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.
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.
Study on equivariant Q-sliceness for strongly invertible knots.
problem Understanding Q-sliceness for strongly invertible knots.
method Constructive and obstructive approaches using Fox-Milnor condition and equivariant concordance.
result Klein amphichiral knots are equivariant Q-slice in a single Q-homology 4-ball.
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 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.
Defines a new homomorphism for strongly invertible knots, proving equivariant algebraic concordance.
problem Equivariant algebraic concordance of strongly invertible knots.
method Defining a homomorphism Φ 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.
This paper gives an algebraic characterization of Alexander polynomials of equivariant ribbon knots and a factorization condition satisfied by Alexander polynomials of equivariant slice 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 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.
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.
We characterize the Murasugi polynomial of an equivariant slice knot by proving a conjecture of J. Davis and S. Naik.
Research classifies knots based on sliceness and amphichirality.
problem Classifying odd-stranded Turk's head knots based on sliceness and amphichirality.
method Constructing commuting pairs of ambient involutions and analyzing the equivariant Fox-Milnor square condition.
result Established a sharp parity dichotomy for equivariant rational sliceness and Klein amphichirality of odd-stranded Turk's head knots.
This paper studies equivariant cohomology of slice groupoids using linearization theorems.
problem Computing equivariant cohomology of groupoids using slice structures.
method Explicitly constructs chain maps between Weil/Cartan models and equivariant cohomologies of slices.
result Induces isomorphisms in equivariant cohomology of groupoids.
Study on 2-bridge knots, proving equivariant concordance order is infinite.
problem Equivariant concordance of 2-bridge knots.
method Formula for butterfly polynomial, two proofs of non-equivariant sliceness, new invariant for strongly invertible knots.
result Equivariant concordance order of 2-bridge knots is infinite.
New invariants prove exotic slice disks for knots.
problem Proving exotic slice disks for knots.
method Involutive Khovanov homology and derived invariants.
result Reprove exotic slice disks for knots Jn. Computes immersions of C2-projective spaces using K-theory.
problem Computing immersions of equivariant projective spaces.
method Geometric filtration and localized slice spectral sequence.
result Obtained equivariant analogue of James periodicity.
Paper explores properties of slice-matching operators for measure transfer.
problem Efficiently transferring measures in high dimensions.
method Examines an associated slice-matching operator with source, target measures and slicing directions.
result Establishes invariance, equivariance, Lipschitz continuity, and error bounds.
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 method constructs translationally equivariant hyperbolic affine spheres.
problem Constructing translationally equivariant hyperbolic affine spheres.
method Noncompact Iwasawa factorization via DPW method and Weierstrass elliptic functions.
result Every translationally equivariant hyperbolic affine sphere is equiaffinely equivalent to one with a circle, hyperbola, or parabola slice curve.
Proves a specific knot is not smoothly slice using real invariants.
problem Determining the smooth sliceness of (2n,1)-cables of the figure-eight knot. method Used real Seiberg-Witten Frøyshov invariant and developed an equivariant lattice homotopy type.
result Proves the (2n,1)-cable of the figure-eight knot is not smoothly slice when n is odd. Normal forms for equivariant maps in infinite dimensions established.
problem Establishing normal forms for equivariant maps in infinite-dimensional manifolds.
method Inspired by Lyapunov-Schmidt reduction and Kuranishi method, uses Slice Theorem for Fréchet manifolds.
result Abstract moduli spaces of equivariant maps are locally modeled on quotient by a compact group.
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.
We prove an extension of a celebrated equivariant bifurcation result of J. Smoller and A. Wasserman, in an abstract framework for geometric variational problems. With this purpose, we prove a slice theorem for continuous affine actions of a (finite-dimensional) Lie group on Banach manifolds. As an application, we discu…
Study of a G2-equivariant octonionic operator and its right spectrum.
problem Understanding the spectrum of a G2-equivariant octonionic operator. method Computed the ordinary real spectrum and analyzed the octonionic right-eigenvalue problem using G2-decomposition and residual symmetry analysis. result Explicit spectral loci (quartic curve and circle) in each complex slice of the octonionic space.
Abstract invariant cannot be expressed using various slice-torus invariants.
problem Cannot express Iida-Taniguchi's slice-torus invariant using other known invariants.
method Analysis of various known invariants and their properties.
result Iida-Taniguchi's slice-torus invariant cannot be realized as a linear combination of other invariants.
