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Matthew Hedden

Publications and source records attributed to Matthew Hedden.

At least 19 recordsLinked to original sources
arXiv2025

Relative Thom Conjectures, symplectic and beyond

We establish a criterion that ensures a bounded almost complex curve in a bounded almost complex 4-manifold minimizes genus amongst all smooth surfaces that share its homology class and the transverse link on its boundary. An immediate corollary affirms the relative symplectic Thom conjecture and, moreover, yields obstructions coming from knot Floer homology to a link bounding a symplectic surface in a symplectic filling. Our results are applicable to knots in manifolds equipped with plane fields that admit no symplectic fillings; for instance, we show that symplectic surfaces in a thickening of any contact 3-manifold with non-zero Ozsvath-Szabo invariant minimize slice genus for their boundary. We conjecture that this phenomenon occurs precisely when the contact structure is tight, which would imply that tightness can be viewed as a symplecto-geometric notion.

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arXiv2025

Knot Floer homology of positive braids

We compute the next-to-top term of the knot Floer homology of any link obtained as the closure of a positive braid, showing in particular that the rank is one for any prime knot in this family. As such knots are fibered, it follows that their monodromies are fixed-point free. We compare the set of positive braids with other classes of knots known to have this property. One such class consists of knots possessing "diagonal" grid diagrams. We provide an example of such a knot that is not a positive braid, providing an answer to a question of Vance and Kubota. We conclude with a number of problems and questions for future study naturally motivated by our theorem.

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arXiv2023

A 4-dimensional rational genus bound

We introduce a 4-dimensional analogue of the rational Seifert genus of a knot $K\subset Y$, which we call the rational slice genus, that measures the complexity of a homology class in $H_2(Y\times [0,1],K;\mathbb{Q})$. Our main theorem is a lower bound for the rational slice genus of a knot in terms of its Heegaard Floer $\tau$ invariants. To prove this, we bound the $\tau$ invariants of any satellite link whose pattern is a closed braid in terms of the $\tau$ invariants of the companion knot, a result which should be of independent value. Our techniques also produce rational PL slice genus bounds. As applications, we use our bounds to prove that Floer simple knots have rational slice genus equal to their rational Seifert genus. We also show that there exist sequences of knots in a fixed 3-manifold whose PL slice genus is unbounded. In addition, we produce stronger bounds for the slice genus of knots relative to the rational longitude, and use these to produce a rational slice-Bennequin bound for knots in contact manifolds with non-trivial contact invariant.

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arXiv2022

Rank-expanding satellites, Whitehead doubles, and Heegaard Floer homology

We show that a large class of satellite operators are rank-expanding; that is, they map some rank-one subgroup of the concordance group onto an infinite linearly independent set. Our work constitutes the first systematic study of this property in the literature and partially affirms a conjecture of the second author and Pinz\'on-Caicedo. More generally, we establish a Floer-theoretic condition for a family of companion knots to have infinite-rank image under satellites from this class. The methods we use are amenable to patterns which act trivially in topological concordance and are capable of handling a surprisingly wide variety of companions. For instance, we give an infinite linearly independent family of Whitehead doubles whose companion knots all have negative $\tau$-invariant. Our also results recover and extend several theorems in this area established using instanton Floer homology.

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arXiv2022

Murasugi sum and extremal knot Floer homology

The aim of this paper is to study the behavior of knot Floer homology under Murasugi sum. We establish a graded version of Ni's isomorphism between the extremal knot Floer homology of Murasugi sum of two links and the tensor product of the extremal knot Floer homology groups of the two summands. We further prove that $\tau=g$ for each summand if and only if $\tau=g$ holds for the Murasugi sum (with $\tau$ and $g$ defined appropriately for multi-component links). Some applications are presented.

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arXiv2021

On naturality of the Ozsvath-Szabo contact invariant

We discuss functoriality properties of the Ozsvath-Szabo contact invariant, and expose a number of results which seemed destined for folklore. We clarify the (in)dependence of the invariant on the basepoint, prove that it is functorial with respect to contactomorphisms, and show that it is strongly functorial under Stein cobordisms.

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arXiv2021

The Fukaya category of the pillowcase, traceless character varieties, and Khovanov Cohomology

For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factorization of Bar-Natan's functor from the tangle cobordism category to chain complexes. In particular, the hom set of our invariant with a particular non-compact Lagrangian associated to the trivial tangle is naturally isomorphic to the reduced Khovanov chain complex of the closure of the tangle. Our construction comes from the geometry of traceless SU(2) character varieties associated to resolutions of the tangle diagram, and was inspired by Kronheimer and Mrowka's singular instanton link homology.

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arXiv2021

A surgery formula for knot Floer homology

Let $K$ be a rationally null-homologous knot in a $3$-manifold $Y$, equipped with a nonzero framing $λ$, and let $Y_λ(K)$ denote the result of $λ$-framed surgery on $Y$. Ozsváth and Szabó gave a formula for the Heegaard Floer homology groups of $Y_λ(K)$ in terms of the knot Floer complex of $(Y,K)$. We strengthen this formula by adding a second filtration that computes the knot Floer complex of the dual knot $K_λ$ in $Y_λ$, i.e., the core circle of the surgery solid torus. In the course of proving our refinement we derive a combinatorial formula for the Alexander grading which may be of independent interest.

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arXiv2021

Satellites of Infinite Rank in the Smooth Concordance Group

We conjecture that satellite operations are either constant or have infinite rank in the concordance group. We reduce this to the difficult case of winding number zero satellites, and use $SO(3)$ gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank subgroup of the smooth concordance group $\mathcal{C}$. Our criterion applies widely; notably to many unknotted patterns for which the corresponding operators on the topological concordance group are zero. We raise some questions and conjectures regarding satellite operators and their interaction with concordance.

