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Jennifer Hom

Publications and source records attributed to Jennifer Hom.

At least 19 recordsLinked to original sources
arXiv2026

Heegaard Floer knot trace invariants, exotic 4-manifolds, and symplectic obstructions

We show that the numerical invariants $\nu$ and $|\varepsilon|$ coming from knot Floer homology are knot $n$-trace invariants for any integer $n$, resolving the remaining case in Hayden-Mark-Piccirillo. This extension allows us to construct new families of exotic pairs using Yasui patterns. Moreover, by studying the invariant $\widehat{\nu}(K)=|\varepsilon(K)|(2\nu(K)-1)$, we give a new topological obstruction to a $4$-manifold being a strong symplectic filling of any contact structure on the boundary.

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arXiv2026

Ribbon concordance and cabling

We study ribbon concordances to cable knots. We formulate a conjecture predicting that any nontrivial knot admitting a ribbon concordance to a (p,q)-cable must itself be a (p,q)-cable. We prove the conjecture when the knot admitting the ribbon concordance is already a cable of the same companion, is a torus knot, or has genus one. We also verify it for a broad class of target cables. The proofs use a minimum-height invariant defined from the immersed-curve formulation of knot Floer homology. This invariant obstructs ribbon concordances and implies that any nontrivial knot admitting a ribbon concordance to a fibered cable knot is prime. We also establish genus bounds for knots admitting ribbon concordances to cable knots.

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arXiv2026

Distinguishing exotic $\mathbb{R}^4$'s with Heegaard Floer homology

Attaching a Casson handle to a slice disk complement yields a smooth 4-manifold that is homeomorphic to $\mathbb{R}^4$. We show that if two slice knots have sufficiently different knot Floer homology, then the resulting exotic $\mathbb{R}^4$'s made using the simplest positive Casson handle are not diffeomorphic, giving us a countably infinite family of pairwise nondiffeomorphic chiral exotic $\mathbb{R}^4$'s. Our main tool is Gadgil's end Floer homology and we use this to produce families of exotic $\mathbb{R}^4$ with various phenomena. As an application, we reprove a result of Bi\v{z}aca-Etnyre that $Y \times \mathbb{R}$, where $Y$ is any closed $3$-manifold, has infinitely many distinct smooth structures.

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arXiv2025

Ribbon knots and iterated cables of fibered knots

We define a knot to be $\gamma_0$-sharp if its Seifert genus is detected by the concordance invariant $\gamma_0$, which arises from the immersed curve formalism in bordered Heegaard Floer homology. We show that a connected sum of $\gamma_0$-sharp fibered knots is ribbon exactly when it is of the form $K \mathbin{\#} -K$. Consequently, either iterated cables of tight fibered knots are linearly independent in the smooth concordance group, or the slice--ribbon conjecture fails.

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arXiv2023

A note on rational band moves

We introduce an oriented rational band move, a generalization of an ordinary oriented band move, and show that if a knot $K$ in the three-sphere can be made into the $(n+1)$-component unlink by $n$ oriented rational band moves, then $K$ is rationally slice.

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arXiv2023

Topologically and rationally slice knots

A knot in $S^3$ is topologically slice if it bounds a locally flat disk in $B^4$. A knot in $S^3$ is rationally slice if it bounds a smooth disk in a rational homology ball. We prove that the smooth concordance group of topologically and rationally slice knots admits a $\mathbb{Z}^\infty$ subgroup. All previously known examples of knots that are both topologically and rationally slice were of order two. As a direct consequence, it follows that there are infinitely many topologically slice knots that are strongly rationally slice but not slice.

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arXiv2023

PL-genus of surfaces in homology balls

We consider manifold-knot pairs $(Y,K)$ where $Y$ is a homology sphere that bounds a homology ball. We show that the minimum genus of a PL surface $\Sigma$ in a homology ball $X$ such that $\partial (X, \Sigma) = (Y, K)$ can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from $(Y, K)$ to any knot in $S^3$ can be arbitrarily large. The proof relies on Heegaard Floer homology.

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arXiv2022

Handle decomposition complexity and representation spaces

We prove that there are homology three-spheres that bound definite four-manifolds, but any such bounding four-manifold must be built out of many handles. The argument uses the homology cobordism invariant $\Gamma$ from instanton Floer homology.

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arXiv2022

Unknotting number and cabling

The unknotting number of knots is a difficult quantity to compute, and even its behavior under basic satelliting operations is not understood. We establish a lower bound on the unknotting number of cable knots and iterated cable knots purely in terms of the winding number of the pattern. The proof uses Alishahi-Eftekhary's bounds on unknotting number from knot Floer homology together with Hanselman-Watson's computation of the knot Floer homology of cables in terms of immersed curves in the punctured torus.

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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.

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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.

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arXiv2021

Homology concordance and knot Floer homology

We study the homology concordance group of knots in integer homology three-spheres which bound integer homology four-balls. Using knot Floer homology, we construct an infinite number of $\mathbb{Z}$-valued, linearly independent homology concordance homomorphisms which vanish for knots coming from $S^3$. This shows that the homology concordance group modulo knots coming from $S^3$ contains an infinite-rank summand. The techniques used here generalize the classification program established in previous papers regarding the local equivalence group of knot Floer complexes over $\mathbb{F}[U, V]/(UV)$. Our results extend this approach to complexes defined over a broader class of rings.

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arXiv2021

A note on PL-disks and rationally slice knots

We give infinitely many examples of manifold-knot pairs (Y, J) such that Y bounds an integer homology ball, J does not bound a non-locally-flat PL-disk in any integer homology ball, but J does bound a smoothly embedded disk in a rational homology ball. The proof relies on formal properties of involutive Heegaard Floer homology.

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arXiv2021

Getting a handle on the Conway knot

A knot is said to be slice if it bounds a smooth disk in the 4-ball. For 50 years, it was unknown whether a certain 11 crossing knot, called the Conway knot, was slice or not, and until recently, this was the only one of the thousands of knots with fewer than 13 crossings whose slice-status remained a mystery. We will describe Lisa Piccirillo's proof that the Conway knot is not slice. The main idea of her proof is given in the title of this article.

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arXiv2020

Linear independence of rationally slice knots

A knot in $S^3$ is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all infinite order. All previously known examples of rationally slice knots were order two.

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arXiv2020

Surgery exact triangles in involutive Heegaard Floer homology

We establish a surgery exact triangle for involutive Heegaard Floer homology by using a doubling model of the involution. We use this exact triangle to give an involutive version of Ozsv\'ath-Szab\'o's mapping cone formula for knot surgery. As an application, we use this surgery formula to give examples of integer homology spheres that are not homology cobordant to any linear combination of Seifert fibered spaces.

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