Tye Lidman's research while affiliated with North Carolina State University and other places
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Publications (56)
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.
We study 4-dimensional homology cobordisms without 3-handles, showing that they interact nicely with Thurston geometries, character varieties, and instanton and Heegaard Floer homologies. Using these, we derive obstructions to such cobordisms. As one example of these obstructions, we generalize other recent results on the behavior of knot Floer hom...
We prove that satellite operations that satisfy a certain positivity condition and have winding number other than one are not homomorphisms. The argument uses the $d$-invariants of branched covers. In the process, we prove a technical result relating $d$-invariants and the Torelli group which may be of independent interest.
We show that there exist infinitely many closed 3-manifolds that do not embed in closed symplectic 4-manifolds, disproving a conjecture of Etnyre-Min-Mukherjee. To do this, we construct L-spaces that cannot bound positive or negative definite manifolds. The arguments use Heegaard Floer correction terms and instanton moduli spaces.
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 numb...
We establish some new relationships between Milnor invariants and Heegaard Floer homology. This includes a formula for the Milnor triple linking number from the link Floer complex, detection results for the Whitehead link and Borromean rings, and a structural property of the $d$-invariants of surgeries on certain algebraically split links.
We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants $\widetilde{\operatorname{Kh}}$ and $\widetilde{\operatorname{BN}}$. We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture...
We prove that any closed simply-connected smooth 4-manifold is 16-fold branched covered by a product of an orientable surface with the 2-torus, where the construction is natural with respect to spin structures. We also discuss analogous results for other families of 4-manifolds with infinite fundamental groups.
We prove that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)-representations. Our methods use instanton Floer homology, and in particular the surgery exact triangle, holonomy perturbations, and a non-vanishing result due to Kronheimer-Mrowka, as well as results abou...
Given a double cover between 3-manifolds branched along a nullhomologous link, we establish an inequality between the dimensions of their Heegaard Floer homologies. We discuss the relationship with the L-space conjecture and give some other topological applications, as well as an analogous result for sutured Floer homology.
Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology, we prove that reduced Khovanov homology (over $\mathbb{Q}$) detects the figure-eight knot.
We establish some new relationships between Milnor invariants and Heegaard Floer homology. This includes a formula for the Milnor triple linking number from the link Floer complex, detection results for the Whitehead link and Borromean rings, and a structural property of the $d$-invariants of surgeries on certain algebraically split links.
When can surgery on a null-homologous knot K in a rational homology sphere produce a non-separating sphere? We use Heegaard Floer homology to give sufficient conditions for K to be unknotted. We also discuss some applications to homology cobordism, concordance, and Mazur manifolds.
We prove that L-space knots do not have essential Conway spheres with the technology of peculiar modules, a Floer theoretic invariant for tangles.
An important class of three-manifolds are L-spaces, which are rational homology spheres with the smallest possible Floer homology. For knots with an instanton L-space surgery, we compute the framed instanton Floer homology of all integral surgeries. As a consequence, if a knot has a Heegaard Floer and instanton Floer L-space surgery, then the theor...
We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb{R}^3$. Our results give new, computable, and effective obstructions to the existence of such cobordisms.
It has recently been shown by several authors that ribbon concordances, or certain variants thereof, induce an injection on knot homology theories. We prove a variant of this for Heegaard Floer homology and homology cobordisms without 3-handles.
We prove a basic inequality for the d-invariants of a splice of knots in homology spheres. As a result, we are able to prove a new relation on the rank of reduced Floer homology under maps between Seifert fibered homology spheres, improving results of the first and second authors. As a corollary, a degree one map between two aspherical Seifert homo...
In this short note, we prove that if a knot in the Poincare homology sphere is homotopically essential, then it does not admit any purely cosmetic surgeries.
We construct infinitely many smooth 4-manifolds which are homotopy equivalent to $S^2$ but do not admit a spine, i.e., a piecewise-linear embedding of $S^2$ which realizes the homotopy equivalence. This is the remaining case in the existence problem for codimension-2 spines in simply-connected manifolds. The obstruction comes from the Heegaard Floe...
