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…
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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…
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…
Knot Floer homology detects ribbon concordance.
problem Detecting ribbon concordance between knots.
method Using knot Floer homology to study ribbon concordance.
result The map induced by ribbon concordance on knot Floer homology is injective.
Study shows knots can have large genus difference from concordance.
problem Understanding genus differences in knots and surfaces.
method Analyzes the topological 4-genus and minimal genus of bounded surfaces.
result Arbitrarily large genus difference between knots and their concordance.
The concordance genus of a knot K is the minimum three-genus among all knots concordant to K. For prime knots of 10 or fewer crossings there have been three knots for which the concordance genus was unknown. Those three cases are now resolved. Two of the cases are settled using invariants of Levine's algebraic concorda…
Study shows concordance invariants bound Turaev genus.
problem Understanding the Turaev genus of knots.
method Using differences between concordance invariants, including Rasmussen's -invariant and -invariants.
result Established lower bounds for Turaev genus and provided examples of quasi-alternating knots with specific genus values.
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.
New homomorphisms from knot Floer homology help classify knots.
problem Classifying knots based on their concordance properties.
method Defined an infinite family of concordance homomorphisms using knot Floer complexes.
result Explicitly computable homomorphisms that are linearly independent.
Obstructs Legendrian knots from being slices of concordances using doubly slice genus.
problem Obstructing Legendrian knots from being slices of concordances.
method Uses Eliashberg and Polterovitch's result on doubly slice genus as an obstruction.
result Obstructs Legendrian knots from being slices of concordances, including examples of Pretzel knots.
The knot concordance invariant Upsilon, recently defined by Ozsvath, Stipsicz, and Szabo, takes values in the group of piecewise linear functions on the closed interval [0,2]. This paper presents a description of one approach to defining Upsilon and of proving its basic properties related to the knot 3-genus, 4-genus, …
New knot invariants from instanton homology.
problem Bounding knot genus and double points.
method Instanton homology groups with local coefficients.
result 1-parameter family of homomorphisms from knot concordance group to reals.
New homomorphism from Khovanov homology gives slice genus bounds.
problem Understanding slice genus of knots.
method A 1-parameter family of concordance homomorphisms from Khovanov homology.
result Can prove linear independence of certain knot families.
New concordance invariant from spectral sequence on Khovanov homology.
problem Concordance invariants from spectral sequences.
method Constructing from spectral sequence on Khovanov homology.
result Provides a bound on the nonorientable slice genus.
In knot concordance three genera arise naturally, g(K), g_4(K), and g_c(K): these are the classical genus, the 4-ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to K. Clearly 0 <= g_4(K) <= g_c(K) <= g(K). Casson and Nakanishi gave examples to show that g_4(K) need not …
Kearton observed that mutation can change the concordance class of a knot. A close examination of his example reveals that it is of 4-genus 1 and has a mutant of 4-genus 0. The first goal of this paper is to construct examples to show that for any pair of nonnegative integers m and n there is a knot of 4-genus m with a…
We modify the construction of knot Floer homology to produce a one-parameter family of homologies for knots in the three-sphere. These invariants can be used to give homomorphisms from the smooth concordance group to the integers, giving bounds on the four-ball genus and the concordance genus of knots. We give some app…
Cha and Kim proved that if a knot K is not algebraically slice, then no iterated Bing double of K is concordant to the unlink. We prove that if K has nontrivial signature , then the n-iterated Bing double of K is not concordant to any boundary link with boundary surfaces of genus less than . The same resul…
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 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.
New invariants improve Heegaard Floer slice genus and clasp number bounds.
problem Improving bounds for knot concordance.
method Using knot Floer homology and involutive correction terms.
result Improved slice genus and clasp number bounds proved.
This paper shows that certain knots are equivalent under a specific relation.
problem Understanding the equivalence classes of genus one knots.
method Using knot Floer complexes and Heegaard Floer theory invariants.
result Any genus one knot is -equivalent to a trefoil, its mirror, or the unknot.
We define a family of link concordance invariants . These link concordance invariants give lower bounds on the slice genus of a link . We compute the slice genus of positive links. Moreover, these invariants give lower bounds on the link splitting number of a link. Especially, t…
Lower bound on stable 4-genus of knots using Casson-Gordon signatures.
problem Finding a lower bound on the stable 4-genus of knots.
method Using Casson-Gordon τ-signatures to compute the lower bound.
result A twist knot is torsion in the knot concordance group if and only if it has vanishing stable 4-genus.
New examples show clasp numbers can be zero yet four-genus can be arbitrarily large.
problem Understanding the relationship between clasp numbers and four-genus for knots.
method Constructing knots with specific clasp numbers and four-genus values.
result Examples of knots with zero clasp numbers but arbitrarily large four-genus.
