New bounds on topological slice genus for torus knots.
problem Estimating the topological slice genus of torus knots.
method Analyzing the locally flat slice genus and applying it to derive bounds.
result Best possible linear estimate of topological slice genus for torus knots with non-maximal signature.
Alexander polynomial degree bounds twice a knot's topological slice genus.
problem Determining the topological slice genus of knots.
method Using Freedman's disc theorem and Alexander polynomial properties.
result The degree of the Alexander polynomial is an upper bound for twice the topological slice genus.
We obtain new lower bounds of the minimal genus of a locally flat surface representing a 2-dimensional homology class in a topological 4-manifold with boundary, using the von Neumann-Cheeger-Gromov ρ-invariant. As an application our results are employed to investigate the slice genus of knots. We illustrate examples …
Local knots can't bound smaller surfaces in rational homology 3-spheres.
problem Understanding local knots and their bounds in rational homology 3-spheres.
method Using Heegaard Floer invariant ν+ and additivity results.
result Local knots from ν+ -sharp knots have rational slice genus equal to the slice genus of the original knot.
Characterizes values of slice-torus invariants related to knot genus.
problem Understanding the values of slice-torus invariants for knots.
method Characterization based on stable smooth slice genus.
result Existence of slice torus invariants without explicit constructions.
The paper characterizes Z-slice genus using algebraic unknotting and linking forms.
problem Characterizing the Z-slice genus of knots and surfaces.
method Balanced algebraic unknotting, Seifert surfaces, Blanchfield pairing, linking pairing.
result Effective lower and upper bounds for the Z-slice genus are derived.
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.
Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.
problem Understanding non-orientable surfaces and their inscribed rectangles.
method Analyzing smooth and locally-flat non-orientable surfaces in 4-ball with specific knots, comparing results.
result Established differences between smooth and locally-flat non-orientable 4-genus of torus knots.
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 g>0. Paper bounds double slice genus of knots.
problem Understanding knot genus complexities.
method Using Casson-Gordon invariants, the paper defines and bounds the double slice genus.
result Double slice genus can be much larger than slice genus.
The paper defines new knot genera and finds bounds for stabilization distances.
problem Finding bounds for stabilization distances of symmetric surfaces.
method Defining new knot genera and using them to find bounds.
result Constructs unknotted symmetric 2-spheres without symmetric 3-ball bounds.
New stabilizing number defined for knots, linking bounds in 4D.
problem Defining a new measure for knot boundaries in 4D.
method Defining stabilizing number sn(K), bounding it by signatures, Casson-Gordon invariants, and 4-genus. result Found examples where stabilizing number is less than 4-genus.
The paper calculates the slice genus for many virtual knots.
problem Determining which virtual knots are slice.
method Computing Turaev's graded genus and developing an algorithm for virtual unknotting operations.
result Many virtual knots with 6 or fewer crossings are slice.
The paper proves a conjecture about satellite knots and their slice genus.
problem The topological slice genus of satellite knots and its bounds.
method Establishes the conjecture for a variant of the topological slice genus, the Z-slice genus.
result The topological slice genus of a satellite knot is bounded above by the sum of the slice genera of the knot and the pattern.
The paper defines a new invariant for links and uses it to show non-sliceness.
problem Determining whether a link is slice or not.
method Defining a new concordance invariant from the Seifert form and using it to bound the slice Euler characteristic.
result The Witt coindex provides an upper bound for the slice Euler characteristic of a link.
The paper generalizes the T-genus to characterize slice knots and slice genus.
problem Characterizing slice knots and slice genus using the T-genus. method Generalizing the T-genus to provide a 3-dimensional characterization of the slice genus. result The difference between the T-genus and the slice genus can be arbitrarily large. New lower bound for doubly slice genus using knot signatures.
problem Finding a lower bound for the doubly slice genus of knots.
method Using the classical signature function to derive a new lower bound.
result Proved that for every nonnegative integer N, there exists a knot with exactly N difference between slice and doubly slice genus.
New bounds on knot complexity defined via band number and surface diagrams.
problem Understanding the complexity of knots and their generalizations.
method Defining and analyzing band number, using broken surface diagrams and 3-manifold embeddings.
result Band number is an upper bound on double slice genus, a knot complexity measure.
Lower bounds on rational slice genus using Heegaard Floer invariants.
problem Measuring complexity of homology classes in 4-manifolds.
method Introducing rational slice genus, bounding Heegaard Floer τ invariants, using satellite links and closed braids.
result Lower bounds on rational slice genus in terms of Heegaard Floer τ invariants.
New link invariant bounds topological slice genus.
problem Bounding the topological slice genus of links.
method Introducing algebraic genus as a new link invariant and using Casson-Gordon invariants.
result Algebraic genus is an upper bound for the topological 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.
New examples show algebraically slice knots with specific genus bounds.
problem Understanding slice genus of algebraic knots and their mirrors.
method Genus bound from Casson-Gordon invariants and cabling formula.
result Examples of algebraically slice knots with specific genus bounds.
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.
New findings on knots that are both topologically and rationally slice.
problem Understanding knots that are both topologically and rationally slice.
method Analyzing the concordance group of knots in S3. result There are infinitely many topologically slice knots that are strongly rationally slice but not slice.
Study of knots in 4-manifolds, proving slice genus is not always an invariant.
problem Slice genus is not always an invariant of X0(K). method Analyzing 0-shake genus and using satellite operations. result Infinitely many knots with 0-shake genus strictly less than slice genus. The paper uses twisting operations to bound knot genus.
problem Bounding the topological slice genus of knots.
method Develops null-homologous twisting operations to study algebraic genus.
result New upper bounds on algebraic genera of torus and satellite knots.