We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen’s connected Seiberg–Witten Floer homology. We use this invariant to study the structure of the homol...
Let $\widehat{\mathcal{C}}_{\mathbb{Z}}$ denote the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms. Answering a question of Matsumoto, the second author previously showed that the natural map from the smooth knot concordance group $\mathcal{C}$ to $\widehat{\mathcal{C}}_{\mathbb{Z}}$ i...
We study lens spaces that are related by distance one Dehn fillings. More precisely, we prove that if the lens space $L(n, 1)$ is obtained by a surgery along a knot in the lens space $L(3,1)$ that is distance one from the meridional slope, then $n$ is in $\{-6, \pm 1, \pm 2, 3, 4, 7\}$. This is proved by studying the behavior of the Heegaard Floer...
In this note, we collect various properties of Seifert homology spheres from the viewpoint of Dehn surgery along a Seifert fiber. We expect that many of these are known to various experts, but include them in one place which we hope to be useful in the study of concordance and homology cobordism.
We correct an error in the statement and proof of Theorem 1.4 of our paper in Acta Mathematica Vietnamica (2014) 39(4), 599-635.
Auckly gave two examples of irreducible integer homology spheres (one toroidal and one hyperbolic) which are not surgery on a knot in the three-sphere. Using Heegaard Floer homology, the authors and Karakurt provided infinitely many small Seifert fibered examples. In this note, we extend those results to give infinitely many hyperbolic examples, as...
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 (for p prime), then Y is a Z/pZ-L-space. Further, we obtain constraints on surgeries on a knot being regular cov...
We show that monopole Floer homology (as defined by Kronheimer and Mrowka) is isomorphic to the S^1-equivariant homology of the Seiberg-Witten Floer spectrum constructed by the second author.
We prove that a positive definite smooth four-manifold with $b_2^+ \geq 2$
and having either no 1-handles or no 3-handles cannot admit a symplectic
structure.
Knot contact homology is an invariant of knots derived from Legendrian
contact homology which has numerous connections to the knot group. We use basic
properties of knot groups to prove that knot contact homology detects every
torus knot. Further, if the knot contact homology of a knot is isomorphic to
that of a cable (respectively composite) knot,...
Quasi-alternating links of determinant 1, 2, 3, and 5 were previously
classified by Greene and Teragaito, who showed that the only such links are
two-bridge. In this paper, we extend this result by showing that all
quasi-alternating links of determinant at most 7 are connected sums of
two-bridge links, which is optimal since there are quasi-alterna...
The cosmetic crossing conjecture (also known as the "nugatory crossing
conjecture") asserts that the only crossing changes that preserve the oriented
isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery
characterization of the unknot to prove this conjecture for knots in integer
homology spheres whose branched double covers...
We give new obstructions to the module structures arising in Heegaard Floer
homology. As a corollary, we characterize the possible modules arising as the
Heegaard Floer homology of an integer homology sphere with one-dimensional
reduced Floer homology. Up to absolute grading shifts, there are only two.
We show that bordered Floer homology provides a categorification of a TQFT
described by Donaldson. This, in turn, leads to a proof that both the Alexander
module of a knot and the Seifert form are completely determined by Heegaard
Floer theory.
We apply results from both contact topology and exceptional surgery theory to
study when Legendrian surgery on a knot yields a reducible manifold. As an
application, we show that a reducible surgery on a non-cabled positive knot of
genus g must have slope 2g-1, leading to a proof of the cabling conjecture for
positive knots of genus 2. Our techniqu...
Using Taubes' periodic ends theorem, Auckly gave examples of toroidal and
hyperbolic irreducible integer homology spheres which are not surgery on a knot
in the three-sphere. We give an obstruction to a homology sphere being surgery
on a knot coming from Heegaard Floer homology. This is used to construct
infinitely many small Seifert fibered exampl...