New knots found that are 4-genus minimal.
problem Finding knots with minimal 4-genus.
method Constructing infinitely many amphichiral knots with specific properties.
result Knots with 4-genus minimal for each .
We define a filtration of the smooth concordance group based on the genus of representative knots. We use the Heegaard Floer epsilon and Upsilon invariants to prove the quotient groups with respect to this filtration are infinitely generated. Results are applied to three infinite families of topologically slice knots.
We define a quasihomomorphism from braid groups to the concordance group of knots and examine its properties and consequences of its existence. In particular, we provide a relation between the stable four ball genus in the concordance group and the stable commutator length in braid groups, and produce examples of infin…
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.
New invariants from framed instanton homology for knot concordance.
problem Concordance of knots and their properties.
method Framed instanton homology to define invariants.
result Computations and bounds on knot concordance invariants.
We derive a linear estimate of the signature of positive knots, in terms of their genus. As an application, we show that every knot concordance class contains at most finitely many positive knots.
New unoriented algebraic concordance group defined using mock Seifert matrices.
problem Understanding unoriented algebraic concordance of knots in thickened surfaces.
method Introducing mock Seifert matrices and using them to define unoriented algebraic concordance.
result The unoriented algebraic concordance group is abelian and infinitely generated.
The study establishes a link between fibered links and their concordance invariants.
problem Determining the conditions for fibered strongly quasi-positive links.
method Proved that an -component fibered link is strongly quasi-positive if and only if .
result Explicitly determined fibered prime links with at most 9 crossings and their properties.
In this note, we combine the recent 4-dimensional light bulb theorem of David Gabai and a recent construction of concordances for knots in due to Eylem Zeliha Yildiz to construct a concordance between the standard surface of genus in and any homologous surface.
Sharp knots and iterated cables lead to ribbon knots or failure of slice-ribbon conjecture.
problem Understanding the concordance properties of knots and their cables.
method Using Seifert genus and concordance invariant γ0 from bordered Heegaard Floer homology.
result Connected sums of γ0-sharp fibered knots are ribbon only if they are of a specific form, or the slice-ribbon conjecture fails.
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…
In this paper we will show that two surfaces of the same genus and homology class in a simply connected 4-manifold are concordant. We will show they are often topologically isotopic when their complements have cyclic fundamental group. Finally, we will show that if they are 0-concordant, then surgery on one is equivale…
We give bounds on knot signature, the Ozsvath-Szabo tau invariant, and the Rasmussen s invariant in terms of the Turaev genus of the knot.
Lagrangian Zigzag Cobordismsmath.SG
Study Lagrangian zigzag cobordisms for Legendrian knots, comparing to smooth concordance.
problem Understanding Legendrian knots through Lagrangian cobordisms.
method Defined an equivalence relation on Legendrian knots using interpolating zigzag Lagrangian cobordisms, studied metric monoid, and compared to other concordance types.
result Proved structural results on torsion and satellite operators in the Lagrangian zigzag concordance classes.
New knots found that can't be turned into L-space after surgery.
problem Finding knots that can't be transformed into L-space after surgery.
method Examined algebraically concordant torus knots and fibred knots.
result Infinitely many knots of genus cannot be turned into L-space.
The generalized Alexander polynomial vanishes for virtually slice knots.
problem Identifying knots that are concordant to homologically trivial knots.
method Using Bar-Natan Zh-correspondence and concordance properties of maps between links and tori.
result The generalized Alexander polynomial vanishes on virtually slice knots.
The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2].…
New invariant defined for unoriented knots, proving no factorization through topological concordance.
problem Defining and proving properties of unoriented slice-torus invariants.
method Introducing and proving properties of unoriented slice-torus invariants.
result Unoriented slice-torus invariants do not factor through the topological concordance group.
Study of knots with generalized Mazur patterns and their invariants.
problem Understanding the invariants and properties of knots with generalized Mazur patterns.
method Computational analysis of and invariants for -twisted satellites.
result None of the -twisted patterns from the family act surjectively on the smooth or rational concordance group.
Proves a concordance version of a 4D light bulb theorem.
problem Proving concordance of embedded spheres in 4-manifolds.
method Analyzes homotopically embedded 2-spheres in 4-manifolds with specific conditions.
result Concordance of spheres under given conditions.
This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relatio…
Using the theory of involutive Heegaard Floer knot theory developed by Hendricks-Manolescu, we define two involutive analogs of the Upsilon knot concordance invariant of Ozsvath-Stipsicz-Szabo. These involutive invariants are piecewise linear functions defined on the interval [0,2]. Each is a concordance invariant and …
It is well-known that generic perturbations of the complex Frobenius algebra used to define Khovanov cohomology each give rise to Rasmussen's concordance invariant s. This gives a concordance homomorphism to the integers and a strong lower bound on the smooth slice genus of a knot. Similar behavior has been observed in…