Figure 11 - uploaded by Christopher M. Herald
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The " " construction and two perturbation curves. The black Hopf link in the center and the arc connecting its components represents the additional data defining the construction relevant to unreduced singular instanton homology. The holonomy perturbation takes place in the neighborhood of the blue circles P 1 and P 2 .
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We introduce explicit holonomy perturbations of the Chern-Simons functional
on a 3-ball containing a pair of unknotted arcs. These perturbations give us a
concrete local method for making the moduli spaces of flat singular SO(3)
connections relevant to Kronheimer and Mrowka's singular instanton knot
homology non-degenerate. The mechanism for this s...
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En esta Tesis Doctoral se desarrollan herramientas geométricas desde el marco teórico de la geometría diferencial y de la teoría de la información para estudiar el caos cuántico y de esta forma dar una visión más completa del mismo. Para ello, en primer lugar estudiamos los fenómenos característicos del caos cuántico y de sus correspondientes escal...
Citations
... (See [33,57,64] for analogous results for closed 3-manifolds.) In a different direction, Artem Kotelskiy, Watson, and Zibrowius showed that, for 4-ended tangles, Bar-Natan's extension of Khovanov homology to tangles can also be interpreted as an immersed curve in a 4-punctured sphere [83], and this immersed curve in fact agrees [84] with an invariant introduced by Hedden, Christopher Herald, Matthew Hogancamp, and Paul Kirk, inspired by instanton link homology [62] (see also [63,82]). ...
Bordered Floer homology is an invariant for 3-manifolds with boundary, defined by the authors in 2008. It extends the Heegaard Floer homology of closed 3-manifolds, defined in earlier work of Zolt\'an Szab\'o and the second author. In addition to its conceptual interest, bordered Floer homology also provides powerful computational tools. This survey outlines the theory, focusing on recent developments and applications.
... Beginning with [HRW16], recent work has interpreted various homology theories in terms of collections of immersed curves on different surfaces. This includes knot and link Floer homology [Zib20,KWZ20], singular instanton knot homology [HHK14] and Khovanov homology [KWZ19]. Although very different in spirit, the classification results of this paper can be restated as another interpretation of link Floer homology, this time in terms of plane curves. ...
We classify isomorphism and chain homotopy equivalence classes of finitely generated $\mathbb{Z} \oplus \mathbb{Z}$ graded free chain complexes over $\frac{\mathbb{F}[U,V]}{(UV)}$. As a corollary, we establish that every link Floer complex $CFL(Y, L)$ over the ring $\frac{\mathbb{F}[U,V]}{(UV)}$ splits as a direct sum of snake complexes and local systems. This generalizes and extends the results of Petkova and Dai, Hom, Stoffregen, and Truong. We give the first example of an essentially infinite knot Floer-like complex, i.e., a complex satisfying all formal properties of link Floer complexes of knots in $S^3$ and whose chain homotopy equivalence class does not admit a representative of the form $C \otimes_{\mathbb{F}} \mathbb{F}[U,V]$. Finally, we also describe the first example of a knot Floer-like complex that does not admit a simultaneously vertically and horizontally simplified basis.
... Along with the abelian arc, χ(S 3 nbd(T p,q )) consists a collection of arcs of conjugacy classes of irreducible representations, mapping to χ(T 2 ) as line segments of slope −pq, with ends limiting to certain points on the abelian arc. The details in the case of T 3,5 are summarized in [HHK14]. For the purposes of this article, we require only the following part of this calculation for T 3,5 , T 2,7 , and T −2,7 . ...
... (see, e.g., [HHK14]). From the subsets of i * X (χ(X)) , h * (i * Y (χ(Y ))) that we have identified and sketched in Figure 3, it is clear that there are two isolated points of intersection. ...
We construct homology 3-spheres for which the (unperturbed) $SU(2)$ Chern-Simons function is not Morse-Bott. In one example, there is a degenerate isolated critical point. In another, a path component of the critical set is not homeomorphic to a manifold. The examples are $+1$ Dehn surgeries on connected sums of torus knots.
... The theta graph can be thought of as analogous to the Hopf link, with an arc connecting its components, employed in defining I ♯ . The computations in Section 5 show that SIK agrees with I ♯ for the empty link and for the unknot in the 3-sphere (see [10,Lemma 8.3]), and we hypothesize that the theories agree in general, as predicted by the Atiyah-Floer conjecture: An Atiyah-Floer-type construction has already been undertaken for the reduced variant I ♮ by Hedden, Herald and Kirk in the form of the their pillowcase homology [6,7]. Their construction decomposes (Y, K) along a Conway sphere, producing an intersection of immersed Lagrangian curves in the pillowcase. ...
