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 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 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 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.
New links can't be smoothly split but have large slice sizes.
problem Non-split topologically slice links with large smooth slice genus.
method Constructed an infinite family of links.
result None of the constructed links is smoothly concordant to a split link and has arbitrarily large smooth slice genus.
Study shows some 2-bridge knots are not topologically slice.
problem Determining which 2-bridge knots are topologically slice.
method Used twisted Alexander polynomials and Casson-Gordon signatures.
result Certain algebraically slice 2-bridge knots are not topologically 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 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…
New knots found with different smooth and topological slice genera.
problem Identifying knots with differing smooth and topological slice genera.
method Examined 2-bridge knots, finding the smallest example with 12 crossings.
result Found infinitely many 2-bridge knots with differing genera.
Characterizes topological sliceness of 3-strand pretzel knots.
problem Determining the topological sliceness of 3-strand pretzel knots.
method Computations of Casson-Gordon 3-manifold signature invariants for double branched covers.
result Odd 3-strand pretzel knots are topologically slice if and only if they are ribbon or have trivial Alexander polynomial.
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.
Study on knots in S1imesS2 with unique smooth concordance class for winding number 1.
problem Understanding concordance classes of knots in S1imesS2. method Defined winding number and used it to classify knots, demonstrated distinctions between smooth and topological categories.
result Unique smooth concordance class for winding number 1, infinitely many topological concordance classes for other winding numbers.
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.
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.
Study shows infinite rank in bipolar filtration of topologically slice knots.
problem Understanding deeper structures in the smooth concordance group of topologically slice knots.
method Used higher order amenable Cheeger-Gromov L2 ρ-invariants and infinitely many Heegaard Floer correction term d-invariants. result Graded quotient of bipolar filtration has infinite rank at each stage greater than one.
Study confirms conjectures for topologically slice knots' concordance group.
problem Primary decomposition conjectures for knot concordance groups.
method Use of amenable L2-signatures, Ozsváth-Szabó d-invariants, and Némethi's Heegaard Floer homology results. result Smooth concordance group of topologically slice knots has large subgroup with true primary decomposition.
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.
Condition for generic sliced spaces to be globally hyperbolic.
problem Global hyperbolicity of sliced spaces without lapse or shift function hypotheses.
method Topological condition for globally hyperbolic sliced spaces.
result Generic sliced spaces are globally hyperbolic under certain conditions.
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.
The paper explores statistical and topological properties of sliced probability divergences.
problem Understanding the topological, statistical, and computational consequences of slicing divergences.
method Deriving theoretical properties of sliced probability divergences, including metric axioms preservation and weak continuity.
result Sliced divergences share similar topological properties and have stable sample complexity.
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 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.
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
Solved A-B slice problem using link-homotopy+.
problem 4D topological surgery conjecture for free groups.
method Geometric applications of 2-Engel relation.
result Solved A-B slice problem.
Study shows knots can be reversed without becoming trivially concordant.
problem Understanding when knots can be reversed without changing their concordance class.
method Constructed an infinite family of knots, analyzed their concordance properties.
result Found an infinitely generated free subgroup in the concordance group of topologically slice knots.
The bipolar filtration introduced by T. Cochran, S. Harvey, and P. Horn is a framework for the study of smooth concordance of topologically slice knots and links. It is known that there are topologically slice 1-bipolar knots which are not 2-bipolar. For knots, this is the highest known level at which the filtration do…
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.
New knots found that can't be smoothly embedded in certain 4D spaces.
problem Existence of knots not smoothly slice in definite 4-manifolds.
method Constructing specific knots and proving their properties.
result Infinitely many topologically slice knots that cannot be smoothly slice in any definite 4-manifold.
We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples…
The existence of topologically slice knots that are of infinite order in the knot concordance group followed from Freedman's work on topological surgery and Donaldson's gauge theoretic approach to 4-manifolds. Here, as an application of Ozsvath and Szabo's Heegaard-Floer theory, we show the existence of an infinite sub…
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.
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.
In the early 1980's Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a sufficient homological condition under which a knot is slice with fundamental group Z …
We show that if K is any knot whose Ozsvath-Szabo concordance invariant tau(K) is positive, the all-positive Whitehead double of any iterated Bing double of K is topologically but not smoothly slice. We also show that the all-positive Whitehead double of any iterated Bing double of the Hopf link (e.g., the all-positive…
New Milnor's invariant condition for topologically slice links.
problem Conditions for topologically slice links involving Milnor's invariant.
method Different Milnor's invariant condition to handle cases not covered by previous theorem.
result New sufficient conditions for topologically slice links.
We consider the issue of the slice invariance of refined topological string amplitudes, which means that they are independent of the choice of the preferred direction of the refined topological vertex. We work out two examples. The first example is a geometric engineering of five-dimensional U(1) gauge theory with a ma…
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…
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.
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 slicing obstruction from knot surgery yields non-trivial results.
problem Detecting non-trivial elements in the smooth concordance group.
method Spin 4-manifold with boundary as 0-surgery on a knot.
result Detects torsion elements and finds non-smoothly slice knots.
The knot Floer complex and the concordance invariant ε can be used to define a filtration on the smooth concordance group. We exhibit an ordered subset of this filtration that is isomorphic to N×N and consists of topologically 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.
As proved by Hedden and Ording, there exist knots for which the Ozsvath-Szabo and Rasmussen smooth concordance invariants, tau and s, differ. The Hedden-Ording examples have nontrivial Alexander polynomials and are not topologically slice. It is shown in this note that a simple manipulation of the Hedden-Ording example…
Let C_T be the subgroup of the smooth knot concordance group generated by topologically slice knots and let C_D be the subgroup generated by knots with trivial Alexander polynomial. We prove the quotient C_T/C_D is infinitely generated, and uncover similar structure in the 3-dimensional rational spin bordism group. Our…
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 proof for a specific class of knots satisfying a weaker Slice-Ribbon Conjecture.
problem Proving a specific class of knots are mutant ribbon.
method Proved by showing infinite order and non-slice properties of knots.
result All slice, odd, 5-stranded pretzel knots are mutant ribbon.
We propose and analyze a structure with which to organize the difference between a knot in the 3-sphere bounding a topologically embedded 2-disk in the 4-ball and it bounding a smoothly embedded disk. The n-solvable filtration of the topological knot concordance group, due to Cochran-Orr-Teichner, may be complete in th…
Alternating knots can't represent certain concordances.
problem Understanding which knots can represent concordances.
method Analyzing properties of alternating knots and their concordances.
result No nontrivial alternating knot can represent certain concordances.