Study constraints on knot surgery invariants using Seiberg-Witten theory.
problem Understanding invariants of knots in the three-sphere.
method Use Manolescu correction terms and Seiberg-Witten theory.
result Constraints on invariants for knots in the three-sphere.
Computes Pin(2)-equivariant monopole Floer homology for certain plumbed 3-manifolds.
problem Computing Pin(2)-equivariant monopole Floer homology for specific 3-manifolds.
method Uses Heegaard Floer/monopole Floer lattice complex and ranks of monopole Floer homology groups.
result Pin(2)-equivariant monopole Floer homology can be determined by ranks of monopole Floer homology groups.
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.
Turaev showed that there is a well-defined map assigning to an oriented link L in the three-sphere a Spin structure t_0 on Sigma(L), the 2-fold cover of S^3 branched along L. We prove, generalizing results of Manolescu-Owens and Donald-Owens, that for an oriented quasi-alternating link L the signature of L equals minus…
Lecture notes on monopole Floer homology, including differential geometry and correction terms.
problem Defining and understanding monopole Floer homology.
method Explains differential geometry, Morse theory, and four-dimensional theory connections.
result Sketches the relation to Manolescu's disproof of the Triangulation Conjecture.
Invariants for connected sums of 3-manifolds, proving Furuta's theorem.
problem Computing and comparing Manolescu invariants of connected sums.
method Using inequalities and monopole theory to compute invariants.
result Proof of Furuta's Theorem using Manolescu invariants.
New formula and properties of inverted Habiro series derived from GM series.
problem Understanding and manipulating knot invariants using series expansions.
method Developed a new formula for the inverted Habiro series (IHS) in terms of GM series and theta functions. Proved a multiplication formula for IHS.
result Established a natural ring structure for IHS and studied its residues, applying them to Dehn surgery formulas.
Estimates Manolescu's κ-invariant using spin 4-orbifolds.
problem Estimating κ-invariant of rational homology 3-spheres.
method Using spin 4-orbifolds bounded by rational homology 3-spheres.
result Restricts and determines the value of κ for specific cases.
Let D(K) be the positively-clasped untwisted Whitehead double of a knot K, and Tp,q be the (p,q) torus knot. We show that D(T2,2m+1) and D2(T2,2m+1) are linearly independent in the smooth knot concordance group C for each m≥2. Further, D(T2,5) and D2(T2,5) generate a $…
Study of knot Floer homology and its relation to Heegaard Floer homology via equivariant surgery.
problem Understanding the action of symmetries on knot Floer homology.
method Relating knot Floer homology to Heegaard Floer homology via equivariant surgeries.
result Identify the action of the involution on Heegaard Floer homology with an action on knot Floer homology.
In this paper, we study Manolescu's construction of the relative Bauer-Furuta invariants arising from the Seiberg-Witten equations on 4-manifolds with boundary. The main goal is to introduce a new gauge fixing condition in order to apply the finite dimensional approximation technique. We also hope to provide a framewor…
We re-derive Manolescu's unoriented skein exact triangle for knot Floer homology over F_2 combinatorially using grid diagrams, and extend it to the case with Z coefficients by sign refinements. Iteration of the triangle gives a cube of resolutions that converges to the knot Floer homology of an oriented link. Finally, …
Study rules out exotic S4 and #nCP2 construction using zero surgery homeomorphisms.
problem Tackles the possibility of constructing exotic S4 or #nCP2 using zero surgery homeomorphisms. method Uses zero surgery homeomorphisms to relate slice properties of knots stably after a connected sum with a 4-manifold.
result Rules out the possibility of constructing exotic S4 or #nCP2 using zero surgery homeomorphisms. Computes Pin(2)-equivariant Seiberg-Witten Floer homology for Seifert fibrations.
problem Homology cobordism between Seifert fiber spaces and integral homology spheres.
method Uses Heegaard Floer homology and Pin(2)-equivariant Seiberg-Witten Floer homology.
result Proves Manolescu's conjecture and identifies new homology cobordism obstructions.
Study shows infinite-rank summand in homology cobordism group.
problem Identifying infinite-rank summands in homology cobordism groups.
method Introduced an algebraic variant of the involutive Heegaard Floer package.
result Demonstrated an infinite-rank summand in the homology cobordism group.