There have been a number of constructions of Lagrangian Floer homology invariants for $3$-manifolds defined in terms of symplectic character varieties arising from Heegaard splittings. With the aim of establishing an Atiyah-Floer counterpart of Kronheimer and Mrowka's singular instanton homology, we generalize one of these, due to H. Horton, to produce a Lagrangian Floer invariant of a knot or link $K \subset Y$ in a closed, oriented $3$-manifold, which we call symplectic instanton knot homology ($\mathrm{SIK}$). We use a multi-pointed Heegaard diagram to parametrize the gluing together of a pair of handlebodies with properly embedded, trivial arcs to form $(Y, K)$. This specifies a pair of Lagrangian embeddings in the traceless $\mathrm{SU}(2)$-character variety of a multiply punctured Heegaard surface, and we show that this has a well-defined Lagrangian Floer homology. Portions of the proof of its invariance are special cases of Wehrheim and Woodward's results on the quilted Floer homology associated to compositions of so-called elementary tangles, while others generalize their work to certain non-elementary tangles.
... We describe here a strategy for constructing Khovanov homology for links in arbitrary lens spaces. Our strategy is inspired by a symplectic interpretation of Khovanov homology for links in S 3 due to Hedden, Herald, Hogancamp, and Kirk [15], which originated as an outgrowth of their project to construct pillowcase homology [13,14], a symplectic counterpart to Kronheimer and Mrowka's singular instanton link homology [18,19,20]. The setup for pillowcase homology is as follows. ...
... Given a knot K in S 3 , Kronheimer and Mrowka show in [18] that there is a spectral sequence whose E 2 page is the reduced Khovanov homology of the mirror knot K m and that converges to the singular instanton homology of K. Singular instanton homology is closely related to Lagrangian Floer homology in character varieties of punctured surfaces. For example, generating sets for the singular instanton homology of knots in a 3-manifold Y can be constructed from Lagrangian intersections in R * (S 2 , 4) when Y = S 3 , as described in [13], and in R * (T 2 , 2) when Y is a lens space, as described in [7]. Thus, one might conjecture that there is a spectral sequence from the cohomology of (C 0 , ∂ 0 ) to the singular instanton homology of the corresponding link in S 2 × S 1 that would generalize Kronheimer and Mrowka's spectral sequence for links in S 3 . ...
... The topology of R(S 2 , 4) is discussed in [13]. As a set, the character variety R(S 2 , 4) consists of conjugacy classes SU (2) representations of the fundamental group of S 2 − {p 1 , p 2 , p 3 , p 4 } that map loops around the punctures to traceless matrices. ...
We describe a strategy for constructing reduced Khovanov homology for links in lens spaces by generalizing a symplectic interpretation of reduced Khovanov homology for links in $S^3$ due to Hedden, Herald, Hogancamp, and Kirk. The strategy relies on a partly conjectural description of the Fukaya category of the traceless $SU(2)$ character variety of the 2-torus with two punctures. From a diagram of a 1-tangle in a solid torus, we construct a corresponding object $(X,\delta)$ in the $A_\infty$ category of twisted complexes over this Fukaya category. The homotopy type of $(X,\delta)$ is an isotopy invariant of the tangle diagram. We use $(X,\delta)$ to construct cochain complexes for links in $S^3$ and some links in $S^2 \times S^1$. For links in $S^3$, the cohomology of our cochain complex reproduces reduced Khovanov homology, though the cochain complex itself is not the usual one. For links in $S^2 \times S^1$, we present results that suggest the cohomology of our cochain complex may be a link invariant.
... Since m is a local diffeomorphism away from (π)ޚ 2 [15], and ∂ ∂ x , ∂ ∂ y span T (x,y) ޒ 2 , the cohomology classes [z x ], [z y ] span 4)). ...
... The proof of the following lemma can be found in [15,Lemma 4.2]. The fibers Stab(ρ 0 )\Stab(ρ 0 | H )/Stab(ρ 1 ) are called gluing parameters. ...
... In particular, for torus knots, previously there were only partial computations of from the related spectral sequences (c.f. [KM14,LZ20]; see also [HHK14] for another approach to obtaining upper bounds from generators), while Corollary 1.1.6 applies to torus knots directly since torus knots admit lens spaces surgeries (c.f. ...
