Searcharxiv⌕ Search

arXiv subjects

Ian Zemke

Publications and source records attributed to Ian Zemke.

At least 19 recordsLinked to original sources
arXiv2026

Link Floer homology, nonformality, and the Borromean rings

Our main result is a computation of the full link Floer complex of the Borromean rings. The Borromean rings are well known to be an L-space link. We prove that the full link Floer complex is not a free-resolution of its homology, in contrast to the situation for L-space knots, plumbed L-space links, and two-component L-space links. We also describe the link involution of the Borromean rings, up to a minor ambiguity. The proof goes by way of studying some general techniques about $A_\infty$-deformations of L-space link Floer complexes. Our techniques give a simple criterion for when an L-space link must have formal link Floer complex, which reproves the existing formality results for one- and two-component L-space links.

math.GT↗
arXiv2026

The link surgery modules of 2-component L-space links

In our earlier work, we studied the link surgery modules of two component L-space links. Therein, we computed two of the four idempotents of such modules. In this article, we use Koszul duality to give an alternate account of this proof, and also to extend it to compute the entire link surgery modules of such links, modulo a technical result which will be proven in a subsequent paper.

math.GT↗
arXiv2025

Applications of the L-space satellite formula

We give a formula for the $\tau$-invariant of a satellite knot $P(K,n)$ when $P$ is an L-space satellite operator. Our formula holds for general L-space satellite operators $P$ when the companion $K$ satisfies $\epsilon(K)=1$. When $\epsilon(K)$ is $0$ or $-1$, we state a formula which requires some additional assumptions on $P$ or $n$. Our main tool is our algorithm which computes the knot Floer complex of satellite knots constructed using L-space satellite operators, which we developed in a previous paper. Our formula for $\tau$ recovers many existing formulas for the behavior of $\tau$ under satellite operators, including for cables. We apply our formula to questions about the slice genus of satellite knots, showing, e.g., that if $K$ is a knot with $\tau(K)=g_4(K)>0$, then satellites of $K$ by L-space satellite operators have the same property. Another application is a proof that L-space satellite operators satisfy a conjecture of Hedden and Pinz\'on-Caicedo: If $P$ is an L-space satellite operator which acts as a group homomorphism on the smooth concordance group, then $P$ is either the zero operator, the identity operator, or the orientation reversing operator.

math.GT↗
arXiv2025

The link surgery formula and equivariant surgeries

We prove an equivariant version of the Heegaard Floer link surgery formula. As a special case, this gives an equivariant knot surgery formula for equivariant knots in $S^3$. Our proof goes by way of a naturality theorem for certain bordered modules described by the last author. As a sample application, we prove the kernel of the forgetful map from the equivariant homology cobordism group to the homology cobordism group contains a $\Z^\infty$-summand.

math.GT↗
arXiv2025

Koszul duality and the link surgery formula

In previous works, the author described an associative algebra whose $A_\infty$-module categories encode the Heegaard Floer Dehn surgery formulas. In this article, we describe the Koszul dual of this algebra. We construct dualizing bimodules, and prove several equivalences of categories. The constructions of this paper have applications to computational problems involving the link surgery formula.

math.GT↗
arXiv2024

L-space satellite operators and knot Floer homology

We consider satellite operators where the corresponding 2-component link is an L-space link. This family includes many commonly studied satellite operators, including cabling operators, the Whitehead operator, and a family of Mazur operators. We give a formula which computes the knot Floer complex of a satellite of $K$ in terms of the knot Floer complex of $K$. Our main tools are the Heegaard Floer Dehn surgery formulas and their refinements. A key step in our computation is a proof that 2-component L-space links have formal knot Floer complexes. We use this to show that the link Floer complexes of 2-component L-space links are determined by their multivariable Alexander polynomials. We implement our satellite formula in Python code, which we also make available.

math.GT↗
arXiv2024

Gompf's cork and Heegaard Floer homology

Gompf showed that for $K$ in a certain family of double-twist knots, the swallow-follow operation makes $1/n$-surgery on $K \# -K$ into a cork boundary. We derive a general Floer-theoretic condition on $K$ under which this is the case. Our formalism allows us to produce many further examples of corks, partially answering a question of Gompf. Unlike Gompf's method, our proof does not rely on any closed 4-manifold invariants or effective embeddings, and also generalizes to other diffeomorphisms.

math.GT↗
arXiv2024

The equivalence of lattice and Heegaard Floer homology

We prove Némethi's conjecture: if $Y$ is a 3-manifold which is the boundary of a plumbing of a tree of disk bundles over $S^2$, then the lattice homology of $Y$ coincides with the Heegaard Floer homology of $Y$. We also give a conjectural description of the $H_1(Y)/\mathrm{Tors}$ action when $b_1(Y)>0$.

math.GT↗
arXiv2024

A note on the involutive invariants of splices

A natural family of potentially 2-torsion elements in the integer homology cobordism group consists of splices of knots with their mirrors. We show that such 3-manifolds have locally trivial involutive Floer homology. We show some related families of splices also have locally trivial involutive Floer homology. Our arguments show that many gauge theoretic invariants also vanish on these 3-manifolds.

