arXiv research

Proves an equivariant version of Heegaard Floer link surgery formula.

Study of knot Floer homology and its relation to Heegaard Floer homology via equivariant surgery.

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(K) of this Blanchfield pairi…

Study uses instanton Floer theory to obstruct knot unknotting operations.

Proves cosmetic surgery conjecture for strongly invertible knots.

We apply Lescop's construction of $\mathbb{Z}$-equivariant perturbative invariant of knots and 3-manifolds to the explicit equivariant propagator of "AL-paths" given in arXiv:1403.8030. We obtain an invariant $\hat{Z}_n$ of certain equivalence classes of fiberwise Morse functions on a 3-manifold fibered over $S^1$, whi…

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the configuration space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(M,K) of this B…

Using the conjugation symmetry on Heegaard Floer complexes, we define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to $\mathbb{Z}_4$-equivariant Seiberg-Witten Floer homology. Further, we obtain two new invariants of homology cobordism, $\underline{d}$ and $\bar{d}$…

Manolescu correction terms are numerical invariants of homology three-spheres arising from $\mathrm{Pin}(2)$-equivariant Seiberg-Witten theory that contain information about homology cobordism. We discuss several constraints on these invariants for homology spheres obtained by Dehn surgery on a knot in the three-sphere…

In a previous article, we constructed an invariant Z for null-homologous knots in rational homology spheres, from equivariant intersections in configuration spaces. Here we present an equivalent definition of Z in terms of configuration space integrals, we prove that Z is multiplicative under connected sum, and we prov…

Invariants for colored links found, with topological protection of certain knots.

Defines band maps in unoriented link Floer homology forming a skein exact triangle.

We study a theory of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries. It is an analogue in the setting of the rational homology of the Goussarov-Rozansky theory for knots in integral homology 3-spheres. We give a partial combinatorial …

The paper classifies involutions on S^4, proving linearities under certain conditions.

Classifies $C_p$-surfaces using equivariant surgery methods.

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

Cosmetic surgeries on pretzel knots are unique.

The paper generalizes index theory for periodic manifolds and proves equivalence of signatures for tori.

Study confirms contact cosmetic surgery for most knots, with exceptions.

New proof for a knot type not admitting certain surgeries.

Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.

Disproves conjectures about shared surgeries for distinct knots.

The study calculates and analyzes alternating surgeries for various knots.

New Heegaard Floer homology findings block chirally cosmetic surgeries.

A knot in the 3-sphere is called an L--space knot if it admits a nontrivial Dehn surgery yielding an L--space. Like torus knots and Berge knots, many L--space knots admit also a Seifert fibered surgery. We give a concrete example of a hyperbolic, L-space knot which has no exceptional surgeries, in particular, no Seifer…

The study examines quasi-alternating surgeries on knots and their properties.

A Seifert surgery is a pair (K, m) of a knot K in the 3-sphere and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K, m) yields Seifert surgeries. We study Seifert surgeries obtained from those o…

The study proves large alternating Montesinos knots cannot have purely cosmetic surgeries.

New findings on cosmetic surgeries for satellite knots.

We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot admits a toroidal Seifert fibered surgery, then the knot is either the trefoil knot and the surgery slope is zero, or the connected sum of a (2,p)-torus knot and a (2,q)-torus kno…

Study shows most odd pretzel knots don't allow chirally cosmetic surgeries.

Unique surgery descriptions found for knots in 3-manifolds.

Two knot families meet cosmetic surgery conjecture.

Study proves nontrivial knots can't undergo cosmetic surgeries.

We study finite type invariants of nullhomologous knots in a closed 3-manifold $M$ defined in terms of certain descending filtration $\{\mathscr{K}_n(M)\}_{n\geq 0}$ of the vector space $\mathscr{K}(M)$ spanned by isotopy classes of nullhomologous knots in $M$. The filtration $\{\mathscr{K}_n(M)\}_{n \geq 0}$ is define…

3-braid knots can't have purely cosmetic surgeries.

New insights into cosmetic surgeries using Heegaard Floer homology.

Added examples of S^1-manifolds with finite 2nd homotopy group and non-zero A-genus.

Infinite hyperbolic knots yield unusual surgeries.

How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the ot…

By obtaining surgery descriptions of knots which lie on the genus one fiber of the trefoil or figure eight knot, we show that these include hyperbolic knots with arbitrarily large volume. These knots admit lens space surgeries and form two families of Berge knots. By way of tangle descriptions we also obtain surgery de…

Enhances knot surgery formulae for instanton Floer homology.

Study computes SL(2,C) Floer cohomology for surgeries on knots.

Surgery on knots always admits a tight contact structure.

Study on reducing surgeries on knots, developing thickness and genus bounds.

New tool: relative Hopf invariant for Poincaré surgery.

The study determines lens spaces that can be obtained from surgeries on knots in the Poincaré homology sphere.

Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not left-orderable. We provide a refinement of this criterion by introducing the notion of a decayed knot; it is shown that Dehn surgery …