Null-homologous Knot Research Articles

Instanton Floer homology, sutures, and Euler characteristics

This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler characteristic \chi_{\textup{gr}} of this homology is fully determined by the axioms we proposed. As a result, we conclude that \chi_{\textup{gr}}(\mathrm{SHI}(M,\gamma))=\chi_{\textup{gr}}(\mathrm{SFH}(M,\gamma)) for any balanced sutured manifold (M,\gamma) . In particular, for any link L in S^3 , the Euler characteristic \chi_{\textup{gr}}(\mathrm{KHI}(S^3,L)) recovers the multi-variable Alexander polynomial of L , which generalizes the knot case. Combined with the authors’ earlier work, we provide more examples of (1,1) -knots in lens spaces whose \mathrm{KHI} and \widehat{\mathrm{HFK}} have the same dimension. Moreover, for a rationally null-homologous knot in a closed oriented 3-manifold Y , we construct canonical \mathbb{Z}_2 -gradings on \mathrm{KHI}(Y,K) , the decomposition of I^\sharp(Y) discussed in the previous paper, and the minus version of instanton knot homology \underline{\mathrm{KHI}}^-(Y,K) introduced by Zhenkun Li.

Dehn surgery and nonseparating two-spheres

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.

On Floer minimal knots in sutured manifolds

Suppose ( M , γ ) (M, \gamma ) is a balanced sutured manifold and K K is a rationally null-homologous knot in M M . It is known that the rank of the sutured Floer homology of M ∖ N ( K ) M\backslash N(K) is at least twice the rank of the sutured Floer homology of M M . This paper studies the properties of K K when the equality is achieved for instanton homology. As an application, we show that if L ⊂ S 3 L\subset S^3 is a fixed link and K K is a knot in the complement of L L , then the instanton link Floer homology of L ∪ K L\cup K achieves the minimum rank if and only if K K is the unknot in S 3 ∖ L S^3\backslash L .

Instanton Floer homology, sutures, and Heegaard diagrams

This paper establishes a new technique that enables us to access some fundamental structural properties of instanton Floer homology. As an application, we establish, for the first time, a relation between the instanton Floer homology of a $3$-manifold or a null-homologous knot inside a $3$-manifold and the Heegaard diagram of that $3$-manifold or knot. We further use this relation to compute the instanton knot homology of some families of $(1,1)$-knots, including all torus knots in $S^3$, which were mostly unknown before. As a second application, we also study the relation between the instanton knot homology $KHI(Y,K)$ and the framed instanton Floer homology $I^\sharp(Y)$. In particular, we prove the inequality $\dim_\mathbb{C} I^\sharp(Y)\le \dim_\mathbb{C}KHI(Y,K)$ for all rationally null-homologous knots $K\subset Y$ and we constructed a new decomposition of the framed instanton Floer homology of Dehn surgeries along $K$ that corresponds to the decomposition along torsion spin$^c$ decompositions in monopole and Heegaard Floer theory.

Upsilon Invariants from Cyclic Branched Covers

We extend the construction of Y-type invariants to null-homologous knots in rational homology three-spheres. By considering m-fold cyclic branched covers with m a prime power, this extension provides new knot concordance invariants of knots in S3. We give computations of some of these invariants for alternating knots and reprove independence results in the smooth concordance group.

Knot homologies in monopole and instanton theories via sutures

In this paper we construct possible candidates for the minus versions of monopole and instanton knot Floer homologies. For a null-homologous knot $K\subset Y$ and a base point $p\in K$, we can associate the minus versions, $\underline{\rm KHM}^-(Y,K,p)$ and $\underline{\rm KHI}^-(Y,K,p)$, to the triple $(Y,K,p)$. We prove that a Seifert surface of $K$ induces a $\mathbb{Z}$-grading, and there is an $U$-map on the minus versions, which is of degree $-1$. We also prove other basic properties of them. If $K\subset Y$ is not null-homologous but represents a torsion class, then we can also construct the corresponding minus versions for $(Y,K,p)$. We also proved a surgery-type formula relating the minus versions of a knot $K$ with those of the dual knot, when performing a Dehn surgery of large enough slope along $K$. The techniques developed in this paper can also be applied to compute the sutured monopole and instanton Floer homologies of any sutured solid tori.