The paper calculates slice genera for knots up to 12 crossings.
problem Determining the slice genera of knots up to 12 crossings.
method Used computer searches for genus one concordances, algebraic genus, and new obstructions from Seifert forms and Donaldson's theorem.
result Completed computation of topological slice genera for all knots up to 11 crossings.
Study of mean curvature flow in de Sitter space, showing convergence to flat slicing.
problem Mean curvature flow in de Sitter space.
method Analysis of mean convex mean curvature flow of local spacelike graphs in de Sitter space.
result As s goes to infinity, Ms becomes graphical in expanding balls, converging to the flat slicing of de Sitter space. New bounds on slice genus from knot invariants.
problem Bounding slice genus of knots in RP3. method Using s-invariant to establish lower bounds. result Proves conjecture on slice genus bounds.
We show that perturbing the definition of sl(n) Khovanov-Rozansky link homology gives a lower bound on the slice genus of a knot. As a corollary this yields another proof of Milnor's conjecture on the slice genus of torus knots.
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.
We define Casson-Gordon sigma-invariants for links and give a lower bound of the slice genus of a link in terms of these invariants. We study as an example a family of two component links of genus h and show that their slice genus is h, whereas the Murasugi-Tristram inequality does not obstruct this link from bounding …
Identifies doubly slice genera for 2909 prime knots with up to 12 crossings.
problem Determining the doubly slice genera of prime knots.
method Identifies the minimal genus g for each knot K that divides a surface in S4. result Identified the doubly slice genera for 2909 prime knots with up to 12 crossings.
The paper finds conditions for slicing knots in 4-manifolds.
problem Conditions for slicing knots in 4-manifolds with boundary the 3-sphere.
method Establishes sufficient and necessary conditions for locally flat discs in 4-manifolds.
result Sufficient and necessary conditions for existence of discs with finite cyclic fundamental group.
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.
New invariants refine link homology, showing large genus differences.
problem Understanding genus differences in equivariant cobordisms.
method Refined Bar-Natan homology for involutive links, constructing new numerical invariants.
result Difference between equivariant and isotopy-equivariant slice genera can be arbitrarily large.
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.
The twisting number of a ribbon knot is at least as large as its doubly slice genus.
problem Proving a lower bound for the twisting number of ribbon knots in terms of their doubly slice genus.
method Analyzing symmetric unions and tangle replacements to establish the bound.
result Ribbon knots have arbitrarily high twisting numbers, matching their doubly slice genus.
New invariant measures doubly slice links, disproving previous bounds.
problem Understanding doubly slice links and their invariants.
method Introduced new invariant gst to measure doubly slice links and disproved previous bounds. result Examples of links with large doubly slice genus but gst=1. Exposes two methods for constructing flat surfaces in 4D spaces.
problem Building locally flat embedded surfaces in 4-manifolds.
method Direct methods and surgery theory.
result Every primitive second homology class in a closed, simply connected 4-manifold is represented by a locally flat embedded torus.
Every genus 1 algebraically slice knot is 1-solvable.
problem Understanding the concordance group of knots and links.
method Developed and applied the solvable filtration to knots and links.
result Proved that every genus 1 algebraically slice knot is 1-solvable.
Study knots in definite 4-manifolds using minimum-genus bounds.
problem Determining whether knots are smoothly slice.
method Minimum-genus bounds on smoothly embedded surfaces in definite 4-manifolds, gauge-theoretic obstructions.
result Alternate proof that (2,1)-cable of figure eight knot is not smoothly slice.
Classifies definite forms from surgeries on knots with small slice genus.
problem Classifying definite forms from surgeries on knots with specific properties.
method Uses Yang--Mills instanton gauge theory and Heegaard Floer correction terms.
result Classifies positive definite intersection forms for surgeries on knots with slice genus at most 2.
Study equivariant 4-genus of knots in symmetric 4-manifolds.
problem Understanding equivariant 4-genus of knots in symmetric 4-manifolds.
method Developed techniques for constructing slice disks via equivariant tubing construction.
result Equivariant 4-genus can differ from standard and equivariant 4-genus of 4-manifolds.
The paper studies link concordance and cobordisms using Floer homology.
problem Link concordance and cobordisms invariants.
method Floer homology and invariants ΥL(t), ν+(L), γ4(L). result Lower bounds for slice genus and 4-dimensional crosscap number.
The slicing number of a knot, us(K), is the minimum number of crossing changes required to convert K to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus gs(K). We show that for many knots, previous bounds on unknotting number obtained by Ozsvath and Szabo and b…
Study on reducing surgeries on knots, developing thickness and genus bounds.
problem Understanding reducible surgeries on knots in S3. method Developed thickness bounds for L-space knots and lower bounds on slice genus; used d-invariants and mapping cone formula from Heegaard Floer homology. result Provided new upper bounds on reducing slopes for fibered, hyperbolic slice knots and on multiple reducing slopes for slice knots; verified the Cabling Conjecture for thin knots.
Study on Whitehead doubles and their sliceness properties.
problem Understanding sliceness of Whitehead doubles of knots.
method Survey of techniques to obstruct sliceness and improve bounds on non-orientable genus.
result Improved bounds on non-orientable 4 genus of Whitehead doubles and genus 1 non-orientable cobordisms to cable knots.