A note on band surgery and the signature of a knot

In this section we state some homological preliminaries and set notation. We assume all singular (co)homology groups have coefficients in $\mathbb{Z}$. Let $K$ be a knot in a rational homology sphere $Y$, and let $M=Y-N(K)$ denote the complement of $K$ in $Y$. A slope is an isotopy class of an unoriented simple closed curve on the boundary of $M$. The Dehn filling $M(\eta)$ is the closed, oriented three-manifold obtained by gluing a solid torus $V$ to $M$ by a homeomorphism which identifies a meridian of $\partial V$ to the slope $\eta$ on $\partial M$. The distance between two surgery slopes on $\partial M$ is the minimal geometric intersection number of the two curves, denoted $\Delta(\eta,\nu)$ for any pair of slopes $\eta,\nu$. In this note we are particularly concerned with surgery slopes which intersect the meridian of $K$ exactly once, i.e. distance one surgeries, or integral surgeries.

When $K$ is null-homologous, there is a standard choice of a meridian $\mu$ and Seifert longitude $\lambda$ on the bounding torus $\partial M$. Here, the meridian $\mu$ is a simple closed curve on $\partial M=\partial N(K)$ that bounds a disk in the solid torus $N(K)$. The preferred longitude is the slope determined by the intersection of a Seifert surface for the knot and $\partial N(K)$. A slope $\eta$ may be written in terms of this standard basis as $\eta=a\mu+b\lambda$. Rational Dehn surgery with the surgery coefficient $a/b$ along $K$ in $Y$ is then well-defined, and the result of surgery is denoted by $Y_{a/b}(K)=M(\eta)$. Indeed, when $K$ is null-homologous a homological argument (see for example [Gai18, Lemma 8.1]) shows that $H_{1}(M)\cong\mathbb{Z}\oplus H_{1}(Y)$, where the $\mathbb{Z}$-summand is generated by the meridian of $K$. This implies that $H_{1}(Y_{a/b}(K))\cong\mathbb{Z}/a\oplus H_{1}(Y)$ by a standard Mayer-Vietoris argument. Notice that the Seifert longitude controls the first homology of the Dehn filling, in the sense that $|H_{1}(Y_{a/b}(K))|=|M(\eta)|=|H_{1}(Y)|\cdot\Delta(\eta,\lambda)$.

When $K$ is not null-homologous, we have that $H_{1}(M)\cong\mathbb{Z}\oplus H$ for some finite group $H$. Let $i_{*}$ denote the map induced by inclusion in the long exact sequence of the pair $(M,\partial M)$,

The kernel and image of $i_{*}$ are both rank one. In particular, the kernel of $i_{*}$ is generated by $k\lambda_{M}$, where $\lambda_{M}$ is a primitive class $[\lambda_{M}]$ in $H_{1}(\partial M)$ uniquely determined up to sign. This class determines a well-defined slope in $\partial M$:

When we work with knots that are non-trivial in $H_{1}(Y)$, we will use the rational longitude to fix a basis $(\mu,\lambda_{M})$ for $H_{1}(\partial M)$, where $\mu$ now denotes some slope with $\Delta(\mu,\lambda_{M})=1$. Just as with the case of the Seifert longitude for null-homologous knots, the rational longitude also controls the order of the first homology of the filling.

In [Wat12, Lemma 3.2], this is proved with a careful analysis of the long exact sequence of the pair. Under the homomorphism $i_{*}:H_{1}(\partial M)\rightarrow H_{1}(M)$, the basis elements $\mu$ and $\lambda_{M}$ are mapped to $(\ell,u)$ and $(0,h)$, respectively, where $|\ell|=\operatorname{ord}_{H}i_{*}(\lambda_{M})$, and $u,h$ are some elements of $H$.

Let us introduce some notation. Consider a decomposition of the finite abelian group $H$ as $\mathbb{Z}/r_{1}\oplus\cdots\oplus\mathbb{Z}/r_{k}$. We will write $\vec{u}=(u_{1},\cdots,u_{k})$ and $\vec{v}=(v_{1},\cdots,v_{k})$ for elements $u,v\in H$ written with respect to this decomposition. The notation $I_{\vec{r}}$ will stand for the $k\times k$ diagonal matrix with $i$-th diagonal entry $r_{i}$ where $r_{1},\dots,r_{k}$ are the invariant factors of $H$.

