# Tye Lidman's research while affiliated with North Carolina State University and other places

## Publications (56)

Article
Full-text available
Preprint
Full-text available
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.
Article
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...
Preprint
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.
Preprint
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.
Preprint
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...
Article
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.
Preprint
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...
Preprint
Full-text available
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.
Preprint
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...
Preprint
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.
Preprint
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.
Preprint
Full-text available
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.
Preprint
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.
Preprint
We prove that L-space knots do not have essential Conway spheres with the technology of peculiar modules, a Floer theoretic invariant for tangles.
Preprint
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...
Preprint
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.
Preprint
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.
Preprint
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...
Preprint
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.
Article
Full-text available
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...
Article
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...
Article
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...
Article
Full-text available
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...
Article
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.
Article
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.
Article
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...
Article
Full-text available
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...
Article
Full-text available
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.
Article
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.
Article
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,...
Article
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...
Article
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...
Article
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.
Article
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.
Article
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...
Article
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...
Article
We study the question of when cyclic branched covers of knots admit taut foliations, have left-orderable fundamental group, and are not L-spaces.
Article
Full-text available
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...
Article
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...
Article
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...
Article
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.
Article
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.
Article
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...
Article
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....
Article
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
Article
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. ...
... 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. ...
... (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. ...
... (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 . ...
... 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. ...
... 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. ...