Heegaard Floer homology and concordance bounds on the Thurston norm

We prove that twisted correction terms in Heegaard Floer homology provide lower bounds on the Thurston norm of certain cohomology classes determined by the strong concordance class of a 2 2 -component link L L in S 3 S^3 . We then specialise this procedure to knots in S 2 × S 1 S^2 \times S^1 and obtain a lower bound on their geometric winding number. We then provide an infinite family of null-homologous knots with increasing geometric winding number on which the bound is sharp.

Finite type invariants of knots in homology 3–spheres with respect to null LP–surgeries

We study a theory of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries. It is an analogue in the setting of the rational homology of the Goussarov-Rozansky theory for knots in integral homology 3-spheres. We give a partial combinatorial description of the graded space associated with our theory and determine some cases when this description is complete. For null-homologous knots in rational homology 3-spheres with a trivial Alexander polynomial, we show that the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant built from integrals in configuration spaces are universal finite type invariants for this theory; in particular it implies that they are equivalent for such knots.

Toward universality in degree 2 of the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant

In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined, respectively, by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity for these maps would imply that Kricker and Lescop invariants are indeed universal invariants; this would prove in particular that these two invariants are equivalent. In the present paper, we investigate the injectivity status of these maps for degree 2 invariants, in the case of knots whose Blanchfield modules are direct sums of isomorphic Blanchfield modules of [Formula: see text]-dimension two. We prove that they are always injective except in one case, for which we determine explicitly the kernel.

Embedded contact knot homology and a surgery formula

Embedded contact knot homology (ECK) is a variation on Embedded contact homology (ECH), defined with respect to an open book decomposition compatible with a contact structure on some 3-manifold, M. The knot in question is given by the (null-homologous) binding of the open book and the chain complex is defined in terms of closed orbits of the Reeb vector field and certain pseudoholomorphic curves in the symplectization of the knot complement. In this thesis we first generalize this construction to the case of rational open book decompositions, allowing us to define ECK for rationally null-homologous knots. In its most general form this is a bi-filtered chain complex whose homology yields ECH of the closed manifold. There is also a hat version of ECK in this situation which is equipped with an Alexander grading equivalent to that in the Heegaard Floer setting, categorifies the Alexander polynomial, and is conjecturally isomorphic to the hat version of knot Floer homology, HFK. The main result of this thesis is a large negative n-surgery formula for ECK. Namely, we start with an (integral) open book decomposition of a manifold with binding K and compute, for large n, ECK of the knot K(-n) obtained by performing (-n)-surgery on K. This formula agrees with Hedden's large n-surgery formula for HFK, providing supporting evidence towards the conjectured equivalence between the two theories. Along the way, we also prove that ECK is, in many cases, independent of the choices made to define it, namely the almost complex structure on the symplectization and the homotopy type of the contact form. We also prove that, in the case of integral open book decompositions, the hat version of ECK is supported in Alexander gradings less than or equal to twice the genus of the knot.

Distance to longitude

Let [Formula: see text] be a hyperbolic knot in the [Formula: see text]-sphere. If a [Formula: see text]-Dehn surgery on [Formula: see text] produces manifold with an embedded Klein bottle or essential [Formula: see text]-torus, then we prove that [Formula: see text], where [Formula: see text] is the genus of [Formula: see text]. We obtain different upper bounds according to the production of a Klein bottle, a non-separating [Formula: see text]-torus, or an essential and separating [Formula: see text]-torus. The well known examples which are the figure eight knot and the pretzel knot [Formula: see text] reach the given upper bounds. We study this problem considering null-homologous hyperbolic knots in compact, orientable and closed [Formula: see text]-manifolds.

Achiral 1-Cusped Hyperbolic 3-Manifolds Not Coming from Amphicheiral Null-homologous Knot Complements

It is experimentally known that achiral hyperbolic 3-manifolds are quite sporadic at least among those with small volume, while we can find plenty of them as amphicheiral knot complements in the 3-sphere. In this paper, we show that there exist infinitely many achiral 1-cusped hyperbolic 3- manifolds not homeomorphic to any amphicheiral null-homologous knot complement in any closed achiral 3-manifold.

A note on the knot Floer homology of fibered knots

We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include new proofs of Krcatovich's result that knots with $L$-space surgeries are prime and Hedden and Watson's result that the rank of knot Floer homology detects the trefoil among knots in the 3--sphere. We also generalize the latter result, proving a similar theorem for nullhomologous knots in any 3--manifold. We note that our method of proof inspired Baldwin and Sivek's recent proof that Khovanov homology detects the trefoils. As part of this work, we also introduce a numerical refinement of the Ozsv\'ath-Szab\'o contact invariant. This refinement was the inspiration for Hubbard and Saltz's annular refinement of Plamenevskaya's transverse link invariant in Khovanov homology.

