Abstract: Proves non-abelian group of equivariant concordance.
problem Equivariant concordance group non-abelian
method Infinite family of nontrivial commutators
result Equivariant concordance group is not abelian
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Defines a new homomorphism for strongly invertible knots, proving equivariant algebraic concordance.
problem Equivariant algebraic concordance of strongly invertible knots.
method Defining a homomorphism from equivariant concordance group to a new equivariant algebraic concordance group, proving it lifts known homomorphisms and provides new obstructions.
result Obtains a new obstruction to equivariant sliceness and novel lower bounds on equivariant slice genus.
Study on 2-bridge knots, proving equivariant concordance order is infinite.
problem Equivariant concordance of 2-bridge knots.
method Formula for butterfly polynomial, two proofs of non-equivariant sliceness, new invariant for strongly invertible knots.
result Equivariant concordance order of 2-bridge knots is infinite.
Study of equivariant ribbon concordance using Khovanov homology.
problem Understanding equivariant ribbon concordance.
method Functoriality of equivariant Khovanov homology under equivariant cobordisms, and induced split injection.
result Equivariant ribbon concordances induce a split injection on equivariant Khovanov homology.
Strongly invertible knots, equivariant slice genera, and an equivariant algebraic concordance groupmath.GT
Study on knots, genera, and algebraic concordance groups.
problem Understanding the equivariant slice genus of strongly invertible knots.
method Using the Blanchfield form to establish lower bounds and formulate an equivariant algebraic concordance group.
result The equivariant slice genus of an equivariant connected sum of a genus one strongly invertible slice knot is at least n/4.
2-knots with symmetry are classified up to equivariant concordance.
problem Classifying 2-knots with symmetry up to equivariant concordance.
method Constructing a new invariant called periodic, based on the Arf invariant.
result The smooth equivariant concordance group of 2-knots in is isomorphic to for all .
Study knot Floer homology to create concordance invariants and slice genus bounds.
problem Developing concordance invariants using knot Floer homology.
method Using knot Floer homology, define and analyze equivariant concordance invariants.
result Showed a family of strongly invertible slice knots with arbitrarily large equivariant slice genus.
Study of Khovanov homology invariants from -equivariant algebra.
problem Understanding concordance invariants from Khovanov homology.
method Analysis of -equivariant Khovanov homology using algebraic filtrations.
result Extracted two families of concordance invariants.
Study on equivariant Q-sliceness for strongly invertible knots.
problem Understanding Q-sliceness for strongly invertible knots.
method Constructive and obstructive approaches using Fox-Milnor condition and equivariant concordance.
result Klein amphichiral knots are equivariant Q-slice in a single Q-homology 4-ball.
Proves non-solvability of concordance groups using Milnor invariants.
problem Non-solvability of concordance groups of 2-string links and strongly invertible knots.
method Using Milnor invariants to prove non-solvability.
result Proves non-solvability of and equivariant concordance groups of strongly invertible knots.
New knot concordance invariants from Seiberg-Witten theory bound slice genus.
problem Bounding the slice genus of knots.
method Equivariant Seiberg-Witten-Floer cohomology applied to cyclic covers.
result Lower bounds on slice genus from knot concordance invariants.
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.
New findings on knot genera using advanced techniques.
problem Understanding the 4-genus of knots, especially strongly invertible and periodic ones.
method Innovative concordance group invariants, Donaldson's theorem, and g-signature.
result Many new examples showing the equivariant 4-genus is larger than the 4-genus.
We construct smooth concordance invariants of knots which take the form of piecewise linear maps from [0,1] to R, one for each n greater than or equal to 2. These invariants arise from sl(n) knot cohomology. We verify some properties which are analogous to those of the invariant Upsilon (which arises from knot Floer ho…
Seidel-Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith-type inequalities. Similar-looking spectral sequences have been defined by Lee, Bar-Natan, Ozsváth-Szabó, Lipshitz-Tre…
The paper introduces new knot invariants using singular instanton gauge theory.
problem Developing new knot invariants using singular instanton gauge theory.
method Using singular instanton gauge theory, the paper constructs invariants and Morse chain complexes.
result The constructions lead to a triad of groups and several concordance invariants.
Study uses instanton Floer theory to obstruct knot unknotting operations.
problem Obstructing knot unknotting operations and ribbon concordance.
method Equivariant singular instanton Floer theory with Chern--Simons filtration.
result For a large class of slice knots, any unknotting sequence must contain both signs.
Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.
problem Finite group actions on rational homology 3-spheres.
method Equivariant version of Seiberg-Witten-Floer stable homotopy type.
result Definition of -invariants with Froyshov-type inequality.
Study cyclic group actions on spin 4-manifolds with boundary using Seiberg-Witten theory.
problem Understanding cyclic group actions on spin 4-manifolds with boundary.
method Using Seiberg-Witten equations and lattice constructions, define equivariant refinements of the invariant κ.
result Equivariant relative 10/8-ths type inequalities for spin equivariant cobordisms between rational homology spheres.
New knot concordance invariants from instantons and Floer theory.
problem Defining and computing concordance invariants of knots.
method Equivariant singular instanton Floer theory with Chern-Simons filtration.
result Computing -invariant and fractional ideal invariants for two-bridge knots.
Manolescu correction terms are numerical invariants of homology three-spheres arising from -equivariant Seiberg-Witten theory that contain information about homology cobordism. We discuss several constraints on these invariants for homology spheres obtained by Dehn surgery on a knot in the three-sphere…
Paper introduces IC-index to evaluate interaction prediction methods.
problem Evaluate interaction prediction methods using IC-index.
method IC-index measures interaction direction prediction performance.
result IC-index complements existing prediction performance estimators.