math.GT↗
arXiv2024

Stabilization distance bounds from link Floer homology

We consider the set of connected surfaces in the 4-ball with boundary a fixed knot in the 3-sphere. We define the stabilization distance between two surfaces as the minimal $g$ such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most $g$. Similarly, we consider a double point distance between two surfaces of the same genus, which is the minimum over all regular homotopies connecting the two surfaces of the maximal number of double points appearing in the homotopy. To many of the concordance invariants defined using Heegaard Floer homology, we construct an analogous invariant for a pair of surfaces. We show that these give lower bounds on the stabilization distance and the double point distance. We compute our invariants for some pairs of deform-spun slice disks by proving a trace formula on the full infinity knot Floer complex, and by determining the action on knot Floer homology of an automorphism of the connected sum of a knot with itself that swaps the two summands. We use our invariants to find pairs of slice disks with arbitrarily large distance with respect to many of the metrics we consider in this paper. We also answer a slice disk analogue of Problem 1.105 (B) from Kirby's problem list by showing the existence of non-0-cobordant slice disks.

math.GT↗
arXiv2024

Lattice homology, formality, and plumbed L-space links

We define a link lattice complex for plumbed links, generalizing constructions of Ozsváth, Stipsicz and Szabó, and of Gorsky and Némethi. We prove that for all plumbed links in rational homology 3-spheres, the link lattice complex is homotopy equivalent to the link Floer complex as an $A_\infty$-module. Additionally, we prove that the link Floer complex of a plumbed L-space link is a free resolution of its homology. As a consequence, we give an algorithm to compute the link Floer complexes of plumbed L-space links, in particular of algebraic links, from their multivariable Alexander polynomial.

math.GT↗
arXiv2024

Connected sums and directed systems in knot Floer homologies

We prove a number of fundamental properties about instanton knot Floer homology. Our arguments rely on general properties of sutured Floer theories and apply also in the Heegaard Floer and monopole Floer settings, where many of our results were already known. Our main result is the connected sum formula for instanton knot Floer homology. An extension of this result proves the oriented skein exact triangle for the minus version of instanton knot Floer homology. Finally, we derive a new model of the minus version of instanton knot Floer homology, which takes the form of a free, finitely generated chain complex over a polynomial ring, as opposed to a direct limit. This construction is new to all of the Floer theories. We explore these results also in the context of Heegaard Floer theory as well.

math.GT↗
arXiv2023

A general Heegaard Floer surgery formula

We give several new perspectives on the Heegaard Floer Dehn surgery formulas of Manolescu, Ozsváth and Szabó. Our main result is a new exact triangle in the Fukaya category of the torus which gives a new proof of these formulas. This exact triangle is different from the one which appeared in Ozsváth and Szabó's original proof. This exact triangle simplifies a number of technical aspects in their proofs and also allows us to prove several new results. A first application is an extensions of the link surgery formula to arbitrary links in closed 3-manifolds, with no restrictions on the link being null-homologous. A second application is a proof that the modules for bordered manifolds with torus boundaries, defined by the author in a previous paper, are invariants. Another application is a simple proof of a version of the surgery formula which computes knot and link Floer complexes in terms of subcubes of the link surgery hypercube. As a final application, we show that the knot surgery algebra is homotopy equivalent to an endomorphism algebra of a sum of two decorated Lagrangians in the torus, mirroring a result of Auroux concerning the algebras of Lipshitz, Ozsváth and Thurston.

math.GT↗
arXiv2022

An involutive dual knot surgery formula

We prove an involutive analog of the dual knot surgery formula of Eftekhary and Hedden-Levine. We also compute a small model for the local equivalence class of the involutive dual knot complex.

math.GT↗
arXiv2022

Naturality and functoriality in involutive Heegaard Floer homology

We prove first-order naturality of involutive Heegaard Floer homology, and furthermore construct well-defined maps on involutive Heegaard Floer homology associated to cobordisms between three-manifolds. We also prove analogous naturality and functoriality results for involutive Floer theory for knots and links. The proof relies on the doubling model for the involution, as well as several variations.

math.GT↗
arXiv2021

Bordered manifolds with torus boundary and the link surgery formula

In this paper, we develop a theory of bordered $\mathit{HF}^-$ using the link surgery formula of Manolescu and Ozsv\'{a}th. We interpret their link surgery complexes as type-$D$ modules over an associative algebra $\mathcal{K}$, which we introduce. We prove a connected sum formula, which we interpret as an $A_\infty$-tensor product over our algebra $\mathcal{K}$. Topologically, this connected sum formula may be viewed as a formula for gluing along torus boundary components. We compute several important examples. We show that the dual knot formula of Hedden--Levine and Eftekhary may be interpreted as the $DA$-bimodule for a particular diffeomorphism of the torus. As another example, if $K_1$ and $K_2$ are knots in $S^3$, and $Y$ is obtained by gluing the complements of $K_1$ and $K_2$ together using an orientation reversing diffeomorphism of their boundaries, then our theory may be used to compute $\mathit{CF}^-(Y)$ from $\mathit{CFK}^\infty(K_1)$ and $\mathit{CFK}^\infty(K_2)$. We additionally compute the type-$D$ modules for rationally framed solid tori. Our theory also computes the Heegaard Floer homology of all 3-manifolds which bound a plumbing of a tree of disk bundles over 2-spheres. In a subsequent article, we use this work to verify N\'{e}methi's conjecture about lattice homology.

math.GT↗