Photo taken by Xiao Zhi
Eureka!
Zhenkun Li
Assistant Professor
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- Institute
- Department of Mathematics and Statistics, University of South Florida
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- Address
- 4202 E. Fowler Avenue, Tampa, FL 33620, USA
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- Office
- CMC 324
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- (my first name)@usf.edu
Photo taken by Xiao Zhi
Instanton Floer homology was first introduced by Floer in [1], as the origin of Floer theory. Floer theory can be thought of as an infinite dimensional Morse theory, and for instanton theory, the critial points are the flat connections on 3-manifolds, which corresponds to $SU(2)$-representations of the fundamental groups of the 3-manifolds. This key feature of instanton theory has led to many important applications. For example, Kronheimer and Mrowka used instanton theory to approved the Property P conjecture in [2]. The followings are my work done in instanton theory:
Though instanton theory was the first to be introduced, there are a lot of fundamental questions in instanton theory not understood. Some of my work is focused on answering those questions in instanton theory.
Sutured instanton Floer homology was introduced by Kronheimer and Mrowka in [3] for balanced sutured manifolds. A balanced sutured manifold is a pair $(M,\gamma)$ where $M$ is a compact oriented 3-manifold with boundary and $\gamma$ is a closed oriented 1-submanifold on $\partial M$ so that $\partial M\backslash\gamma$ can be divided into two parts of equal Euler characteristics. Sutured instanton Floer homology, denoted by $SHI(M,\gamma)$, has many important applications, such as an alternative proof of Property P conjecture (See [3]) and the establishment of the fact that Khovanov homology detects the unknots (See [4]). Instanton theory fits into a (3+1)-TQFT setup, i.e., roughly speaking, it associates a closed oriented 3-manifold a vector space and associates a cobordism between 3-manifolds a linear map between the vector spaces associated to the boundary 3-manifolds. Roughly speaking, instanton Floer homology can be viewed as a functor from the 3-dimensional cobordism category to the category of complex vector spaces. However, previously it is not known whether sutured instanton Floer homology shares similar features.
In [5], Juhász introduced a notion for cobordisms between sutured manfolds, which is called the sutured cobordism. In [6], I constructed maps between sutured instanton Floer homologies associated to sutured cobordisms, and proved basic functorialities of such cobordism maps. Thus it makes sutured instanton Floer homology a functor between suitably chosen categories.
If $(M,\gamma)$ is a balanced sutured manifold and $S\subset (M,\gamma)$ is a properly embedded surface, then it is natural to ask if $SHI(M,\gamma)$ can decompose according to $S$. In [7], Kronheimer and Mrowka proposed a way to construct such a decomposition, or equivalently, a $\mathbb{Z}$-grading on $SHI(M,\gamma)$ associated to $S$, but they didn't prove that the construction is indepedent of various auxiliary choices. In [8], Baldwin and Sivek proved the independence in a special case. Finally, in [9], I proved that the construction of grading is well-defined in general. Note this grading does not only depend on the isotopy class of the surface $S\subset M$, but also depends on the ways $S$ intersects the suture $\gamma$. For different isotopies of the surface, I also determine the grading shifting between the gradings associated to them. This grading shifting turns out to be important to figure out the grading shifts for bypass maps defined by Baldwin and Sivek in [10], which leads to many interesting applications. The grading would be an essential ingredient for most of my work presented below.
For a closed oriented 3-manifold $Y$, its monopole Floer homology and Heegaard Floer homology both natural decompose along spin${}^c$ structures on $Y$. The set of spin${}^c$ structures on $Y$ is an affine space over $H^2(Y;\mathbb{Z})$. On the other hand, there is no such natural decompositions in instanton theory. Instead, if $\alpha\in H_2(Y;\mathbb{Z})$ is a homology class, then there is an action $\mu(\alpha)$ on $I^{\sharp}(Y)$, the framed instanton Floer homology defined by Kronheimer and Mrowka in [11], and actions associates to any two homology classes commute with each other. By looking at the simultaneous eigenspaces, we can obtained a decomposition of $I^{\sharp}(Y)$ along $(H_2(Y;\mathbb{Z}))^*$ which is isomorphic to $H^2(Y;\mathbb{Z})$ module torsion. The missing of the torsion part would be crucial in some cases. For example, if $Y$ is obtained form a knot $K\subset S^3$ by a non-zero surgery, then $b_1(Y)=0$, which means that $H_2(Y;\mathbb{Z})=0$, and hence previously there is no effective way to decompose $I^{\sharp}(Y)$. Similar questions arise in sutured instanton homology as well. For a balanced sutured manifold $(M,\gamma)$, the sutured (Heegaard) Floer homology, introduced by Juhász in [12], decomposes along relative spin${}^c$ structures on $(M,\gamma)$, the set of which is an affine space over $H^2(M,\partial M;\mathbb{Z})$. On the other hand, using the grading associated to surfaces on $SHI(M,\gamma)$, one can obtain a decomposition of $SHI(M,\gamma)$ along $(H_2(M,\partial M;\mathbb{Z}))^*$, which again misses the torsion part of $H^2(M,\partial M;\mathbb{Z})$. To resolve this problem, in [13] and [14], my collaborator and I constructed decompositions of $I^{\sharp}(Y)$ and $SHI(M,\gamma)$, which can distinguish torsion parts of $H^2(Y;\mathbb{Z})$ and $H^2(M,\partial M;\mathbb{Z})$. The torsion contains some subtle informations which can help us to compute the instanton knot homology of some families of (1,1)-knots, detect unknot among null-homologous knot inside instanton L-space knots, and contribute to the proof that 3-surgery on a non-trivial knot always admits an irreducible representation.