We study the question of when cyclic branched covers of knots admit taut
foliations, have left-orderable fundamental group, and are not L-spaces.
Let $P(K)$ be a satellite knot where the pattern, $P$, is a Berge-Gabai knot
(i.e., a knot in the solid torus with a non-trivial solid torus Dehn surgery),
and the companion, $K$, is a non-trivial knot in $S^3$. We prove that $P(K)$ is
an L-space knot if and only if $K$ is an L-space knot and $P$ is sufficiently
positively twisted relative to the g...
We establish three rank inequalities for the reduced flavor of Heegaard Floer
homology of Seifert fibered integral homology spheres. Combining these
inequalities with the known classifications of non-zero degree maps between
Seifert fibered spaces, we prove that a map f from Y' to Y between Seifert
homology spheres yields the inequality that |deg f...
In this paper, we use Heegaard Floer homology to study reducible surgeries.
In particular, suppose K is a non-cable knot in the three-sphere with an
L-space surgery. If p-surgery on K is reducible, we show that p divides
2g(K)-1. This implies that any knot with an L-space surgery has at most one
reducible surgery, a fact that we show additionally f...
A rational homology sphere whose Heegaard Floer homology is the same as that
of a lens space is called an L-space. We classify pretzel knots with any number
of tangles which admit L-space surgeries. This rests on Gabai's classification
of fibered pretzel links.
We construct an infinite family of knots in rational homology spheres with
irreducible, non-fibered complements, for which every non-longitudinal filling
is an L-space.
For any three-manifold presented as surgery on a framed link (L,\Lambda) in
an integral homology sphere, Manolescu and Ozsv\'ath construct a hypercube of
chain complexes whose homology calculates the Heegaard Floer homology of
\Lambda-framed surgery on Y. This carries a natural filtration that exists on
any hypercube of chain complexes; we study th...
We show that every irreducible toroidal integer homology sphere graph manifold has a left-orderable fundamental group. This is established by way of a specialization of a result due to Bludov and Glass [Proc. Lond. Math. Soc. 99 (2009) 585–608] for the amalgamated products that arise, and in this setting work of Boyer, Rolfsen and Wiest [Ann. Inst....
We give a complete calculation of the infinity flavor of Heegaard Floer homology with mod 2 coefficients for all three-manifolds and torsion Spin^c structures. The computation agrees with the conjectured calculation of Ozsvath and Szabo. This therefore establishes an isomorphism with Mark's cup homology mod 2. Comment: 29 pages, 3 figures
Ozsvath and Szabo construct a spectral sequence with E_2 term \Lambda^*(H^1(Y;Z))\otimes Z[U,U^{-1}] converging to HF^\infty(Y,s) for a torsion Spin^c structure s. They conjecture that the differentials are completely determined by the integral triple cup product form via a proposed formula. In this paper, we prove that HF^\infty(Y,s) is in fact de...
Citations
... Given a property that L-space knots exhibit, it is natural to ask if almost L-space knots too exhibit that property. For example, L-space knots do not have essential Conway spheres by a result of Lidman-Moore-Zibrowius [LMZ20], so it is natural to ask the following; Question 1.8. Do almost L-space knots have essential Conway spheres? ...
... Ribbon cobordism and concordance.Our techniques extend to the finer relation of ribbon cobordism between branched double covers of alternating links. Recall that a rational homology cobordism W : Y 1 → Y 2 is a ribbon cobordism if it admits a handle decomposition rel Y 1 composed of 1-and 2-handles[4]. A rational homology cobordism W :Y 1 → Y 2 is a quasi-ribbon cobordism if the restriction map H 2 (W ; Z) → H 2 (Y 1 ; Z) surjects[18, Definition 2.1.2]. ...