Study monopole h-invariants from a topological viewpoint.
problem Understanding the h-invariants of 3-manifolds.
method Using Lidman and Manolescu's description of monopole Floer homology.
result Prove several properties of the h-invariants.
We iterate Manolescu's unoriented skein exact triangle in knot Floer homology with coefficients in the field of rational functions over Z/2Z. The result is a spectral sequence which converges to a stabilized version of delta-graded knot Floer homology. The (E2,d2) page of this spectral sequence …
Study maps on link Floer homology for unoriented link cobordisms.
problem Defining maps on link Floer homology induced by unoriented link cobordisms.
method Introduced disoriented link cobordism to track critical points, constructed a map on unoriented link Floer homology.
result Comparison with existing constructions for unoriented band moves.
Computes homology of an obstruction chain complex in grid homology.
problem Computing the homology of an obstruction chain complex in grid homology.
method Defined and computed the homology of the obstruction chain complex of the full grid.
result Results about the existence of sign assignments in grid homology.
New link homology theories for RP3 yield distinct invariants.
problem Extending link homologies for links in RP3. method Introducing new Lee and Bar-Natan spectral sequences for links in RP3. result New Lee and Bar-Natan invariants are distinct from existing ones.
Formulae for 1-3 handle attachments in 4-manifolds.
problem Understanding handle attachments in 4-dimensional manifolds.
method Developed general formulae for 1-3 handle attachments using skein modules.
result Derived complete description of gluing homomorphism on skein modules.
We define a new combinatorial complex computing the hat version of link Floer homology over Z/2Z, which turns out to be significantly smaller than the Manolescu-Ozsvath-Sarkar one.
We derive a closed formula for the Heegaard Floer correction terms of lens spaces in terms of the classical Dedekind sum and its generalization, the Dedekind-Rademacher sum. Our proof relies on a reciprocity formula for the correction terms established by Ozsvath and Szabo. A consequence of our result is that the Casso…
New theorem connects tangle complements in 3-manifolds via Floer homology.
problem Understanding tangle complements in arbitrary 3-manifolds.
method Adapting holomorphic polygon counts to bordered-sutured Floer homology.
result Exact triangle relation for tangle complements in arbitrary 3-manifolds.
Study knot invariants to deduce Hopf invariant and propose a slope conjecture.
problem Understanding the topological significance of knot invariants and their relations.
method Analyzing the Gukov-Manolescu knot series and its coefficients, relating to Hopf invariant and colored Jones polynomials.
result Explicit formula for the Hopf invariant in terms of colored Jones polynomials for fibered knots up to 12 crossings.
If the Bing double of a knot K is slice, then K is algebraically slice. In addition, Heegaard--Floer concordance invariants developed by Ozsvath-Szabo and by Manolescu-Owens vanish on K.
Researchers find a Steenrod square for link Floer homology.
problem Computing the second Steenrod square for link Floer homology.
method Explicitly framing moduli spaces and constructing a framed 1-flow category.
result An algorithm for computing the second Steenrod square for all grid homology versions.
The paper constructs homology spheres with large correction terms.
problem Finding homology spheres with large correction terms.
method Constructing families of homology spheres that bound 4-manifolds with intersection forms isomorphic to -E8.
result Homology spheres have arbitrarily large correction terms.
Estimates Heegaard Floer correction term for (2,q)-cable knots.
problem Estimating Heegaard Floer correction term for (2,q)-cable knots. method Uses Heegaard Floer theory and (+1)-surgeries. result Equality holds for certain knots in a specific class.
Study proposes a new method to estimate bias-correction term for ATE estimation.
problem Estimating the bias-correction term for ATE estimation.
method Directly estimating the bias-correction term by minimizing Bregman divergence.
result Automatic covariate balancing property achieved through specific model choices.
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.
Extends knot surgery to exotic four-manifolds.
problem Constructing exotic definite four-manifolds.
method Generalizes RBG construction to n-surgery. result Produces pairs of knots with same n-surgery. Lattices embeddability determined by correction terms.
problem Embeddability of nonunimodular definite lattices.
method Using Elkies' theorem and lattice correction terms.
result Embeddability of lattices is determined by correction terms.