Instanton Floer homology admits a canonical $\mathbb{Z}_2$ grading, so it is natural to ask what is the Euler characteristics of the instanton Floer homology. For a closed 3-manifold $Y$, the Euler characteristic of $I^{\sharp}(Y)$ has been understood by Scaduto in [15]. However, his proof was built on surgery exact triangles and the maps in the triangle might not commute with the $\mu$-action associated to surfaces inside the manifold, when the surgery curve intersects the surface. As a result, Scaduto's proof cannot be generalized to sutured instanton Floer homology and its graded parts. Previously, only a special case where the manifold $M$ is the complement of a knot $K\subset S^3$ and the suture $\gamma$ consists of two meridians of $K$ is understood. In this case, $SHI(M,\gamma)$ is defined to be the instanton knot homology of $K\subset S^3$ by Kronheimer and Mrowka [3], and is also denoted by $KHI(K)$. The Seifert surface of $K$ induces a $\mathbb{Z}$-grading on $KHI(K)$ which is called the Alexander grading. Kronheimer and Mrowka [7] proved in this case that the graded Euler characteristics of $KHI(K)$ corresponds to the coefficients of the symmetrized Alexander polynomial of $K$. Similar results holds true for links with a single Alexander $\mathbb{Z}$-grading. However, it is previously not known for the case of knots inside other 3-manifolds or links inside $S^3$ with multi Alexander grading. In [16] and [14], my collaborator and I established a more general result that the graded Euler characteristcs of sutured instanton Floer homology and sutured (Heegaard) Floer homology coincide up to an overall grading shift and up to a sign. This result helps us to compute the instanton knot homology of some families of (1,1)-knots, detect unknot among null-homologous knot inside instanton L-space knots, and contribute to the proof that 3-surgery on a non-trivial knot always admits an irreducible representation. It is also worth mentioning that Xie and Zhange [17] also used this result to study special types of $SU(2)$ representations for link complements.
Besides instanton Floer homology, there are other types of Floer theory. Two major brances are monopole theory introduced by Kronheimer and Mrowka [18] and Heegaard Floer theory by Ozsváth and Szabó [19]. Monopole theory and Heegaard Floer thoery are known to be isomorphic by work of Colin, Ghiggini, and Honda [20] and Taubes [21] (and subsequent papers), or by Kutluhan, Lee, and Taubes [22] (and subsequent papers). However, the relation between instanton theory and other Floer theories remains illusive. Kronheimer and Mrowka [3] conjectured that for a balanced sutured manifold $(M,\gamma)$ we always have $$\dim_{\mathbb{C}} SHI(M,\gamma)={\rm rk}_{\mathbb{Z}} SFH(M,\gamma),$$ where $SFH$ denotes the sutured (Heegaard) Floer homology defined by Juhász in [12]. However, previously there is no evidence beyond some computational examples. The identification of the graded Euler characteristics between $SHI$ and $SFH$ may provide more evidence of this conjecture.
Different brances of Floer theory has its own merits. For example, in order to prove the fact that 3-surgery on a non-trivial knot always admits an irreducible representation, we need to start with instanton theory, which has closed connection with the representation varieties of the fundamental groups of knots and 3-manifolds, and we will end up with Heegaard Floer homology, whose Euler characteristics is known to be related to Taurev-torsion-type invariant of the 3-manifold, by work of Friedl, Juhász, and Rasmussen [23] and a bridge to connected instanton side and Heegaard Floer side by work of my collaborators and I in [16] and [14].
Heegaard diagram is a useful tool to describe 3-manifolds and knots. It is well known that any (knot inside) 3-manifold is described and determined by its Heegaard diagrams. The Heegaard Floer theory constructed by Ozsváth and Szabó [19] and the sutured Floer homology constructed by Juhász [12] are both built on Heegaard diagrams. Intrinsically, Heegaard diagrams determine 3-manifolds and knots, and hence determine instanton Floer homology as well. However, for a long time, an explicit relation between instanton theory and Heegaard diagrams remains illusive. Motivated by this question, and motivated by the conjectural isomorphism between $SHI$ and $SFH$, my collaborator and I in [13] introduce a way to extract information about instanton theory from the Heegaard diagrams of 3-manifolds and knots. This idea was enhanced in [24], where we show that if $(M,\gamma)$ is a balanced sutured manifold and $(\Sigma,\alpha,\beta)$ is an admissible Heegaard diagram of it, then we can construct a new balanced sutured manifold $(H,\Gamma)$ so that we have the following relation: $$\dim_{\mathbb{C}}SHI(M,\gamma)\leq \dim_{\mathbb{C}}SHI(H,\Gamma)= {\rm rk}_{\mathbb{Z}}SFC(\Sigma,\alpha,\beta).$$ Here $SFC(\Sigma,\alpha,\beta)$ is the chain complex constructed out of the Heegaard diagram $(\Sigma,\alpha,\beta)$ whose homology computes $SFH(M,\gamma)$. Also, the manifold $H$ here is a handle body. This relation can help us to compute the instanton knot homology of some families of (1,1)-knots and all strong Heegaard Floer L-spaces.