... We discovered [4] a workaround in that setting, however, by a significantly more complicated argument which involves cabling and our framed instanton contact invariant [2]. We then used that instanton contact class to prove the r ≥ 2g(K )−1 bound, in a manner very similar to the proof of Proposition 15 here (also using results from [3] and [9]). The same difficulties and solutions apply in monopole Floer homology, using our contact invariant from [1]. 1 Figure 1. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/305228774084609-1449783618876_Q64/Juanita-Pinzon-Caicedo.jpg)
... Performing the above type of Seifert fibered surgeries, the items (2) and (4) in Theorem L can be constructed from the items (1) and (3) in Theorem L respectively. We know that the d-invariant remains same under this special Seifert fiber surgery, consult the articles of Lidman and Tweedy [LT18], Karakurt, Lidman, and Tweedy [KLT21], and Seetharaman, Yue, and Zhu [SYZ21] for this result. Relying on the computations in [Twe13] and [KŞ20] again, we have the following result. ...
Reference: A survey of the homology cobordism group
... (6) If Λ admits an augmentation, Λ does not admit a Lagrangian cap, as the augmentation implies the non-acyclicity of the DGA A(Λ) [EES09, Theorem 5.5], and from [DR15, Corollary 1.9] if a Legendrian admits a Lagrangian cap then its DGA A(Λ) (with Z 2 coefficients) is acyclic. There are additional obstructions, obtained through Heegaard Floer Theory, that can be used to obstruct Lagrangian concordances and cobordisms [BSar,GJ19,BLWar]. Some of these will be discussed more in Section 5.3. ...
Reference: Constructions of Lagrangian cobordisms
... (In fact, the larger group of annular braids acts.) More generally, a tangle T in a strip R × [0, 1] with 2n bottom and 2m top points (a (2m, 2n)-tangle) induces a Z[q, q −1 ]-linear map (10) K(T ) : Kau n → Kau m . ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/839313700966402-1577119388848_Q64/Radmila-Sazdanovic.jpg)
... The corresponding invariants in the context of Heegaard Floer homology are known as correction terms [42], and the identity dpY, sq "´2hpY, sq holds under the isomorphism between the theories (see [46], [15], [25] and subsequent papers). These invariants have been applied in recent years to a wide range of problems in three and four dimensional topology, see among the many [43], [44], [26]. Despite this, their computation in specific examples is still a very challenging problem, even under the assumption that Y is an L-space. ...
... The group Θ 3 Z has a Z ∞ summand generated by the family of Brieskorn spheres {Σ(2n + 1, 4n + 1, 4n + 3)} ∞ n=1 . Their proof subsumes several approaches and techniques that consecutively appeared in the literature of involutive Heegaard Floer homology [HMZ18], [DM19], [DS19], and [HHL21]. Moreover, involutive Floer theoretic invariants have provided a major change for the understanding of the structure of Θ 3 Z and its subgroups. ...
Reference: A survey of the homology cobordism group
... A. Levine proved the surprising result that there exist knots in homology spheres which are not smoothly concordant to any knot in S 3 , even if one allows for concordances in homology cobordisms. Since then subsequent works due to Hom-Levine-Lidman and Zhou [HLL18,Zho21] have strengthened this result showing that there are many knots in homology spheres which are not smoothly concordant to knots in S 3 . In this paper we present evidence that the opposite is true topologically. ...
... Many authors have researched various properties of band surgery and related constructions over the past decade. See [40,42,43,45] for more details on the relationship between band surgery and polynomial invariants, [13,39,41,46,77] for the H(2)-Gordian distance, [1,2,6,7,44,62] for calculation of the H(2)-Gordian distance to the unknot, [59,63] for the relationship between band surgery and lens spaces, [14] for fibered links band surgery, [33] for a description of cosmetic surgery, [35,79,80] for details on graphs and complexes associated with band surgery, and [34,64] for applications in biology. ...
Reference: Lernaean knots and band surgery
![[object Object]](https://i1.rgstatic.net/ii/profile.image/272365108068359-1441948310225_Q64/Mariel-Vazquez.jpg)
