Ciprian Manolescu
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View article: Heegaard Floer homology and integer surgeries on links
Heegaard Floer homology and integer surgeries on links Open
Let L be a link in an integral homology three-sphere. We give a description of the Heegaard Floer homology of integral surgeries on L in terms of some data associated to L, which we call a complete system of hyperboxes for L. Roughly, a co…
View article: Searching for ribbons with machine learning
Searching for ribbons with machine learning Open
We apply Bayesian optimization and reinforcement learning to a problem in topology: the question of when a knot bounds a ribbon disk. This question is relevant in an approach to disproving the four-dimensional smooth Poincaré conjecture; u…
View article: Real Heegaard Floer Homology
Real Heegaard Floer Homology Open
We define an invariant of three-manifolds with an involution with non-empty fixed point set of codimension $2$; in particular, this applies to double branched covers over knots. Our construction gives the Heegaard Floer analogue of Li's re…
View article: From zero surgeries to candidates for exotic definite 4‐manifolds
From zero surgeries to candidates for exotic definite 4‐manifolds Open
One strategy for distinguishing smooth structures on closed 4‐manifolds is to produce a knot in that is slice in one smooth filling of but not slice in some homeomorphic smooth filling . In this paper, we explore how 0‐surgery homeomorphis…
View article: Searching for ribbons with machine learning
Searching for ribbons with machine learning Open
We apply Bayesian optimization and reinforcement learning to a problem in topology: the question of when a knot bounds a ribbon disk. This question is relevant in an approach to disproving the four-dimensional smooth Poincaré conjecture; u…
View article: Skein lasagna modules and handle decompositions
Skein lasagna modules and handle decompositions Open
The skein lasagna module is an extension of Khovanov-Rozansky homology to the setting of a four-manifold and a link in its boundary. This invariant plays the role of the Hilbert space of an associated fully extended (4+epsilon)-dimensional…
View article: A knot Floer stable homotopy type
A knot Floer stable homotopy type Open
Given a grid diagram for a knot or link K in $S^3$, we construct a filtered spectrum whose homology is the knot Floer homology of K. We conjecture that the filtered homotopy type of the spectrum is an invariant of K. Our construction does …
View article: A two-variable series for knot complements
A two-variable series for knot complements Open
The physical 3d \mathcal N = 2 theory T[Y] was previously used to predict the existence of some 3 -manifold invariants \widehat{Z}_{a}(q) that take the form of power series with integer coefficients, converging in the unit disk. Their radi…
View article: From zero surgeries to candidates for exotic definite four-manifolds
From zero surgeries to candidates for exotic definite four-manifolds Open
One strategy for distinguishing smooth structures on closed $4$-manifolds is to produce a knot $K$ in $S^3$ that is slice in one smooth filling $W$ of $S^3$ but not slice in some homeomorphic smooth filling $W'$. In this paper we explore h…
View article: Relative genus bounds in indefinite four-manifolds
Relative genus bounds in indefinite four-manifolds Open
Given a closed four-manifold $X$ with an indefinite intersection form, we consider smoothly embedded surfaces in $X \setminus $int$(B^4)$, with boundary a knot $K \subset S^3$. We give several methods to bound the genus of such surfaces in…
View article: Skein lasagna modules for 2-handlebodies
Skein lasagna modules for 2-handlebodies Open
Morrison, Walker, and Wedrich used the blob complex to construct a generalization of Khovanov-Rozansky homology to links in the boundary of a 4-manifold. The degree zero part of their theory, called the skein lasagna module, admits an elem…
View article: A generalization of Rasmussen's invariant, with applications to surfaces in some four-manifolds
A generalization of Rasmussen's invariant, with applications to surfaces in some four-manifolds Open
We extend the definition of Khovanov-Lee homology to links in connected sums of $S^1 \times S^2$'s, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in $S^1 \times S^2$, we compute th…
View article: The Knight Move Conjecture is false
The Knight Move Conjecture is false Open
The Knight Move Conjecture claims that the Khovanov homology of any knot decomposes as direct sums of some “knight move” pairs and a single “pawn move” pair. This is true for instance whenever the Lee spectral sequence from Khovanov homolo…
View article: A sheaf‐theoretic SL(2,C) Floer homology for knots
A sheaf‐theoretic SL(2,C) Floer homology for knots Open
Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three-manifolds, similar to the three-manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat c…
View article: Floer homology and covering spaces
Floer homology and covering spaces Open
We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, we deduce that if a 3-manifold Y admits a p^n-sheeted regular cover that is a Z/pZ-L-space (…
View article: A sheaf-theoretic model for SL(2,C) Floer homology
A sheaf-theoretic model for SL(2,C) Floer homology Open
Given a Heegaard splitting of a three-manifold Y, we consider the SL(2,C) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth open subset corresponding to irreducibl…