It is worth mentioning that the relation between $SHI(M,\gamma)$ and $SHI(H,\Gamma)$ is not merely a dimension inequality. A simplified model of this pair of sutured manifold is when we look at a knot $K$ inside a closed oriented 3-manifold $Y$. Let $Y(1)$ be the sutured manifold obtained from $Y$ by removing a $3$-ball and choosing a connected simple closed curve on the boundary as the suture. We can find a properly embedded arc $T$ inside $Y(1)$ so that the two end points $\partial T\subset\partial Y(1)$ are separated by the suture and that when removing a neighborhood of $T$ from $Y(1)$, we obtain the knot complement $Y-N(K)$. Write $Y(K)$ the sutured manifold whose underline manifold is the knot complement $Y-N(K)$ and whose sure consists of two meridians of $K$, then $Y(K)$ can also be thought of as obtained from $Y(1)$ by removing a neighborhood of the tangle $T$ and adding a meridian of $T$ to the suture. In Heegaard Floer theory, the knot $K$ leads to a filtration on the chain complex $SFC(Y(1))$ that computes $SFH(Y(1))$. This filtration leads to a spectral sequence whose $E_1$-page is $SFH(Y(K))$ and which converges to $SFH(Y(1))$. One could seek for similar relations in instanton theory. According to [3], the framed instanton Floer homology of $Y$ is defined to be $I^{\sharp}(Y)=SHI(Y(1))$ and the instanton knot homology of $K\subset Y$ is defined to be $KHI(Y,K)=SFH(Y(K))$. However, in instanton theory, it is hardly possible to study this problem at chain levels, so in [13] my collaborator and I constructed differentials on $KHI(Y,K)$, making it into a chain complex whose homology is isomorphic to $I^{\sharp}(Y)$.
In general, if $(M,\gamma)$ is a balanced sutured manifold and $(H,\Gamma)$ is obtained from an admissible Heegaard diagram of $(M,\gamma)$, then there is a set of tangles $T$ so that if we remove a neighborhood of $T$ from $M$ and add one meridian of each component of $T$ to the suture, then we obtain the manifold $(H,\Gamma)$. One can expect, as the above special case where we deal with only a connected tangle, there should be a differential on $SHI(H,\Gamma)$ whose homology computes $SHI(M,\gamma)$. In an on-going project, my collaborators and I are working on constructing the differential and compare it with the one in the Heegaard Floer chain complex.
As mentioned above, in [16] and [14], my collaborator and I identified the graded Euler characteristics in sutured instanton theory and sutured (Heegaard) Floer theory. The proof was to setup a set of axioms, and then prove that any Floer theory satisfying the set of axioms must have the same Euler characteristic. This idea was motivatied by the isomorphism between monopole homology and Heegaard Floer homology, and the conjectural isomorphism between sutured instanton Floer homology and sutured (Heegaard) Floer homology. The ultimate goal in this direction is to find a set of axioms that not only determines the Euler characteristics, but also determines the whole homology group. It is worth mentioning that the construction of $(H,\Gamma)$ from $(M,\gamma)$ is topological, and the Floer homology of $(H,\Gamma)$ is independent of theories. So our goal is to find a set of axioms that also determines the differential on the Floer homology of $(H,\Gamma)$ that lead to the Floer homology of $(M,\gamma)$.
Roughly speaking, the instanton Floer homology of a 3-manifold $Y$ is build on sets of partial differentail equations on some suitably chosen bundles over $Y$ and $\mathbb{R}\times Y$. In general, it is impossible to solve the equations and compute the instanton Floer homology directly. Using various method, my collaborators and I were able to compute the instanton Floer homology of the following families of 3-manifolds and knots.
Non-trivial sutures on a solid torus $M$ are parametrized by the number of components and slope. If a suture $\gamma$ has slope $p/q$ and has $2n$ components, then in [9], I showed that $$\dim_{\mathbb{C}}SHI(M,\gamma)=2^{n-1}\cdot |p|.$$ This result relies on the grading in sutured instanton Floer homolog and how bypass maps shifts the gradings.
The idea above to compute the dimension of the sutured instanton Floer homology of a sutured solid torus can be generalized to deal with some sutured handle bodies as well. For a general sutured handle body $(M,\gamma)$ in [25], my collaborator and I proposed a combinatorial way to compute an upper bound for the dimension of $SHI(M,\gamma)$. This bound is sometimes sharp as it coincides with the Euler characteristic of the sutured Floer homology, which serves as a lower bound for $SHI(M,\gamma)$ as well.
(1,1)-knots are a family of knots that have good Heegaard diagrams. They are the analogue of 2-bridge knots in lens spaces. For any (1,1)-knot $K$ in a lens space $Y$, there is a special genus-one Heegaard diagram $(\Sigma,\alpha,\beta)$ so that the corresponding chain complex $SFC(\Sigma,\alpha,\beta)$ has zero differential, i.e., $${\rm rk}_{\mathbb{Z}_2}SFH(Y(K))={\rm rk}_{\mathbb{Z}_2}SFC(\Sigma,\alpha,\beta).$$ Together with the result that relates instanton theory with Heegaard diagrams as mentioned above, we obtain that for any (1,1)-knot $K\subset Y$, there is an inequality $$\dim_{\mathbb{C}}KHI(Y,K)\leq\dim_{\mathbb{C}}SHI(Y(K))={\rm rk}_{\mathbb{Z_2}}SFH(Y(K))={\rm rk}_{\mathbb{Z_2}}\widehat{HFK}(Y,K).$$ Here $\widehat{HFK}$ is the hat version of knot Floer homology defined by Ozsváth and Szabó in [26]. As in [12], $SFH(Y(K))$ is by construction equivalent to $\widehat{HFK}(Y,K)$.