Study transverse knots and symplectic surfaces using Seiberg-Witten monopole equations.
problem Investigate transverse knots and symplectic surfaces via Seiberg-Witten monopole equations.
method Equivariant Seiberg-Witten theory on branched covers, introducing a novel slice-torus invariant.
result Determine the value of slice-torus invariant for certain Montesinos knots within a deviation of 2.
In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an a…
We present a framework for studying the dynamics of equivariant vector fields near relative equilibria. To overcome the lack of linearization at a relative equilibrium or the possible non-smoothness of the orbit space, we categorify the space of equivariant vector fields. A category where the objects are equivariant ve…
2-knots with S4 symmetry are classified up to equivariant concordance.
problem Classifying 2-knots with S4 symmetry up to equivariant concordance. method Constructing a new invariant called periodic, based on the Arf invariant.
result The smooth equivariant concordance group of 2-knots in S4 is isomorphic to Z/2Z for all d≥2. In his seminal work \cite{pal:61}, R. Palais extended a substantial part of the theory of compact transformation groups to the case of proper actions of locally compact groups. Here we extend to proper actions some other important results well known for compact group actions. In particular, we prove that if H is a co…
We prove that for a compact subgroup H of a locally compact Hausdorff group G, the following properties are mutually equivalent: (1) G/H is a manifold, (2) G/H is finite-dimensional and locally connected, (3) G/H is locally contractible, (4) G/H is an ANE for paracompact spaces, (5) G/H is a metrizable $G…
Study knot invariants to answer questions about slice genus and clasp numbers.
problem Whether the difference between the four-dimensional clasp number and the slice genus can be arbitrarily large.
method Equivariant singular instanton theory and Chern--Simons functional.
result Answers a conjecture by Livingston about slicing numbers and provides a lower bound for the unoriented slice genus.
We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry). The result is also a generalization of Conn's linearization theorem from one-point leaves to arbitrary symplectic leaves (however, we do not make use of Conn's theorem).
A long-standing conjecture due to Michael Freedman asserts that the 4-dimensional topological surgery conjecture fails for non-abelian free groups, or equivalently that a family of canonical examples of links (the generalized Borromean rings) are not A-B slice. A stronger version of the conjecture, that the Borromean r…
Unified classification of equivariant principal bundles using higher homotopy theory.
problem Unified classification of equivariant principal bundles.
method Smooth Oka principle, singular-cohesive homotopy theory, internally describing principal bundles.
result Unified classification results for equivariant principal bundles.
Let G a semisimple Lie group of non-compact type and let XG be the Riemannian symmetric space associated to it. Suppose XG has dimension n and it has no factor isometric to either H2 or SL(3,R)/SO(3). Given a closed n-dimensional Riemannian manifold $…
We study two special cases of the equivariant index defined in part I of this series. We apply this index to deformations of Spinc-Dirac operators, invariant under actions by possibly noncompact groups, with possibly noncompact orbit spaces. One special case is an index defined in terms of multiplicities of discrete…
The paper classifies involutions on S^4, proving linearities under certain conditions.
problem Classifying involutions on S^4 with specific fixed-point sets.
method Combining surgery theory, Schoenflies theorem, and equivariant topology.
result Linear involutions on S^4 with 1-dimensional fixed-point sets are proven.
Paper defines half-Conway polynomial and computes it for knots up to 12 crossings.
problem Computing and characterizing half-Conway polynomials of knots.
method Normalized Conway polynomial, equivariant skein relation, diagrammatic interpretation.
result First examples of non-slice strongly negative amphichiral knots with determinant one.
Study Lie 2-group actions on Riemannian groupoids, proving existence and developing geometric Killing vector fields.
problem Understanding isometric actions of Lie 2-groups on Riemannian groupoids.
method Exhibit properties, prove existence, construct bi-invariant metrics, provide infinitesimal description.
result Existence of 2-equivariant Slice Theorem and Equivariant Tubular Neighborhood Theorem.
Study uses instanton Floer theory to obstruct knot unknotting operations.
problem Obstructing knot unknotting operations and ribbon concordance.
method Equivariant singular instanton Floer theory with Chern--Simons filtration.
result For a large class of slice knots, any unknotting sequence must contain both signs.
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.
Regular sliceness implies once-stably decomposable sliceness in symplectizations.
problem Relationship between regular and decomposable Lagrangian cobordisms in symplectizations.
method Stabilization-free strategy and satellite operations.
result Regular sliceness implies once-stably decomposable sliceness.
We consider linear slices of the space of Kleinian once-punctured torus groups; a linear slice is obtained by fixing the value of the trace of one of the generators. The linear slice for trace 2 is called the Maskit slice. We will show that if traces converge `horocyclically' to 2 then associated linear slices converge…
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.