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.
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.
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.
We show that if the connected sum of two knots with coprime Alexander polynomials is doubly slice, then the Ozsváth-Szabó correction terms as smooth double sliceness obstructions vanish for both knots. Recently, Jeffrey Meier gave smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice. …
A knot is said to be slice if it bounds a smooth properly embedded disk in the 4-ball. We demonstrate that the Conway knot, 11n34 in the Rolfsen tables, is not slice. This completes the classification of slice knots under 13 crossings, and gives the first example of a non-slice knot which is both topologically slice an…
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 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.
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.
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.
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 \(\mathbb{R}^4\).
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.
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.
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.
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.
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 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.
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. 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…
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…
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…
In [D.A. Fedoseev, V.O. Manturov, A sliceness criterion for odd free knots,arXiv:1707.04923], the authors proved a sliceness criterion for odd free knots: free knots with odd chords. In the present paper we give a similar criterion for stably odd free knots. Some additional results on knot sliceness and cobordism are g…
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
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.
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.
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.
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.
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.
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.
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…
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.
Research on knots and their 4-manifold covers.
problem Determining if knots are slice.
method Analyzing double branched covers and computer searches.
result Methods to decide if knots are slice.
A crucial step in the surgery-theoretic program to classify smooth manifolds is that of representing a middle--dimensional homology class by a smoothly embedded sphere. This step fails even for the simple 4-manifolds obtained from the 4-ball by adding a 2-handle with framing r along some knot K in S^3. An r-shake slice…
In this paper, we compute the slice genus for many low-crossing virtual knots. For instance, we show that 1295 out of 92800 virtual knots with 6 or fewer crossings are slice, and that all but 248 of the rest are not slice. Key to these results are computations of Turaev's graded genus, which we show extends to give an …
New complexity measure for shake-slice knots established.
problem Defining and measuring complexity for shake-slice knots.
method Using dualizable patterns and studying knot signatures.
result Existence of n-shake-slice knots with specified complexity. Algorithm finds ribbon disks for alternating knots, resolving sliceness for most prime knots.
problem Finding ribbon disks for alternating knots to determine sliceness.
method Algorithm based on Donaldson's diagonalization theorem, computer implementation.
result Successfully finds ribbon disks for many prime knots, resolves sliceness for most.
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.
We prove that certain fibered, −amphicheiral knots are rationally slice. Moreover, we show that the concordance invariants ν+ and Υ(t) from Heegaard Floer homology vanish for a class of knots that includes rationally slice knots.