For some special families of (1,1)-knots, the above inequality maybe upgraded to an equality. For example, since we have identified the graded Euler characteristics of $SHI$ and $SFH$, if the graded Euler characteristic of $SFH(Y(K))$ has already been equal to the rank of it, then the above inequality becomes equality. There are a few families of (1,1)-knots that satisfies this condition: thin knots (including all alternating), and knots that admits (Heegaard Floer) L-space surgeries (including all torus knots inside $S^3$).
Similar idea applies to another family of 3-manifolds, namely Strong Heegaard Floer L-space. A rational homology sphere $Y$ is called a strong Heegaard Floer L-space if there is a Heegaard diagram $(\Sigma,\alpha,\beta)$ of $Y$ so that the following equalitys hold: $${\rm rk}_{\mathbb{Z}_2}\widehat{CF}(\Sigma,\alpha,\beta)={\rm rk}_{\mathbb{Z}_2}\widehat{HF}(Y)=|H_1(Y;\mathbb{Z})|.$$ Here $\widehat{HF}$ is the hat version of Heegaard Floer homology introduced by Ozsváth and Szabó in [19], and $\widehat{CF}$ is the chain complex of the homology constructed via a Heegaard diagram. For a strong Heegaard Floer L-space $Y$, my collaborators and I proved in [24] that $${\rm dim}_{\mathbb{C}}I^{\sharp}(Y)={\rm rk}_{\mathbb{Z}_2}\widehat{HF}(Y).$$
As an application of the large surgery formula in instanton theory, in [27], my collaborator and I show that if $K\subset S^3$ is a genus-one alternating knot and $r\in\mathbb{Q}\backslash\{0\}$, we have $${\rm dim}_{\mathbb{C}}I^{\sharp}(S^3_r(K))={\rm rk}_{\mathbb{Z}_2}\widehat{HF}(S^3_r(K)).$$
Suppose $K\subset S^3$ is a knot. The minus version of the knot, $KHI^{-}(S^3,K)$ can be viewed as a $\mathbb{C}[U]$-module. We define the torsion order of $K$ to be $${\rm ord}_U(K)=\max\{k~|~\exists~x\in KHI^{-}(S^3,K)~{\rm s.t.}~U^{k-1}x\neq0~{\rm and}~U^{k}x=0\}.$$ In [60], my collaborator and I proved that if $K\subset S^3$ is a knot having torsion order one, then the bent complex is determined by $KHI(S^3,K)$ and $\tau_I(K)$. Here $\tau_I(K)$ is the instanton tau invariant of $K$. In [60], my collaborator and I proved that some families of Pretzel type alternating knots has torsion order one. In [4], Kronheimer and Mrowka showed that the instanton knot homology of an alternating knot $K\subset S^3$ is determined by its Alexander polynomial. Also, in [31], Baldwin and Sivek showed that for alternating knots, $\tau_I(K)=-\frac{1}{2}\sigma(K)$. Here $\sigma(K)$ is the signature of the knot. As a result, we are able to compute the instanton Floer homology of this family of knots via the integral surgery formula.
Twisted Whitehead double is a special family of genus-one knots. For genus-one knots, we can use techniques from our large surgery formula to compute the framed instanton Floer homology of Dehn surgeries. Two ingredients are required: $KHI(S^3,K,1)$ and $\tau_I(K)$. The former one can be understood via sutured manifold decompositions, and the second one can be understood using oriented skein relation and an inductive argument. For more details, see [61].
The study of twisted Whitehead double can also help us to understand some splicings. Suppose $K\subset S^3$ is a twist knot, or equivalently, the twisted Whitehead double of the unknot, and $J\subset S^3$ be arbitrary. Then the splicing of the complement of $K$ and $J$ can be understood as the Dehn surgery along the twisted Whitehead double of $J$. As a result, the instanton Floer homology of the splicing can be understood.
Take the $3$-component Borromean link. Perform $0$-surgeries along two of the three components. The rest component becomes a knot $K\subset Y=(S^1\times S^2)\#(S^1\times S^2)$. We can also form the connected sums of $K$: $K^g\subset Y^g=\#^{2g}(S^1\times S^2)$. In this special case, we can compute $KHI(Y^g,K^g)\cong I^{\sharp}(Y^g)$ so all the differentials in the bent complexes are trivial. In the integral surgery formula, we also need to understand the map $\sigma_*$. In the case of $Y^g=\#^{2g}(S^1\times S^2)$, as is explained later, $\sigma_*=id$. As a result, we could apply the integral surgery formula to compute all non-zero integral surgeries of $K^g$. Note these surgeries give rise to non-trivial circle bundles over a connected closed oriented surface of genus-$g$. Also, we can form the connected sum of $K^g$ with a seqeunce of core knots in lens spaces. The integral surgery will give rise to a general Seifert fibred surfaces. For more details, see [61].
My work mainly focuses on Floer-theoritic invariants, Khovanov homology, and their applications to understand properties of knots, representation varieties, Dehn surgeries, and knot detections. Besides, I also studied some classical knot invariants that come from a Morse function on $S^3$.
