Study on r−shake slice knots and proves 0-shake slice knots are slice.
problem Understanding and characterizing r−shake slice knots. method Exploring the relation to corks and proving slice properties.
result Proves 0-shake slice knots are slice.
The Conway knot is not slice, resolving a knot classification problem.
problem Determining which knots are slice in 4-dimensional space.
method Demonstrated through a proof involving knot classification and properties of slice knots.
result The Conway knot is the first example of a non-slice knot that is topologically slice and a positive mutant of a slice knot.
Proves certain knots are slice without shaking.
problem Identifying slice knots without using traditional methods.
method Direct proof for 0−shake slice knots. result Proves 0−shake slice knots are slice. Proves a special knot type is slice.
problem Characterizing slice knots.
method Proof by contradiction and algebraic topology.
result 0-shake slice knots are indeed slice.
Study shows vanishing correction terms for doubly slice knots.
problem Understanding doubly slice knots and their properties.
method Analyzing connected sums of knots with coprime Alexander polynomials and using Ozsváth-Szabó correction terms.
result Correction terms vanish for doubly slice knots, providing new insights.
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.
New knots found with tough, unsliceable discs.
problem Finding tough knots that can't be sliced smoothly.
method Constructed infinitely many knots with non-approximable slice discs.
result Smoothly sliceable knots have non-approximable slice discs.
New knots show linear independence in slice concordance.
problem Understanding the structure of rationally slice knots.
method Provided an infinite family of knots that are linearly independent.
result Found knots that are linearly independent and infinite order.
Defines slice depth for 2-knots and sets upper bounds for specific knots.
problem Determining the minimum dimension for a 2-knot to be slice.
method Introduces slice depth, defines it for 2-knots, and provides upper bounds for specific knot types.
result Upper bounds for slice depth of certain 2-knots.
The paper provides a sliceness criterion for stably odd knots.
problem Sliceness of stably odd knots
method Analyzes cobordisms and free knots to derive a sliceness criterion
result A sliceness criterion for stably odd knots
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.
Study on slicing knots in 4-manifolds, focusing on CP^2-slicing numbers.
problem Understanding the slicing properties of knots in 4-manifolds.
method Lower and upper bounds on CP^2-slicing numbers using double branched covers and Seifert forms.
result Findings on the finite and distinct CP^2-slicing numbers for certain knots.
Lisa Piccirillo solved the mystery of the Conway knot's sliceness.
problem Determining if the Conway knot is slice.
method The main idea of the proof is given in the title.
result The Conway knot is not slice.
New proof for some knots being topologically slice.
problem Understanding which knots are topologically slice.
method Equivariant topological slice disks for strongly negative amphichiral knots.
result Strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice.
Study uses knot Floer homology to distinguish slice disks.
problem Classifying slice disks of knots up to isotopy and diffeomorphism.
method Invariants in knot Floer homology to compute and distinguish slice disks.
result Invariant can distinguish non-isotopic slice disks with diffeomorphic complements.
New knots found that can be cut topologically but not smoothly.
problem Existence of (1,1)-knots that are topologically slice but not smoothly slice. method Proof of existence of infinitely many (1,1)-knots. result Infinitely many (1,1)-knots topologically slice but not smoothly slice. New knot found with unique property.
problem Existence of hyperbolic fibered slice knots with specific monodromy.
method Constructed a specific type of hyperbolic fibered slice knot.
result Negative answer to a question posed by Hubbard et al.
Proves conditions for odd pretzel knots to be doubly slice.
problem Identifying conditions for odd pretzel knots to be doubly slice.
method Analyzes knots with specific twist parameters and odd integers.
result Identifies all doubly slice odd pretzel knots.
Study shows certain knots can't be sliced using 2-fold branched covers.
problem Determining which algebraically slice knots are actually slice.
method Used d invariants of 2-fold branched covers to show nonsliceness.
result Shows nonsliceness of a set of algebraically slice knots.
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.
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.
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 \(\mathbb{R}^4\).
New obstructions for knots in high dimensions prevent double sliceness.
problem Preventing double sliceness of knots in high dimensions.
method Developed L(2)-signature obstructions for metabelian knot groups. result Found infinite families of knots not obstructed by previous invariants.