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arXiv2020

4-dimensional aspects of tight contact 3-manifolds

In this article we conjecture a 4-dimensional characterization of tightness: a contact structure is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in Yx[0,1]. An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: if a fibered link L induces a tight contact structure on Y then its fiber surface maximize Euler characteristic amongst all surfaces in Yx[0,1] with boundary L. We provide evidence for both conjectures by proving them for contact structures with non-vanishing Ozsv\'ath-Szab\'o contact invariant. We also show that any subsurface of a page of an open book inducing a contact structure with non-trivial invariant maximize "slice" Euler-characteristic for its boundary, and conjecture that this holds more generally for open books inducing tight contact structures.

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arXiv2020

Knot Floer homology and relative adjunction inequalities

We establish inequalities that constrain the genera of smooth cobordisms between knots in 4-dimensional cobordisms. These "relative adjunction inequalities" improve the adjunction inequalities for closed surfaces which have been instrumental in many topological applications of gauge theory. The relative inequalities refine the latter by incorporating numerical invariants of knots in the boundary associated to Heegaard Floer homology classes determined by the 4-manifold. As a corollary, we produce a host of concordance invariants for knots in a general 3-manifold, one such invariant for every non-zero Floer class. We apply our results to produce analogues of the Ozsv\'ath-Szab\'o-Rasmussen concordance invariant for links, allowing us to reprove the link version of the Milnor conjecture, and, furthermore, to show that knot Floer homology detects strongly quasipositive fibered links.

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arXiv2020

Corks, involutions, and Heegaard Floer homology

Building on the algebraic framework developed by Hendricks, Manolescu, and Zemke, we introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution over any homology ball, rather than a particular contractible manifold. Unlike previous approaches, we do not utilize any closed 4-manifold topology or contact topology. Instead, we adapt the formalism of local equivalence coming from involutive Heegaard Floer homology. As an application, we define a modification $\Theta^{\tau}_{\mathbb{Z}}$ of the homology cobordism group which takes into account an involution on each homology sphere, and prove that this admits a $\mathbb{Z}^\infty$-subgroup of strongly non-extendable corks. The group $\Theta^{\tau}_{\mathbb{Z}}$ can also be viewed as a refinement of the bordism group of diffeomorphisms. Using our invariants, we furthermore establish several new families of corks and prove that various known examples are strongly non-extendable. Our main computational tool is a monotonicity theorem which constrains the behavior of our invariants under equivariant negative-definite cobordisms, and an explicit method of constructing such cobordisms via equivariant surgery.

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arXiv2019

Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot

We give two infinite families of examples of closed, orientable, irreducible 3-manifolds $M$ such that $b_1(M)=1$ and $π_1(M)$ has weight 1, but $M$ is not the result of Dehn surgery along a knot in the 3-sphere. This answers a question of Aschenbrenner, Friedl and Wilton, and provides the first examples of irreducible manifolds with $b_1=1$ that are known not to be surgery on a knot in the 3-sphere. One family consists of Seifert fibered 3-manifolds, while each member of the other family is not even homology cobordant to any Seifert fibered 3-manifold. None of our examples are homology cobordant to any manifold obtained by Dehn surgery along a knot in the 3-sphere.

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arXiv2018

On the functoriality of Khovanov-Floer theories

We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2 to a link diagram such that (1) the E_2 page of the resulting spectral sequence is naturally isomorphic to the Khovanov homology of the link; (2) this filtered complex behaves nicely under planar isotopy, disjoint union, and 1-handle addition; and (3) the spectral sequence collapses at the E_2 page for any diagram of the unlink. We prove that a Khovanov-Floer theory naturally yields a functor from the link cobordism category to the category of spectral sequences. In particular, every page (after E_1) of the spectral sequence accompanying a Khovanov-Floer theory is a link invariant, and an oriented cobordism in R^3 \times [0,1] between links in R^3 induces a map between each page of their spectral sequences, invariant up to smooth isotopy of the cobordism rel boundary. We then show that the spectral sequences relating Khovanov homology to Heegaard Floer homology and singular instanton knot homology are induced by Khovanov-Floer theories and are therefore functorial in the manner described above, as has been conjectured for some time. We further show that Szabo's geometric spectral sequence comes from a Khovanov-Floer theory, and is thus functorial as well. In addition, we illustrate how our framework can be used to give another proof that Lee's spectral sequence is functorial and that Rasmussen's invariant is a knot invariant. Finally, we use this machinery to define some potentially new knot invariants.

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arXiv2017

On the geography and botany of knot Floer homology

This note explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a new constraint for knots admitting lens space surgeries, as well as a proof that the rank of knot Floer homology detects the trefoil knot. For the second, we show that any non-trivial band sum of two unknots gives rise to an infinite family of distinct knots with isomorphic knot Floer homology. We also prove that the fibered knot with identity monodromy is strongly detected by its knot Floer homology, implying that Floer homology solves the word problem for mapping class groups of surfaces with non-empty boundary. Finally, we survey some conjectures and questions and, based on the results described above, formulate some new ones.

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arXiv2015

Splicing knot complements and bordered Floer homology

We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology.

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arXiv2015

The Upsilon function of L-space knots is a Legendre transform

Given an L-space knot we show that its Upsilon function is the Legendre transform of a counting function equivalent to the d-invariants of its large surgeries. The unknotting obstruction obtained for the Upsilon function is, in the case of L-space knots, contained in the d-invariants of large surgeries. Generalizations apply for connected sums of L-space knots, which imply that the slice obstruction provided by Upsilon on the subgroup of concordance generated by L-space knots is no finer than that provided by the d-invariants.

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