The knot Floer homology introduced by Ozsváth and Szabó has a few different flavors, namely infinity, plus, minus, and hat. Among them, the hat version, $\widehat{HF}$ is conjecturally isomorphic to the instanton knot homology $KHI$. However, previously there is no construction of other flavors in instanton theory. Inspired by work of Etnyre, Vela-Vick, and Zarev in [28], in [9] I defined a minus version of instanton knot homology for knots $K\subset Y$, which is denoted by $KHI^-(Y,K)$. The minus version is equipped with a $U$-action, and is closedly related to the large surgery formula in instanton theory. It is worth mentioning that a veriation of $KHI^-$ is an essential ingredient in Wang's proof that split link satisfies the cosmetic crossing conjecture in [29].
In [30], my collaborators and I proved that for a knot $K\subset S^3$, the minus version $KHI^-(S^3,K)$ always has a unique free $U$ tower. So minus the top Alexander grading of any homogeneous element in the free $U$-tower is a knot invariant and is denoted by $\tau_I$. In that paper we showed that $\tau_I$ is a concordance homomorphism to $\mathbb{Z}$, and computed $\tau_I$ for all twisted knots.
On the other hand, in [31], Baldwin and Sivek studied the Dehn surgeries along knots in $S^3$, and defined a concordance invariant $\nu^{\sharp}$ from some surgery slopes that have special properties. The also define a new invariant $\tau^{\sharp}$ as the homogenization of $\nu^{\sharp}$. They showed that $\tau^{\sharp}$ is a concordance homomorphism to $\mathbb{R}$.
In [30], my collaborators and I also identified the $\tau_I$ invariant I defined with the $\tau^{\sharp}$ invariant Baldwin and Sivek defined. A direct corollary is that, $\tau^{\sharp}$ must take integral values as opposed to potentially taking values in $\mathbb{R}$. Also, since $\tau_I$ and $\tau^{\sharp}$ comes from different aspects, the identification of them might be useful. For example, in [31] Baldwin and Sivek proved that $\tau^{\sharp}$ of an alternating knot is determined by its knot signature. On the other hand, $\tau_I$ is closed related to $KHI^-$ and hence is closedly related to the large surgery formula in instanton theory. So combining all these work may help us a lot to compute the instanton Floer homology of Dehn surgery along any alternating knots.
In Heegaard Floer theory, a null-homologous knot $K\subset Y$ gives rise to a doubly-pointed Heegaard diagram. Each base point leads to a filtration on the chain complex $\widehat{CFK}(Y,K)$ of the knot, and further leads to a spectral sequence whose first page is $\widehat{HFK}(Y,K)$ and which converges to $\widehat{HF}(Y)$. It is natural to ask whether a similar thing happens in instanton theory. However, the construction of the instanton knot homology is based on sutured instanton Floer homology and the sutured instanton Floer homology is constructed via the closures. Different closures lead to isomorphic homology groups, but it is unclear how to effectively describe the relation between the instanton chain complex of different closures. As a result, it is hard to construct a filtration on the chain level in instanton theory. In [27], my collaborator and I constructed differentials on $KHI(Y,K)$ whose homology computes $I^{\sharp}(Y)$. In particular, we constructed differentials $$d^{i}_{j}:KHI(Y,K,i)\to KHI(Y,K,j)$$ for any $i\neq j$, and define the bent complexes $(A_s,d_s)$ where $$A_s=KHI(Y,K),~{\rm and~}d_s=\sum_{j>i \geq s}d^{i}_j+\sum_{j < i\leq s}d^i_j.$$ In particular, since $KHI(Y,K,i)$ is only non-trivial for $|i|\leq g(K)$, we know that $$(A_s,d_s)=(A_{s+1},d_{s+1})~{\rm if}~s\geq g(K)~{\rm and~}(A_s,d_s)=(A_{s-1},d_{s-1})~{\rm if}~s\leq -g(K).$$ As a result, we can define $$(B^{\pm},d^{\pm})=(A_s,d_s)~{\rm for~}\mp s\geq g(K).$$ In [27], we proved the following. $$H_*(B^{\pm},d^{\pm})\cong I^{\sharp}(Y).$$
In [26], Ozsváth and Szabó proved a large surgery formula that relates the Heegaard Floer homology of large integral Dehn surgeries of a knot with the knot Floer homology of the knot. This formula is powerful in dealing with problems related to Dehn surgeries and is the foundation of the more general integral (and rational) surgery formula. In [27], my collaborator and I developed a similar large surgery formula in instanton theory: For a null-homologous knot $K\subset Y$ and an integer $n\geq 2g(K)+1$, we have $$I^{\sharp}(Y_n(K))\cong\bigoplus_{s=\lfloor -\frac{n-1}{2}\rfloor}^{\lfloor\frac{n-1}{2}\rfloor}H_*(A_s,d_s).$$ As for applications, on the compuational side, the formula is practical, as we used it to computed the instanton Floer homology of all genus-one-alternating knots. With our knowledge of tau invariants it may also helps to compute the Dehn surgery of any alternating knots.