New local equivalence groups refine Rasmussen's s-invariant.
problem Refining Rasmussen's s-invariant for knot concordance.
method Introducing local equivalence groups combining Khovanov homologies.
result A refined s-invariant that determines triviality of knot images.
Abstract invariant cannot be expressed using various slice-torus invariants.
problem Cannot express Iida-Taniguchi's slice-torus invariant using other known invariants.
method Analysis of various known invariants and their properties.
result Iida-Taniguchi's slice-torus invariant cannot be realized as a linear combination of other invariants.
Stability theorem for concordance embeddings with applications.
problem Stability of concordance embeddings.
method Proving a stability theorem.
result Various applications to diffeomorphisms and homeomorphisms.
We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo concordance in Yx[0,1] and the action of the concordance group of knots in the 3-sphere that ties in local knots. We prove that the trivial fre…
The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11--crossing prime knots. This analysis resolves the concordance ge…
This work shows non-abelian quotient groups in string link concordance.
problem Understanding the structure of string link concordance groups.
method Analyzing the quotient groups formed by string link concordance and pure braid groups.
result The quotient of each string link concordance group by its pure braid subgroup is non-abelian.
Khovanov homology shows (4,5) torus knot is a summand in its concordance class.
problem Understanding the global ribbon minima in knot concordance classes.
method Application of Khovanov homology to ribbon concordance.
result Khovanov homology of (4,5) torus knot is a summand in any knot's homology.
Study shows knots with similar Blanchfield forms can be homotopy ribbon concordant.
problem Understanding homotopy ribbon concordance for knots.
method Using Blanchfield pairings and twisted Alexander polynomials.
result Existence of infinite families of knots with same Blanchfield form but not homotopy ribbon concordant.
New homomorphism from Khovanov homology for knot concordance.
problem Understanding smooth concordance of knots.
method Modulo equivalence relation on Khovanov chain complex.
result Strictly stronger than Rasmussen invariants.
Study knot invariants to answer questions about slice genus and clasp numbers.
problem Whether the difference between the four-dimensional clasp number and the slice genus can be arbitrarily large.
method Equivariant singular instanton theory and Chern--Simons functional.
result Answers a conjecture by Livingston about slicing numbers and provides a lower bound for the unoriented slice genus.
The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that ther…
Kirby and Lickorish showed that every knot in the 3-sphere is concordant to a prime knot, equivalently, every concordance class contains a prime knot. We prove here that their result can be strengthened: Every knot in the 3-sphere is invertibly concordant to a prime knot. A consequence is that every double concordance …
The Stable Concordance Genusmath.GT
The concordance genus of a knot is the least genus of any knot in its concordance class. Although difficult to compute, it is a useful invariant that highlights the distinction between the three-genus and four-genus. In this paper we define and discuss the stable concordance genus of a knot, which describes the behavio…
Introduces slice knots and concordance, linking to exotic smooth structures.
problem Understanding slice knots and their equivalence through concordance.
method Explains connections and introduces filtrations, satellite operations.
result Connections between slice knots and exotic smooth structures on \(\mathbb{R}^4\).
We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second homology of the 4-manifold. This naturally extends r-shake slice, a…
New hyperbolic knots not concordant to algebraic ones found.
problem Identifying knots not concordant to algebraic knots.
method Constructing hyperbolic L-space knots.
result Found hyperbolic knots that are not concordant to algebraic knots.
We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove the existence of non-trivial almost-concordance classes in all non-abelian 3-manif…
We define an infinite family of linearly independent, integer-valued smooth concordance homomorphisms. Our homomorphisms are explicitly computable and rely on local equivalence classes of knot Floer complexes over the ring . We compare our invariants to other concordance homomorphisms coming fr…
Virtual Covers of Links IImath.GT
A fibered concordance of knots, introduced by Harer, is a concordance between fibered knots that is well-behaved with respect to the fibrations. We consider semi-fibered concordance of two component ordered links with fibered. These are concordances that restrict to fibered concordances on the first …
Knots can be ordered by ribbon concordance, solving a long-standing question.
problem Ordering knots by ribbon concordance.
method Representation varieties of knot groups to and relations induced by ribbon concordance.
result Ribbon concordance forms a partial ordering on the set of knots.
We define a new smooth concordance homomorphism based on the knot Floer complex and an associated concordance invariant, epsilon. As an application, we show that an infinite family of topologically slice knots are independent in the smooth concordance group.
It was shown by Jim Davis that a 2-component link with Alexander polynomial one is topologically concordant to the Hopf link. In this paper, we show that there is a 2-component link with Alexander polynomial one that has unknotted components and is not smoothly concordant to the Hopf link, answering a question of Jim D…
Adding a braid closure to a fibered knot makes a link ribbon concordance minimal.
problem Finding the minimal ribbon concordance for links in .
method Link Floer homology combined with classical techniques.
result Any link in can be made ribbon concordance minimal by adding a single unknot.
The Mazur pattern acts by the identity up to topological concordance.
problem Understanding the action of the Mazur pattern up to topological concordance.
method Comparing satellite operators and using topological concordance properties.
result Evidence that the Mazur pattern acts by the identity up to topological concordance.
Links with nontrivial Alexander polynomial which are topologically concordant to the Hopf linkmath.GT
We give infinitely many -component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any -component link with trivial Alexander polynomial. Our examples are pairwise non-concordant.
New findings on knot concordance show limitations to primary decompositions.
problem Understanding the structure of knot concordance groups.
method Analyzing homomorphisms and polynomial factorizations of Alexander polynomials.
result Primary decompositions of topologically slice knots cannot exist.