Searcharxiv⌕ Search

arXiv subjects

Robert Lipshitz

Publications and source records attributed to Robert Lipshitz.

At least 19 recordsLinked to original sources
arXiv2026

Real bordered Floer homology

Fix a 3-manifold $Y$ with boundary $F\amalg F$ and an orientation-preserving involution $\tau: Y\to Y$ exchanging the boundary components, with nonempty fixed set. To an appropriate kind of Heegaard diagram for $Y$, we describe how to associate a module over the bordered Heegaard Floer algebra of $F$. These modules satisfy a gluing, or pairing, theorem, and extend the "hat" variant of Guth-Manolescu's real Heegaard Floer homology, $\widehat{HFR}(Y,\tau)$. Using these modules, we give a practical algorithm to compute $\widehat{HFR}(Y,\tau)$ for real 3-manifolds $(Y,\tau)$ with connected fixed set.

math.GT↗
arXiv2025

Detecting Heegaard Floer homology solid tori

We show that a rational homology solid torus is a Heegaard Floer homology solid torus if and only if it has a Dehn filling with a non-separating 2-sphere. Using this, we characterize Seifert fibered Heegaard Floer solid tori.

math.GT↗
arXiv2025

Bordered Floer homology, handlebody detection, and compressing diffeomorphisms

We show that, up to connected sums with integer homology $L$-spaces, bordered Floer homology detects handlebodies, as well as whether a mapping class extends over a given handlebody or compression body. Using this, we combine ideas of Casson-Long with the theory of train tracks to give an algorithm using bordered Floer homology to detect whether a mapping class extends over any compression body.

math.GT↗
arXiv2024

Torsion in linearized contact homology for Legendrian knots

We present examples of Legendrian knots in $\mathbb{R}^3$ that have linearized Legendrian contact homology over $\mathbb{Z}$ containing torsion. As a consequence, we show that there exist augmentations of Legendrian knots over $\mathbb{Z}$ that are not induced by exact Lagrangian fillings, even though their mod $2$ reductions are.

math.SG↗
arXiv2023

Local equivalence and refinements of Rasmussen's s-invariant

Inspired by the notions of local equivalence in monopole and Heegaard Floer homology, we introduce a version of local equivalence that combines odd Khovanov homology with equivariant even Khovanov homology into an algebraic package called a local even-odd (LEO) triple. We get a homomorphism from the smooth concordance group $C$ to the resulting local equivalence group $C_{LEO}$ of such triples. We give several versions of the $s$-invariant that descend to $C_{LEO}$, including one that completely determines whether the image of a knot $K$ in $C_{LEO}$ is trivial. We discuss computer experiments illustrating the power of these invariants in obstructing sliceness, both statistically and for some interesting knots studied by Manolescu-Piccirillo. Along the way, we explore several variants of this local equivalence group, including one that is totally ordered.

math.GT↗
arXiv2023

Bordered HF- for three-manifolds with torus boundary

We define an invariant for bordered 3-manifolds with torus boundary, taking the form of a module over a weighted A-infinity algebra associated to a torus defined in previous work. On setting U=0, we obtain the bordered three-manifold invariants with torus boundary constructed earlier.

math.GT↗
arXiv2023

Floer homology beyond borders

Bordered Floer homology is an invariant for 3-manifolds with boundary, defined by the authors in 2008. It extends the Heegaard Floer homology of closed 3-manifolds, defined in earlier work of Zoltán Szabó and the second author. In addition to its conceptual interest, bordered Floer homology also provides powerful computational tools. This survey outlines the theory, focusing on recent developments and applications.

math.GT↗
arXiv2022

A Khovanov stable homotopy type

Given a link diagram L we construct spectra X^j(L) so that the Khovanov homology Kh^{i,j}(L) is isomorphic to the (reduced) singular cohomology H^i(X^j(L)). The construction of X^j(L) is combinatorial and explicit. We prove that the homotopy type of X^j(L) depends only on the isotopy class of the corresponding link.

math.GT↗
arXiv2022

Khovanov homology of strongly invertible knots and their quotients

We construct a spectral sequence relating the Khovanov homology of a strongly invertible knot to the annular Khovanov homologies of the two natural quotient knots. Using this spectral sequence, we re-prove that Khovanov homology distinguishes certain slice disks. We also give an analogous spectral sequence for the Heegaard Floer homology of the branched double cover.

math.GT↗
arXiv2021

The cube and the Burnside category

In this note we present a combinatorial link invariant that underlies some recent stable homotopy refinements of Khovanov homology of links. The invariant takes the form of a functor between two combinatorial 2-categories, modulo a notion of stable equivalence. We also develop some general properties of such functors.

math.GT↗
arXiv2021

On transverse invariants from Khovanov homology

O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskaya's invariant, one valued in Bar-Natan's deformation of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum. We show that the first of these refinements is invariant under negative flypes and SZ moves; this implies that Plamenevskaya's class is also invariant under these moves. We go on to show that for small-crossing transverse knots K, both refinements are determined by the classical invariants of K.

math.GT↗
arXiv2021

A bordered HF- algebra for the torus

We describe a weighted $A_\infty$-algebra associated to the torus. We give a combinatorial construction of this algebra, and an abstract characterization. The abstract characterization also gives a relationship between our algebra and the wrapped Fukaya category of the torus. These algebras underpin the (unspecialized) bordered Heegaard Floer homology for three-manifolds with torus boundary, which will be constructed in forthcoming work.

math.GT↗
arXiv2021

Bordered Heegaard Floer homology: Invariance and pairing

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A-infinity module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A-infinity tensor product of the type D module of one piece and the type A module from the other piece is HF^ of the glued manifold. As a special case of the construction, we specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for HF^. We relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.

math.GT↗