On the theoritical side, the advantage of instanton theory is that it is closedly related to the representation variety of 3-manifolds and knots. So information we obtain for the instanton Floer homology of Dehn surgeries along knots can tell us something about the representation variety of these manifolds. For example, in [32], Baldwin and Sivek proved that, under some conditions, if $Y$ is the $n$-surgery along a knot $K\subset S^3$ and there is no irreducible $SU(2)$-representations of $\pi_1(Y)$, then $Y$ is an instanton L-space, i.e., $$\dim_{\mathbb{C}}I^{\sharp}(Y)=|H_1(Y)|=|n|.$$ Using the large surgery, we can impose a strong restriction on the knot $K$, for example, we have $$\dim_{\mathbb{C}}KHI(S^3,K,i)\leq 1$$ for any (Alexander) grading $i\in\mathbb{Z}$. Note by work of Kronheimer and Mrowka [7], coeffients of the symmetrized Alexander polynomial of the knot $K\subset S^3$ equals minus the graded Euler characteristics of $KHI(S^3,K)$. Hence, by examining the Alexander polynomial we know that the fundamental groups of Dehn surgeries of many families of knots must admit irreducible $SU(2)$-representations.
It is worth mentioning that whether small surgeries have irreducible representations is of particular interests. The property P conjecture is a corollary of the fact that for any non-trivial knot $K\subset S^3$, $\pi_1(S^3_1(K))$ admits an irreducible $SU(2)$-representation. In [33], Kronheimer and Mrowka showed that if $K\subset S^3$ is a non-trivial knot and $r\in\mathbb{Q}$ satisfies $|r|\leq 2$, then $\pi_1(S^3_r(K))$ always admits an irreducible $SU(2)$-representation. Since the $5$-surgery of a right-handed trefoil is a lens space, whose fundamental group does not admits any irreducible $SU(2)$-representations, they also asked whether $\pi_1(S^3_r(K))$ always admits an irreducible $SU(2)$-representation for $r=3$ and $4$.
In [32], Baldwin and Sivek answered the question affirmatively for the case $r=4$. In [34], my collaborators and I proved that for any non-trivial knot $K\subset S^3$, $\pi_1(S^3_r(K))$ indeed admits an irreducible $SU(2)$-representation for $r=3$ and many other rational slopes in the interval $(3,5)$.
The argument can also be used to study almost L-space knots. A knot $K\subset S^3$ is called an almost L-space knot if it is not an L-space knot but there exists an integer $n$ so that ${\rm dim}_{\mathbb{C}}I^{\sharp}(S^3_n(K))=|n|+2$. In [61], my collaborator and I showed that a genus-one almost L-space knot must be either $4_1$ or $5_2$ in the Rolfsen's knot table, and almost L-space knot of genus at least two must be fibred and strongly quasi-positive (i.e., the fibration of the knot complement gives rise to an open book decomposition of the unique tight contact structure on $S^3$.) This result also helps us to classify all knots $K\subset S^3$ so that $\dim_{\mathbb{C}}I^{\sharp}(K)=3$.
Another application to study the large surgery formula is to study knots of genus one. In this case, the surgery slope need not to be really 'large' to satisfy the condition for us to apply large surgery formula. In particular, we proved that $I^{\sharp}(S^3_{n}(K))$ for a genus-one knot $K\subset S^3$ and any non-trivial integer $n$ is determined by $KHI(S^3,K,1)$ and $\tau_I(K)$.
In the large surgery formula, we need the requirement that $n\geq 2g(K)+1$ in order to apply the formula. It is natural to ask if we could also have a formula for smaller $n$. In [60], we prove an integral surgery formula as follows. Recall we have constructed the bent complexes $(A_s,d_s)$. Write $(B^{\pm}_s,d^{\pm}_s)$ the identical copies of $(B^{\pm},d^{\pm})$. For each $s$, define two maps $$\pi^{\pm}_s:A_s\to B^{\pm}_s,~\pi^{\pm}_s(x)=\begin{cases} x & x\in KHI(S^3,K,i)~{\rm for~}\pm i\geq s\\ 0 & x\in KHI(S^3,K,i)~{\rm for~}\pm i < s \end{cases}.$$ Define a map $\tau^{\pm}_n:B^{\pm}_s\to B^{\pm}_{s+n}$ to be the identity map. we proved that there exists a chain homotopy equivalence $\sigma: (B^+,d^+)\to (B^-,d^-)$ which leads to $\sigma_s: (B^+_s,d^+_s)\to (B^-_s,d^-_s)$ so that $$I^{\sharp}(Y_n(K))\cong H_*\bigg({\rm Cone}\big((\sum_{s\in\mathbb{Z}}\pi^-_s+\tau_n^-\circ\sigma_s\circ\pi^+_s):\bigoplus_{s\in\mathbb{Z}}A_s\to\bigoplus_{s\in\mathbb{Z}}B^-_s\big)\bigg)$$ for any non-zero integer $n$. The surgery formula is a powerful tool for us to compute the instanton Floer homology of Dehn surgeries. In some cases, the bent complexes $(A_s,d_s)$ and hence the maps $\pi^{\pm}_s$ are possible to compute, yet we also need to compute the mysterious map $\sigma_s$. A good news is that it is enough to compute the map $\sigma$ on the homology level. In the following few cases one can indeed compute the map $\sigma$ on the level of homology.
Note we have already known that $H_*(B^{\pm},d^{\pm})\cong I^{\sharp}(Y)$. If $Y=S^3$ then $I^{\sharp}(Y)\cong\mathbb{C}$. As a result, the isomorphism $\sigma_*$ is determined up to a non-zero scalar. As a fact in holomorphic algebra, if $n\neq0$, we know that the (isomorphism type of) mapping cone is independent of the choice of scalar.