We compute the Khovanov lasagna module of S²×S², confirming a conjecture.
problem Computing the Khovanov lasagna module of S²×S².
method Interpreting Manolescu-Neithalath's formula as a homotopy colimit, using categorified projectors.
result The Khovanov lasagna module of S²×S² is trivial.
fkcompute calculates a knot invariant from a braid presentation.
problem Computing the Gukov-Manolescu invariant for complex knots and links.
method Three-phase pipeline: braid presentation search, state space encoding, R-matrix multiplication.
result fkcompute efficiently computes the invariant for knots and links up to 12 crossings.
New invariants defined for 3-manifolds, linking knot surgeries to polynomial relations.
problem Defining and studying new invariants for 3-manifolds based on root systems.
method Defined Z^G and FKG for specific cases, derived recurrence relations. result Found a connection between knot surgeries and polynomial relations for symmetric representations.
The paper connects quivers to knot complements and studies their BPS states and 3d N=2 theories.
problem Understanding the relationship between quivers and knot complements.
method Assigning quivers to knot complements and exploring their physical interpretation.
result Proposed a physical interpretation of quivers in terms of BPS states and 3d N=2 theories.
Design experiments to correct misspecified dynamical models efficiently.
problem Efficiently correct misspecified dynamical models with limited data.
method Model the correction term as a Gaussian Process and use submodular optimization for optimal experimental design.
result Approximately optimal designs can be efficiently derived for Gaussian Process models.
New homology theory for instantons, defined and analyzed.
problem Understanding instanton structures in symplectic geometry.
method Definition and analysis of twisted Symplectic Instanton homology, fitting into Floer Field theory.
result Properties of the new homology invariant for connected sums and Dehn surgery.
A new correction term improves sample efficiency in deep reinforcement learning.
problem Momentum accumulation in TD learning leads to doubly stale gradients.
method Proposed a correction term to address the issue of doubly stale gradients.
result Improves sample efficiency in policy evaluation.
In this small note we use results derived in Berestycki et al. to correct the celebrated formulae of Hagan et al. We derive explicitly the correct zero order term in the expansion of the implied volatility in time to maturity. The new term is consistent as β→1. Furthermore, numerical simulations show that it reduc…
New invariant recovers known contact element and considers finite coverings.
problem Defining and studying new contact invariants in Seiberg-Witten Floer spectra.
method Cohomotopy set of Seiberg-Witten Floer spectrum, equivariant Borel cohomology.
result New invariant recovers known contact element and considers finite coverings.
New invariants defined for 3-manifolds, generalizing existing work.
problem Defining new invariants for 3-manifolds.
method Defining unfolded Seiberg-Witten Floer spectra for closed oriented 3-manifolds.
result Invariants generalize existing work and are computable for some examples.
New invariants from twisted Heegaard Floer homology restrict 4-manifold topology.
problem Defining numerical invariants for arbitrary 3-manifolds with torsion spinc structures. method Using Heegaard Floer homology with twisted coefficients.
result Twisted correction terms provide restrictions on 4-manifold topology.
Study constraints on diffeomorphisms and homeomorphisms of 4-manifolds with boundary.
problem Constraints on smooth families of 4-manifolds with boundary.
method Use Manolescu's Seiberg-Witten Floer stable homotopy type.
result Inclusion map between diffeomorphisms and homeomorphisms is not a weak homotopy equivalence.
Remark on Pin(2)-equivariant Floer homology and its connections.
problem Understanding the relationship between Pin(2)-equivariant Floer homology and other invariants.
method Analyzing the monopole Frøyshov invariant and Involutive Heegaard Floer correction terms.
result Identifying the only interesting correction terms as coming from subgroups Z/4, S^1, and Pin(2).
New knot invariants from covering involutions help deduce linear independence results.
problem Understanding the smooth concordance group of knots.
method Covering involutions and adapted ideas from previous research.
result Novel linear independence results in the smooth concordance group of knots.
We make use of the action of H1(Y) in Heegaard Floer homology to generalize the Ozsváth-Szabó correction terms for 3-manifolds with standard HF∞. We establish the basic properties of these invariants: conjugation invariance, behavior under orientation reversal, additivity, and spinc ratio…