arXiv research

Refines Ozsváth-Szabó d-invariants for knot concurrence.

Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.

The study uses Heegaard Floer homology to analyze and bound the splitting number of links.

Study improves Heegaard Floer homology relations for knots in homology spheres.

Method calculates $d$-invariants for a family of Brieskorn spheres.

Kim shows a polynomial splitting property for Ozsvath and Szabo's d-invariants.

Study Heegaard Floer homology for two-component L-space links with zero linking number.

Legendre transform connects Upsilon function of L-space knots to large surgeries d-invariants.

Study shows certain knots can't be sliced using 2-fold branched covers.

Study $d$-invariants of L-space double branched covers of arborescent links.

Study surgeries between lens spaces using Heegaard Floer d-invariant.

Satellite operations with winding number ≠ 1 are not homomorphisms.

Notes on applying d-invariant and Donaldson's theorem to knot theory.

Algorithm computes knot invariants for surgeries on prime knots.

Paper proves a conjecture about a Heegaard Floer invariant for certain rational homology spheres.

Study rational cuspidal curves in projective surfaces with topological and algebraic obstructions.

The Fourier transform of Heegaard Floer d-invariants helps classify 3-manifolds.

We investigate constraints on embeddings of a non-orientable surface in a $4$-manifold with the homology of $M \times I$, where $M$ is a rational homology $3$-sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsváth--Sazbó $d$-invariants or …

We define a "reduced" version of the knot Floer complex $CFK^-(K)$, and show that it behaves well under connected sums and retains enough information to compute Heegaard Floer $d$-invariants of manifolds arising as surgeries on the knot $K$. As an application to connected sums, we prove that if a knot in the three-sphe…

New combinatorial proof confirms lens spaces' homology cobordism classification.

Study distance one surgeries between specific lens spaces.

New 4D shapes without spine found.

Paper determines 2-adjacent knots up to 12 crossings.

Classifies knots that bound equivariant surfaces with free symmetries.

The d-invariant of an integral, positive definite lattice L records the minimal norm of a characteristic covector in each equivalence class mod 2L. We prove that the 2-isomorphism type of a connected graph is determined by the d-invariant of its lattice of integral cuts (or flows). As an application, we prove that a re…

We prove that for particular infinite families of $L$-spaces, arising as branched double covers, the $d$-invariants defined by Ozsváth and Szabó are arbitrarily large and small. As a consequence, we generalise a result by Greene and Watson by proving, for every odd number $Δ\geq 5$, the existence of infinitely many non…

Floer homology linked to Milnor fibers for certain singularities.

Study shows infinite rank in bipolar filtration of topologically slice knots.

Method constructs knots with diffeomorphic 4-manifolds, proving non-concordance.

We give an $O(p^{2})$ time algorithm to compute the generalized Heegaard Floer complexes $A_{s_{1},s_{2}}^{-}(\overrightarrow{L})$'s for a two-bridge link $\overrightarrow{L}=b(p,q)$ by using nice diagrams. Using the link surgery formula of Manolescu-Ozsváth, we also show that ${\bf HF}^{-}$ and their $d$-invariants of…

This paper explores how many positive integer surgeries on a knot produce a manifold rational homology cobordant to a lens space.

The paper reconfirms a lower bound on rational genus using Heegaard Floer homology.

For a fixed p, there are only finitely many elliptic 3-manifolds given by p/q-surgery on a knot in S^3. We prove this result by using the Heegaard Floer correction terms (d-invariants) to obstruct elliptic manifolds from arising as knot surgery.

Proves conditions for surgery on trefoil knot in lens space.

Band surgery affects knot signatures by 0 or 8.

We apply Heegaard Floer homology to study deformations of singularities of plane algebraic curves. Our main result provides an obstruction to the existence of a deformation between two singularities. Generalizations include the case of multiple singularities. The obstruction is formulated in terms of a semicontinuity p…

Geography problem for nonorientable surfaces bounded by knots.

The paper examines lattice homology invariants of Seifert homology spheres.

A flat complex vector bundle (E,D) on a compact Riemannian manifold (X,g) is stable (resp. polystable) in the sense of Corlette [C] if it has no D-invariant subbundle (resp. if it is the D-invariant direct sum of stable subbundles). It has been shown in [C] that the polystability of (E,D) in this sense is equivalent to…

New invariants from Seiberg-Witten theory for 3-spheres with involution.

New formulas link Milnor invariants to Heegaard Floer homology.

Let T denote the group of smooth concordance classes of topologically sice knots. We show that the first quotient in the bipolar filtration of T (i.e. 0-bipolar knots modulo 1-bipolar knots) has infinite rank, even modulo Alexander polynomial one knots. Any 0-bipolar knot has vanishing tau-, epsilon-, and s-invariants.…

Let D be a self-adjoint differential operator of Dirac type acting on sections in a vector bundle over a closed Riemannian manifold M. Let H be a closed D-invariant subspace of the Hilbert space of square integrable sections. Suppose D restricted to H is semibounded. We show that every element u in H has the weak uniqu…

The paper shows infinite rank in T_1/T_2 and new insights into Alexander polynomials and concordance.

For any knot $K$ which bounds non-orientable and null-homologous surfaces $F$ in punctured $n\mathbb{C}P^2$, we construct a lower bound of the first Betti number of $F$ which consists of the signature of $K$ and the Heegaard Floer $d$-invariant of the integer homology sphere obtained by $1$-surgery along $K$. By using …

We compute the Heegaard Floer homology of $S^3_1(K)$ (the (+1) surgery on the torus knot $T_{p,q}$) in terms of the semigroup generated by $p$ and $q$, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsvath--Szabo d-invariant. We relate the result to known knot invariants of $T_{p,q}$ as …

New invariants from twisted Heegaard Floer homology restrict 4-manifold topology.

Researchers compute bounds and formulas for non-orientable 4-genus of torus knots.