The argument for instanton L-spaces is similar to that of $S^3$. The idea is that if $Y$ is an instanton L-space, then we can have a spin${}^c$ type decomposition of $Y$ so that each summand has dimension one. One can try to show that the map $\sigma_*$ preserves this spin${}^c$ type decomposition and hence the one-dimensional argument for $S^3$ works again.
For a $3$-manifold $Y$, the homology group $H_1(Y)$ acts on $I^{\sharp}(Y)$. For the special case $Y=\#^g(S^1\times S^2)$, Scaduto proved in [15] that $$I^{\sharp}(Y)\cong\Lambda^*H_1(Y;\mathbb{C}).$$ In this case, one can use the $H_1(Y)$ action to pin down the map $\sigma_*$ to be just the identity map.
The rational surgery along a knot $K\subset Y$ can be thought of as the integral surgery along the connected sum of the knot $K\subset Y$ and a core knot in a suitable lens space. Using this obvervation, we also proved in [61] a rational surgery formula.
It is an important question how knot homologies detects varaious properties of knots, and even the knot itself. For example, the study on $3$-surgeries of the knot above finally reduced to classifying knots in $L(3,1)$ that has minimal dimension of its instanton knot homology. I have done the following work along this direction of knot detections.
In [14], my collaborator and I showed that, in an instanton L-space $Y$, if a knot satisfies $$[K]=0\in H_1(Y;\mathbb{Z})~{\rm and~}KHI(Y,K)\cong I^{\sharp}(Y),$$ then the knot must be the unknot. An instanton $L$-space is a rational homology sphere $Y$ so that $$\dim_{\mathbb{C}}(Y)=|H_1(Y)|.$$
In order to obtain this detection result, we need to study the decomposition of $I^{\sharp}(Y)$ and $KHI(Y,K)$ along the torsion group $H_1(Y;\mathbb{Z})\cong H^2(Y;\mathbb{Z})$. In Heegaard Floer theory, corresponding to each element of $H_1(Y;\mathbb{Z})$, the summand of $\widehat{HFK}(Y,K)$ has euler characteristic 1. By identifying the Euler characteristics of instanton and Heegaard Floer theory, we know that each summand of $KHI(Y,K)$ has an odd dimension. Since by assumption $$\dim_{\mathbb{C}}KHI(Y,K)=\dim_{\mathbb{C}}I^{\sharp}(Y)=|H_1(Y;\mathbb{Z})|,$$ we know that each summand of $KHI(Y,K)$ has dimension one. Hence, $KHI(Y,K)$ is fully determined by its Euler characteristic, and we could further show that the whole $KHI(Y,K)$ is supported in a single Alexander grading and hence $K$ must be the unknot.
In [35], my collaborators and I proved the following: Suppose $(M,\gamma)$ is a taut balanced sutured manifold and $K\subset (M,\gamma)$ satisfying some homotopical conditions, and $K\subset {\rm int}(M)$ is a knot. Let $M_K=M\backslash N(K)$ be the complement of $K$ and $\gamma_K$ is the union of $\gamma$ with two meridians of $K$. Then $K$ is the unknot if and only if $$\dim_{\mathbb{C}}SHI(M_K,\gamma_K)=2\cdot \dim_{\mathbb{C}}SHI(M,\gamma).$$ As a corollary, the dimension of the instanton knot homology of links detects the unlink.
The proof of this detection results uses singular instanton Floer homology, which was introduced by Kronheimer and Mrowka in [11]. For singular instanton Floer homology, a point on the knot gives rise to a base-point action whose square is zero. Call this action $X$. In the paper we studied the $\mathbb{C}[X]/(X^2)$-structure associated to this action, and how it is changed through sutured manifold hierarchies, and used these properties to prove our unknot-detection result.
Since the milestone result that Khovanov homology detects the unknot established by Kronheimer and Mrowka [4], many detection results for khovanov homology have been established: the trefoil by Baldwin and Sivek [8], the Hopf link by Baldwin, Sivek, and Xie [35], the forests of unknots by Xie and Zhang [36], the link L4a1 and L6n1 in the Thistlethwaite link table by Xie and Zhang [37], the $T_(2,6)$ torus link by Gage [38], the figure-eight knot by Baldwin, Dowlin, Lenine, Lidman, and Sazdanovic [39], and the $T_(2,5)$ torus knot by Baldwin, Hu, and Sivek [40].
In [41], my collaborators and I added two links to the above list: the link L7n1 and the trefoil with its meridian. See the following figure.
The proof makes use the spectral sequence from Khovanov homology either to instanton knot homology by Kronheimer and Mrowka [4], or to knot Floer homology by Dowlin [42]. Then information about Floer homologies give us information about the multi-variable Alexander polynial of the link and hence some information about link numbers between components of the link. With this information we managed to prove the detection result.
In [58], defined a Khovanov type homology for links inside thickened surfaces. In particular, if a knot $K\subset [-1,1]\times \Sigma$ is isotopic to a knot in $\{0\}\times\Sigma$, then we have $APS(\Sigma, K)\cong\mathbb{Z}_2^2$. It is a natural question whether the inverse is true, i.e., if $APS(\Sigma, K)\cong\mathbb{Z}_2^2$, then can we always isotope $K$ into a surface knot? In [59], my collaborators and I used singular instanton Floer homology to prove this for planar surfaces $\Sigma$. Note for a surface knot we could further use the gradings of APS to determine the isotopy class of the knot. An interesting point to mention is the following. In the original proof by Kronheimer and Mrowka that Khovanov homology detects the unknot, they constructed a spectral sequence whose second page is the Khovanov homology of the knot, and which converges to the singular instanton Floer homology of the knot. Now for APS homology, we can use a similar idea to construct a spectral sequence. However, the second page is not exact the APS homology (but we can relate it to the APS homology). The construction of the second page of the spectral sequence can be generalized to surface of arbitrary genera. As a result, it is interesting to study whether the second page is a knot invariant. If so, then we would potentially construct a new combinatorial knot invariants that detects surface knots.