Smoothly slice a knot with specific properties.
problem Existence of certain types of knots.
method Proving the existence of a specific knot with given properties.
result Existence of a smoothly doubly slice, amphicheiral knot with Alexander polynomial 1 and unknotting number 5.
The paper solves sliceness for odd free knots.
problem Determining if odd free knots have a spanning disc.
method Using the pairing of knot diagram chords.
result An explicit criterion for sliceness of odd free knots.
Study shows most knots in a family are not slice.
problem Determining which 3-stranded pretzel knots are slice.
method Analyzing a specific infinite family of knots and proving their non-slice properties.
result Four-fifths of the remaining knots in the family are not slice.
We give a new construction of slice knots via annulus twists. The simplest slice knots obtained by our method are those constructed by Omae. In this paper, we introduce a sufficient condition for given slice knots to be ribbon, and prove that all Omae's knots are ribbon.
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.
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 tackles deeply slice knots via immersed curves.
problem Whether there exists a homology 3-sphere with certain knot properties.
method Constructing knots in contractible 4-manifolds and using Heegaard Floer homology invariants.
result Proves the existence of a class of deeply slice knots.
Study proves obstructions to equivariantly slice strongly negative amphichiral knots.
problem Proving obstructions for equivariantly slice strongly negative amphichiral knots.
method Using determinant, Spinc-structures, Donaldson's theorem, and Heegaard Floer correction terms.
result 8 out of 16 strongly negative amphichiral knots with 12 or fewer crossings are not equivariantly 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.
New findings show infinitely many knots cannot be smoothly round handle slices.
problem Understanding the smoothability of knots in 4-dimensional space.
method Analyzing the properties of knots under surgery conjectures and cobordism conjectures.
result Infinitely many knots fail to be smoothly round handle slices.
We construct an infinite family of smoothly slice knots that we prove are topologically doubly slice. Using the correction terms coming from Heegaard Floer homology, we show that none of these knots is smoothly doubly slice. We use these knots to show that the subgroup of the double concordance group consisting of smoo…
Study shows (2,1)-cable of figure-eight knot can't be smoothly sliced.
problem Determining if a knot can be smoothly sliced.
method Showed that the branched double cover of the (2,1)-cable of the figure-eight knot bounds no equivariant homology ball. result The (2,1)-cable of the figure-eight knot is not smoothly slice. 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.
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…
We use recently introduced Rasmussen invariant to find knots that are topologically locally-flatly slice but not smoothly slice. We note that this invariant can be used to give a combinatorial proof of the slice-Bennequin inequality. Finally, we compute the Rasmussen invariant for quasipositive knots and show that most…
A knot in the three-sphere is doubly slice if it is the cross-section of an unknotted two-sphere in the four-sphere. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We extend that work through 12 crossings, resolving all but four cases among the 2…
New knots bound rational homology balls, using Alexander polynomials.
problem Finding sliceness obstructions for knots.
method Computing twisted Alexander polynomials and simplifying their calculation.
result New non-slice knots with rational homology ball bounds.
Explicit formulas for pretzel knots' Alexander polynomials.
problem Alexander polynomial of pretzel knots
method Provided explicit formulas
result Characterization of pretzel knots with trivial Alexander polynomial
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.
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.
New method detects sliceness of knots in a torus.
problem Detecting sliceness of knots in a torus.
method Group-theoretical combinatorial techniques.
result Constructs many sliceness obstructions.
Study on prime knots, slice obstructions, and ribbon concordances.
problem Determining which prime knots are slice in smooth and topological categories.
method Use of a wide range of tools and techniques, including new or refined methods for probing these properties.
result About 1.6 million prime knots are smoothly slice (ribbon), and 350.5 million are not even topologically slice.
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 4-manifold accounts for rationally slice knots.
problem Characterize rationally slice knots.
method Show independence of construction of rational homology ball VK. result Single 4-manifold accounts for all known rationally slice knots.
A knot in the 3-sphere is called doubly slice if it is a slice of an unknotted 2-sphere in the 4-sphere. We give a bi-sequence of new obstructions for a knot being doubly slice. We construct it following the idea of Cochran-Orr-Teichner's filtration of the classical knot concordance group. This yields a bi-filtration o…