For a filling slope $\eta=p\mu+q\lambda_{M}$, Watson observes that $H_{1}(M(\eta))=H_{1}(M)/(p\ell,pu+qh)$ has a block form presentation matrix

The statement of the lemma then follows.

The reader may notice that the proof of Lemma 2.3 below is adapted from the argument given in [LM17, Theorem 2.4]. In general, we will use the Smith normal form of the presentation matrix (2). Recall that the Smith normal form is a diagonal matrix $I_{\vec{\delta}}$ where $\delta_{i}|\delta_{i+1}$. Each $\delta_{i}=\Gamma_{i}/\Gamma_{i-1}$, where $\Gamma_{i}$ is the greatest common divisor of the the $i\times i$ minors of $A$, and $\Gamma_{0}=1$.

For the remainder of the article, we will assume that $K$ is a null-homologous knot in $Y$ with meridian $\mu$ and longitude $\lambda$. We will be primarily concerned with slopes of the form $n\mu+\lambda$, for $n\in\mathbb{Z}$, which are framing curves on the boundary of a neighborhood of $K$. These slopes intersect the meridian once transversely and inherit an orientation from $K$. The notation $Y_{n}(K)$ will denote the result of integral surgery along $K$, and $K_{n}$ will denote the core of the surgery $Y_{n}(K)$.

The set $\text{Spin}^{c}(Y)$ is the space of nowhere vanishing vector fields on $Y$ modulo homotopy outside of a ball. There is a non-canonical correspondence $\text{Spin}^{c}(Y)\cong H^{2}(Y)$ [Tur97]. We write $\mathfrak{t}$ to denote an element of $\text{Spin}^{c}(Y)$, and write $\underline{\mathrm{Spin}^{c}}(Y,K)$ to denote the relative $\text{Spin}^{c}$ structures on $(M,\partial M)$. Note that $\underline{\mathrm{Spin}^{c}}(Y,K)$ has an affine identification with $H^{2}(Y,K)$, which by excision is isomorphic with $H^{2}(M,\partial M)\cong H_{1}(M)$. In particular, because $K$ is null-homologous and $H_{1}(M)\cong H_{1}(Y)\oplus\mathbb{Z}$, we may label an element $\xi$ of $\underline{\mathrm{Spin}^{c}}(Y,K)$ by $\xi=(\mathfrak{t},s)$, where $\mathfrak{t}\in\text{Spin}^{c}(Y)$ and $s\in\mathbb{Z}$. The correspondence $\underline{\mathrm{Spin}^{c}}(Y,K)\equiv H^{2}(M,\partial M)$ is non-canonical, and the labelling that we adopt follows closely to that of [Gai18]. When $H_{1}(Y)$ has odd order, there is a unique self-conjugate $\text{Spin}^{c}$ structure $\mathfrak{t}_{0}$, distinguished from the others by the requirement that it satisfies $c_{1}(\mathfrak{t}_{0})=0\in H^{2}(Y)$ (see for example [Section 2][MO07]).

We will assume some familiarity with Heegaard Floer homology, referring the reader to [OS03, OS11] for more information. We take all Heegaard Floer complexes with coefficients in the field $\mathbb{F}=\mathbb{Z}/2$. One of the main components of the Heegaard Floer package we will use is the $d$-invariant $d(Y,\mathfrak{t})$, or correction term, which is a rational number associated to a $\text{Spin}^{c}$ rational homology sphere $(Y,\mathfrak{t})$. More specifically, given a rational homology sphere $Y$, the Heegaard Floer module $HF^{+}(Y)$ splits over $\text{Spin}^{c}$ structures, and in each summand we have

as $\mathbb{F}[U]$-modules. The first summand is abbreviated $\mathcal{T}_{d}^{+}$ and referred to as the tower. The second summand is a torsion $\mathbb{F}[U]$-module. An L-space is a rational homology sphere whose Heegaard Floer homology $HF^{+}(Y)$ is a free $\mathbb{F}[U]$-module with rank $|H^{2}(Y;\mathbb{Z})|$. That is, the torsion summand in (3) vanishes, leaving only a tower in each $\text{Spin}^{c}$ structure. The $d$-invariant $d:=d(Y,\mathfrak{t})$ is defined to be the minimal Maslov grading of the tower. The $d$-invariants switch sign under orientation-reversal [OS03].