We know that Floer homology for knots detects fibredness, due to work of Ni [55] and Kronheimer and Mrowka [3] and [7]. In particular, if $K\subset S^3$ is a knot, then $K$ is fibred if and only if $$KHI(S^3,K,g(K))\cong \mathbb{C}~{\rm or}~\widehat{HFK}(S^3,K,g(K))\cong\mathbb{Z}_2.$$ So a natural question to ask, is what happens if $$KHI(S^3,K,g(K))\cong \mathbb{C}^2~{\rm or}~\widehat{HFK}(S^3,K,g(K))\cong\mathbb{Z}_2^2?$$ The knot won't be fibre again, but it is expected that the knot is not too far from being a fibred knot. In [56], Baldwin and Sivek call such knots nearly fibred knots, and they classified all genus-one nearly fibred knots. In [57], my collaborator and I established an if and only if condition for a knot to be nearly fibred in terms of the Seifert surface complement of the knot.
Suppose $h:S^3\to \mathbb{R}$ is a Morse function that only has two critical points. Any knot $K\subset S^3$ can be made into a position so that $h|_{K}$ is Morse and all critical points have distinct values under $h$. Using the Morse function, we can construct many knot invariants. One mose commonly know such invariant is the bridge index or the bridge number of the knot. In [43], Gabai introduced another invariant called the width of the knot, which we denote by $\omega$. Width has some important applications, such as the establishment of the Property R conjecture by Gabai [43] and the proof of the important fact that knots are determined by their complements by Gordon and Luecke [44]. The relation between the widths of a satellite knot and its companion remains illusive. In [45], Zupan conjectured the following: $$\omega(K)\geq m^2\cdot\omega(J),$$ where $K$ is a satellite knot with non-trivial companion $J$ and wrapping number $m$.
In [46], My collaborator and I proved a weaker version of Zupan's conjecture that $$\omega(K)\geq n^2\cdot\omega(J),$$ where $n$ is the winding number instead of wrapping number. In [47], we also verify Zupan's conjecture in the special case where $K$ is the whitehead double of $J$.
It is also worth mentioning that in one of my PRIMES project, my student and I also prove the following inequality for trunk number instead of width: $${\rm trunk}(K)\geq \frac{1}{2}\cdot m\cdot{\rm trunk}(J),$$ where $K$ is a satellite knot with non-trivial companion $J$ and wrapping number $m$.
Suppose $(M,\gamma)$ is a balanced sutured manifold and $(Z,\xi)$ is an oriented $3$-manifold with boundary, so that $\partial Z$ is convex with respect to $\xi$. Suppose we have an embedding $$f:\partial M\hookrightarrow \partial Z$$ which is orientation reversing and which maps the suture $\gamma$ to the dividing curves on $\partial Z$. Let $(M',\gamma')$ be the balanced sutured manifold obtained by gluing $M$ and $Z$ using $f$ and taking the dividing curve on the rest part of the boundary, then Honda, Kazez, and Matić [49] Constructed a gluing map $$\Phi_{\xi}:SFH(-M,-\gamma)\to SFH(-M',-\gamma')$$ in Heegaard Floer theory.
In [6], I constructed a similar gluing map in instanton theory: $$\Phi_{\xi}:SHI(-M,-\gamma)\to SHI(-M',-\gamma').$$ This gluing map composes well under composition of the contact pieces $Z$ and preserves contact elements. It plays an essential role in the construction of cobordism maps for sutured instanton Floer homology. It also helps to prove that the contact element of contact structures that contains a Giroux torsion is zero, which solves some technical issue in establishing a satisfactory version of large surgery formula in instanton theory.
The classification of tight contact structures is a fundamental problem in contact geometry, though not too much is known, especially on 3-manifolds with boundary. Previously the only irreducible 3-manifold with boundary on which the tight contact structures are fully classified is the 3-ball, on which there is a unique tight contact structure by work of Eliashberg [50]. The next 3-manifold to work on is the solid torus. Dividing sets on the boundary of a solid torus can be described by their slope and number of components. Honda [51] classified all tight contact structures on a solid torus whose dividing set on the boundary has two components.
In one of my PRIMES project, my student and I were able to classify all tight contact structures on a solid torus. See [52]. To resolve some technical aspect of the problem, we make use of the computation of $SHI$ of a sutured solid torus I have done before.
In [43], Gabai proved that any taut balanced sutured manifold $(M,\gamma)$ admits a finite depth taut foliation. However, there was previously no bounds on how finite the depth of a taut foliation could be. In [53], Juhas conjectured that the depth can be bounded by $2{\rm log}_2\big({\rm rk}_{\mathbb{Z}}SFH(M,\gamma)\big)$. In [53], I provide the first known bound for the depth of taut foliations: There is a constant $C$ (less than 100) so that any taut balanced sutured manifold $(M,\gamma)$ admits a taut foliation of depth at most $C\cdot \dim_{\mathbb{C}}SHI(M,\gamma)$.
The references are indexed in the order they appeared in this page.