-quandles.
result The second quandle homology group of knot n-quandles provides more information than the knot quandle's homology group.
New symplectic annular Khovanov homology connects knot theory to Floer homology.
problem Understanding the relationship between knot theory and Floer homology.
method Introducing a new version of symplectic annular Khovanov homology and establishing spectral sequences.
result Established spectral sequences linking different knot homologies.
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…
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.
Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured …
Knot lattice homology invariant of smooth knot type in rational homology spheres.
problem Invariance of knot lattice homology in rational homology spheres.
method Proving knot lattice homology invariant through doubly-filtered homotopy type.
result Knot lattice homology invariant of smooth knot type in rational homology spheres.
Lecture notes on Heegaard Floer homology for 3-manifolds and knots.
problem Understanding Heegaard Floer homology for 3-manifolds and knots.
method Explains Heegaard diagrams, Heegaard Floer homology, and knot Floer homology.
result Relationship between knot and 3-manifold invariants.
New technique connects instanton Floer homology to Heegaard diagrams for knots and 3-manifolds.
problem Accessing structural properties of instanton Floer homology.
method Developed a new technique using Heegaard diagrams and sutures.
result Established a relation between instanton knot homology and Heegaard diagrams for (1,1)-knots. A new surgery formula for knot lattice homology.
problem Developing a new surgery formula for knot lattice homology.
method Provided an iterable version of the surgery formula using doubly-filtered spaces and involutive data.
result Computed knot lattice spaces for specific knots and three-manifolds.
Algebraic methods prove knot primality using Floer homology.
problem Proving the primality of knots.
method Knot Floer homology, metacyclic representations, and twisted homology.
result Primality tests have proven primality for over 99.6% of knots.
We show that the knot lattice homology of a knot in an L-space is equivalent to the knot Floer homology of the same knot (viewed these invariants as filtered chain complexes over the polynomial ring Z/2Z [U]). Suppose that G is a negative definite plumbing tree which contains a vertex w such that G-w is a union of rati…
Let K be a rationally null-homologous knot in a three-manifold Y. We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot K. As an application, we express the Heegaard Floer homology of rational surgerie…
Persistent homology can recognize knotting in curves.
problem Recognizing knotting in curves
method Compute one-dimensional persistent homology, extract cycle representatives, and assign a hypergraph curvature-based score.
result Systematic differences between knotted and unknotted structures are revealed.
Formula for Heegaard Floer multicurves of double tangles from knot complements.
problem Computing Heegaard Floer multicurve invariants of double tangles.
method Using a simple formula derived from knot complements, comparing with Khovanov homology.
result New characterisation of L-space knots and first example of thin knot Floer homology.