Associated to $(Y,K)$ and each relative $\text{Spin}^{c}$ structure $\xi$ in $\underline{\mathrm{Spin}^{c}}(Y,K)$ is the knot Floer chain complex $C_{\xi}=CFK^{\infty}(Y,K,\xi)$ [Ras03, OS04, OS11]. The complex is $\mathbb{Z}$-filtered over $\mathbb{F}[U,U^{-1}]$ with a second filtration induced by the action of the variable $U$. We denote these two filtrations as $(i,j)=$ (algebraic, Alexander). The chain complex also has a homological Maslov grading that we suppress in the notation. Multiplication by $U$ decreases the Maslov grading by two and the Alexander filtration by one, and the action of $PD[\mu]$ on $\xi$ shifts the Alexander filtration by one, i.e. $C_{\xi+PD[\mu]}=C_{\xi}[(0,-1)]$. Recall that $\mu$ denotes the (class of a) meridian.

As in [OS11], for each $\xi$ in $\underline{\mathrm{Spin}^{c}}(Y,K)$ there are complexes $A^{+}_{\xi}=C_{\xi}\{\max\{i,j\}\geq 0\}$ and $B^{+}_{\xi}=C_{\xi}\{i\geq 0\}$. The complex $B^{+}_{\xi}$ is $CF^{+}(Y,G_{Y,K}(\xi))$. The map $G_{Y,K}:\underline{\mathrm{Spin}^{c}}(Y,K)\rightarrow\text{Spin}^{c}(Y)$ is defined in [OS11, Section 2.2], and sends a relative $\text{Spin}^{c}$ structure to a $\text{Spin}^{c}$ structure in the target manifold indicated by the subscript. The complex $A^{+}_{\xi}$ represents the Heegaard Floer complex of a large surgery $Y_{N}(K)$ in a certain $\text{Spin}^{c}$ structure, where $Y_{N}(K)$ is obtained by Dehn surgery along a framing curve $N\mu+\lambda$ with $N>>0$. There are analogous complexes in the ‘hat’ version of Heegaard Floer homology. In particular we have $\widehat{A}_{\xi}=C_{\xi}\{\max\{i,j\}=0\}$ and $\widehat{B}_{\xi}=C_{\xi}\{i=0\}\cong\widehat{CF}(Y,G_{Y,K}(\xi))$. The complexes are related by chain maps

where $K_{n}$ is the push-off of $K$ inside $Y-N(K)$ using framing $n\mu+\lambda$. In [OS11, Theorem 4.1], it is shown that the maps $v^{+}_{\xi}$ and $h^{+}_{\xi}$ correspond with a negative definite cobordism $W^{\prime}_{N}:Y_{N}(K)\rightarrow Y$ equipped with the $\text{Spin}^{c}$ structures $\mathfrak{v}$ and $\mathfrak{h}=\mathfrak{v}+PD[\widehat{F}]$, respectively, which extend a given $\text{Spin}^{c}$ structure on $Y_{N}(K)$. Here, $[\widehat{F}]$ generates $H_{2}(W^{\prime}_{N},Y)$ and is represented by a capped-off Seifert surface for $K$.

The complexes $A^{+}_{\xi}$ and $B^{+}_{\xi}$ each contain a non-torsion summand, i.e. a tower. On homology, each of the maps $v^{+}_{\xi}$ and $h^{+}_{\xi}$ induces an endomorphism of the towers, which is multiplication by $U^{V}_{\xi}$ or $U^{H}_{\xi}$ for integers $V_{\xi}\geq 0$ and $H_{\xi}\geq 0$. This defines the knot invariants $V_{\xi}$ and $H_{\xi}$, which appeared originally as the local h-invariants in [Ras03]. We remark for later use that when $V_{\xi}>0$, the corresponding map $\widehat{v}_{\xi}$ is identically zero, and similarly $H_{\xi}>0$ implies $\widehat{h}_{\xi}$ is zero.