A bound for rational Thurston–Bennequin invariants

In this paper, we introduce a rational $$\tau $$ invariant for rationally null-homologous knots in contact 3-manifolds with nontrivial Ozsvath–Szabo contact invariants. Such an invariant is an upper bound for the sum of rational Thurston–Bennequin invariant and the rational rotation number of the Legendrian representatives of the knot. In the special case of Floer simple knots in L-spaces, we can compute the rational $$\tau $$ invariants by correction terms.

Formula omitted]-character varieties and Seifert fibered cosmetic surgeries

We study small Seifert possibly chiral cosmetic surgeries on not necessarily null-homologous knot in rational homology spheres. We define an absolute semi-norm derived from the Culler–Shalen semi-norm of PSL2(C)-character variety theory. We prove that two cosmetic slopes must have the same absolute semi-norm and use this result to give a sharp bound on the number of slopes producing the same small Seifert manifold if the ambient manifold satisfies some representation theoretic conditions.

Heegaard Floer homology and knots determined by their complements

In this paper we investigate the question of when different surgeries on a knot can produce identical manifolds. We show that given a knot in a homology sphere, unless the knot is quite special, there is a bound on the number of slopes that can produce a fixed manifold that depends only on this fixed manifold and the homology sphere the knot is in. By finding a different bound on the number of slopes, we show that non-null-homologous knots in certain homology $\mathbb{R}P^3$'s are determined by their complements. We also prove the surgery characterisation of the unknot for null-homologous knots in $L$-spaces. This leads to showing that all knots in some lens spaces are determined by their complements. Finally, we establish that knots of genus greater than $1$ in the Brieskorn sphere $\Sigma(2,3,7)$ are also determined by their complements.

Equivariant triple intersections

Given a null-homologous knot $K$ in a rational homology 3-sphere $M$, and the standard infinite cyclic covering $\tilde{X}$ of $(M,K)$, we define an invariant of triples of curves in $\tilde{X}$, by means of equivariant triple intersections of surfaces. We prove that this invariant provides a map $\phi$ on $\Al^{\otimes 3}$, where $\Al$ is the Alexander module of $(M,K)$, and that the isomorphism class of $\phi$ is an invariant of the pair $(M,K)$. For a fixed Blanchfield module $(\Al,\bl)$, we consider pairs $(M,K)$ whose Blanchfield modules are isomorphic to $(\Al,\bl)$, equipped with a marking, {\em i.e.} a fixed isomorphism from $(\Al,\bl)$ to the Blanchfield module of $(M,K)$. In this setting, we compute the variation of $\phi$ under null borromean surgeries, and we describe the set of all maps $\phi$. Finally, we prove that the map $\phi$ is a finite type invariant of degree 1 of marked pairs $(M,K)$ with respect to null Lagrangian-preserving surgeries, and we determine the space of all degree 1 invariants of marked pairs $(M,K)$ with rational values.

Turaev torsion invariants of 3-orbifolds

We construct a combinatorial invariant of 3-orbifolds with singular set a link that generalizes the Turaev torsion invariant of 3-manifolds. We give several gluing formulas from which we derive two consequences. The first is an understanding of how the components of the invariant change when we remove a curve from the singular set. The second is a formula relating the invariant of the 3-orbifold to the Turaev torsion invariant of the underlying 3-manifold in the case when the singular set is a nullhomologous knot.

Computing the Thurston–Bennequin invariant in open books

We give explicit formulas and algorithms for the computation of the Thurston-Bennequin invariant of a nullhomologous Legendrian knot on a page of a contact open book and on contact Heegaard surfaces. Furthermore, we extend the results to rationally nullhomologous knots in arbitrary 3-manifolds.

Classical invariants of Legendrian knots in the 3-dimensional torus

All knots in R3 possess Seifert surfaces, and so the classical Thurston–Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on R3 can be defined. The definitions extend easily to null-homologous knots in any 3-manifold M endowed with a contact structure ξ. We generalize the definition of Seifert surfaces and use them to define these invariants for all Legendrian knots, including those that are not null-homologous, in a contact structure on the 3-torus T3. We show how to compute the Thurston–Bennequin and rotation invariants in a tight oriented contact structure on T3 using projections.