Kronheimer-Mrowka conjectured that sutured instanton Floer homology $SHI(M,\gamma)$ has the same dimension as the sutured Floer homology $SFH(M,\gamma)$ constructed by Juh\'{a}sz for any balanced sutured manifold $(M,\gamma)$. Motivated by their conjecture, we introduce new techniques for calculations of sutured instanton Floer homology, some of which are inspired by analogous results in Heegaard Floer theory. The first technique is based on Heegaard diagrams of balanced sutured manifolds, from which we obtain an upper bound on the dimension of $SHI$. For any rationally null-homologous knot $K$ in a closed 3-manifold $Y$, we prove the dimension of the instanton knot homology $KHI(Y,K)$ is greater than or equal to the dimension of the framed instanton homology $I^\sharp(Y)$. We also use this technique to compute the instanton knot homology of $(1,1)$-knots that are also L-space knots. In particular, we calculate the homologies for all torus knots in $S^3$. The second technique is based on the identification of Euler characteristics of $SFH$ and $SHI$, from which we obtain a lower bound on the dimension of $SHI$. We construct a decomposition of $SHI$ analogous to the spin$^c$ structure decomposition of $SFH$, and prove that the enhanced Euler characteristic defined by this decomposition equals to the Euler characteristic of $SFH$. We introduce a family of $(1,1)$-knots called \textbf{constrained knots} and show that the upper bound from the first technique coincides with the lower bound from the second technique. The third technique relates $KHI(S^3,K)$ to $I^\sharp(S^3_n(K))$ by a large surgery formula, where $S^3_n(K)$ is obtained from a knot $K\subset S^3$ by $n$-Dehn surgery. As an application, we show that $S^3_r(K)$ admits an irreducible SU(2) representation for a dense set of slopes $r$ unless $K$ is a prime knot and the coefficients of the Alexander polynomial $\Delta_K(t)$ lie in $\{-1,0,1\}$. In particular, any hyperbolic alternating knot satisfies this property.
... Later, several groups of people studied the representation varieties of some special families of knots to write down an explicit set of generators of the chain complex of the instanton knot homology and thus obtained some better upper bounds than the one coming from Khovanov homology. See [HHK14,DS19,LZ20]. ...
In a recent paper, the first author and his collaborator developed a method to compute an upper bound of the dimension of instanton Floer homology via Heegaard Diagrams of 3-manifolds. For a knot inside S3, we further develop an algorithm that can compute an upper bound of the dimension of instanton knot homology from knot diagrams. We test the effectiveness of the algorithm and found that for all knots up to seven crossings, the algorithm provides sharp bounds. In the second half of the paper, we show that, if the instanton knot Floer homology of a knot has a specified form, then the knot must an instanton L-space knot.
... Sincem is a local diffeomorphism away from (πZ) 2 [HHK14], and ∂ ∂x , ∂ ∂y span T (x,y) R 2 , the cohomology classes [z x ], [z y ] span H 1 (S 2 − 4D 2 ; su(2) adm(x,y) ) = T [m(x,y)] (χ (S 2 , 4)). ...
We prove that the restriction map from the subspace of regular points of the holonomy perturbed SU(2) traceless flat moduli space of a tangle in a 3-manifold to the traceless flat moduli space of its boundary marked surface is a Lagrangian immersion. A key ingredient in our proof is the use of composition in the Weinstein category, combined with the fact that SU(2) holonomy perturbations in a cylinder induce Hamiltonian isotopies. In addition, we show that $(S^2,4)$, the 2-sphere with four marked points, is its own traceless flat SU(2) moduli space.
... Previously, Hedden, Herald, and Kirk [HHK14] studied explicit sets of generators of the instanton chain complexes for some special families of knots, which would also offer upper bounds of dim C KHI. However, their bounds might not always be sharp. ...
In this paper, we introduce a method to extract information about instanton Floer homology of rational homology spheres and knots in them from Heegaard diagrams, based on sutured manifolds and bypass maps. As an application, we prove that for a (1,1)-knot $K$ in a lens space $Y$, we have $\dim_\mathbb{C}KHI(Y,K)\le {\rm rk}_\mathbb{Z} \widehat{HFK}(Y,K)$. If $Y=S^3$ and $K$ is also a Heegaard Floer L-space knot, for example, $K$ is a torus knot, then $\dim_\mathbb{C}KHI(Y,K)= {\rm rk}_\mathbb{Z} \widehat{HFK}(Y,K)$. Also, we prove that all simple knots in lens spaces are instanton Floer simple knots and admit instanton L-space surgeries. Another application is that for a rationally null-homologous knot $K$ in a general 3-manifold $Y$, we prove the dimension inequality $\dim_\mathbb{C} I^\sharp(Y)\le \dim_\mathbb{C}KHI(Y,K)$. In the second part of this paper, for a rationally null-homologous knot $K\subset Y$, we construct a decomposition of $I^\sharp(Y)$ as a candidate for the counterpart of the torsion spin$^c$ decompositions in $\widetilde{HM}(Y)$ and $\widehat{HF}(Y)$. Moreover, we construct a spectral sequence from $KHI(Y,K)$ to $I^\sharp(Y)$ respecting this decomposition. As an application, we derive the relation $\tau_I(\bar{K})=-\tau_I(K)$, where $K\subset S^3$ is a knot, $\bar{K}$ is its mirror, and $\tau_I$ is the instanton tau invariant defined by the first author. Also, we use this decomposition to derive a dimension formula of framed instanton Floer homology for large surgeries on a null-homologous knot in a 3-manifold.