Recall that because $K$ is null-homologous, $\underline{\mathrm{Spin}^{c}}(Y,K)\cong\text{Spin}^{c}(Y)\oplus\mathbb{Z}$, where the $\mathbb{Z}$-summand is generated by the meridian of $K$ [Gai18, Lemma 8.1]. We write $\xi=(\mathfrak{t},s)$ for $\mathfrak{t}\in\text{Spin}^{c}(Y)$ and $s\in\mathbb{Z}$. Fixing $\mathfrak{t}\in\text{Spin}^{c}(Y)$, we have the subgroup $\{\mathfrak{t}\}\oplus\mathbb{Z}\cong\mathbb{Z}$. Because the following properties have appeared in various forms throughout the literature, we provide just a sketch of the proof.

The proof of Theorem 3.5 below will require the mapping cone formula for the Heegaard Floer homology of the integral surgery $Y_{n}(K)$. We give only a terse review here, sending the reader to Ozsváth and Szabó for details [OS08, OS11]. The generalization of the mapping cone formula specific to rational surgeries on null-homologous knots in L-spaces is also reviewed in [NW15, Gai18]. Note that for our applications, we only require the formulation for integral surgery and ‘hat’ homology.

For each $\mathfrak{t}\in\text{Spin}^{c}(Y)$ and $0\leq i<n$, sum up the complexes $\widehat{A}_{\xi}=\widehat{A}_{(\mathfrak{t},s)}$ and $\widehat{B}_{\xi}=\widehat{B}_{(\mathfrak{t},s)}$ into

where the first component of each tuple records the index of the summand. Using the maps $\widehat{v}_{\xi}$ and $\widehat{h}_{\xi}$ from equation (4) above, we define the map $\widehat{\mathbb{D}}_{(\mathfrak{t},i)}:\widehat{\mathbb{A}}_{(\mathfrak{t},i)}\rightarrow\widehat{\mathbb{B}}_{(\mathfrak{t},i)}$ by

The mapping cone complex of $\widehat{\mathbb{D}}_{(\mathfrak{t},i),n}$ is denoted $\widehat{\mathbb{X}}_{(\mathfrak{t},i),n}$. Let us omit the surgery coefficient ‘$n$’ and write the summand of the cone corresponding to the equivalence class of $\xi=(\mathfrak{t},i)$ more concisely as $\widehat{\mathbb{X}}_{\xi}$.

Given a $\text{Spin}^{c}$ structure in $\text{Spin}^{c}(Y_{n}(K))\cong\text{Spin}^{c}(Y)\oplus\mathbb{Z}/n$, there is a unique $\text{Spin}^{c}$ structure on $Y$ which extends over the two-handle cobordism $W_{n}(K):Y\rightarrow Y_{n}(K)$ and this defines the projection from $\text{Spin}^{c}(Y_{n}(K))$ to $\text{Spin}^{c}(Y)$. This projection appears in the following result of Ni and Wu, which will allow us to describe the $d$-invariants of surgeries in terms of the invariants $V_{\mathfrak{t},i}$, which crucially, are non-negative integers.

Proposition 3.3 was originally proved for knots in the three-sphere, but it generalizes immediately to the case of a null-homologous knot in an L-space [NW15], which is the version stated here (see also [LMV19]). The term on the right includes the $d$-invariants of the lens space $L(n,1)$, which are made explicit in section 3.3.

The following proposition will be useful in the proofs of the main results. A proof of Proposition 3.4 is given in [LMV19, Proposition 2.10].

In this section we prove the main result. The requirement that $Y$ and $Y^{\prime}$ are L-spaces in the statement of Theorem 3.5 is necessary both to calculate the differences in their $d$-invariants via the surgery formula of Proposition 3.3, and to use the surgery formula directly to determine the value of the invariant $V_{\mathfrak{t},0}$. The L-space condition will surface again when we apply Theorem 3.5 to alternating and quasi-alternating knots to obtain Corollary 4.3 and Theorem 4.4. The significance in this context is that quasi-alternating knots have branched double covers that are L-spaces, and there is a connection between the signature of the knot and a certain $d$-invariant of the branched double cover.