# Stellar Wind – Magnetosphere Interactions in Hot Jupiters

Derek L. Buzasi1 1Dept. of Chemistry and Physics, Florida Gulf Coast
University
10501 FGCU Boulevard S., Fort Myers, FL 33965 USA
email<EMAIL_ADDRESS>
(2015)

###### Abstract

One potential star-planet interaction mechanism for hot Jupiters involves
planetary heating via currents set up by interactions between the stellar wind
and planetary magnetosphere. Early modeling results indicate that such
currents, which are analogous to the terrestrial global electric circuit
(GEC), have the potential to provide sufficient heating to account for the
additional radius inflation seen in some hot Jupiters. Here we present a more
detailed model of this phenomenon, exploring the scale of the effect, the
circumstances under which it is likely to be significant, implications for the
planetary magnetospheric structure, and observational signatures.

###### keywords:

planets and satellites: general, solar system: formation

††volume: 320††journal: Solar and Stellar Flares and Their Effects on
Planets††editors: A.G. Kosovichev, S.L. Hawley, & P. Heinzel, eds.

##  1 Introduction

Hot Jupiters frequently have radii which are “inflated” relative to the
expected mass-radius relation [Fortney & Nettlemann (2010), (Fortney &
Nettleman 2010)]. The degree of inflation is correlated with orbital semimajor
axis (Demory & Seager 2011) and typically becomes significant for $F>2\times
10^{8}\rm~{}erg~{}cm^{-2}~{}s^{-1}$, which corresponds to approximately the
Alfvén radius $a=r_{A}$, and there is some indication that it exhibits a
dependence on stellar activity level [Buzasi (2013), (Buzasi 2013)]. Proposed
models for this behavior include tidal heating [Gu, Bodenheimer, & Lin (2004),
(Gu et al. 2004)] and “Ohmic heating” due to interactions between atmospheric
flows and the planetary magnetic field [Batygin, Stevenson, & Bodenheimer
(2011), (Batygin et al. 2011)]. [Buzasi (2013), Buzasi (2013)] proposed
interactions between the stellar wind and the planetary magnetic field as an
alternate mechanism; in this work we explore the ramifications of that
suggestion in the context of a more complete planetary model.

##  2 Model and Results

The model calculates currents internal to the planet generated by the electric
potential difference produced by the stellar wind/magnetosphere interaction
and mapped down to the outer layers of the planet, analogous to the “global
electric circuit” (GEC; [Tinsley, Burns, & Zhou (2007), Tinsley et al. 2007]).
A solar wind model [Suzuki (2006), (Suzuki 2006)] is taken as input, and the
conductivity is calculated for the planetary interior, which in turn is
calculated using the MESA code [Paxton11, (Paxton et al. 2011)]. The presence
of a $\sim 10\rm G$ planetary magnetic field renders conductivity
inhomogeneous. The magnetic dipole moment is presumed coaligned with the
rotation axis and ecliptic poles, and the potential difference drives Pedersen
and parallel (field-aligned) currents, resulting in Joule heating of the
planetary interior.

A series of models with solar composition were calculated for
$M_{PL}=10^{-3}M_{\odot}$ ($\sim M_{J}$) using MESA [Paxton11, (Paxton et al.
2011)] and evolved to $t=5\rm~{}Gyr$. Model electron densities were combined
with the adopted $10\rm~{}G$ dipole planetary magnetic field to calculate the
classical ($\sigma_{0}$), Pedersen ($σ\sigma_{P}$), and Hall ($\sigma_{H}$)
conductivities. The ionospheric electric potential was taken as zero except at
two regions in each hemisphere located at invariant latitude
$\Lambda=\cos^{-1}\sqrt{\frac{R_{PL}}{R_{M}}}$ and separated by $\pi/2$ in
longitude. Regions were circular with diameter $\Lambda$. The potential
adopted was the minimum of the wind-induced field $E=v_{w}B_{w}R_{M}$, or the
value leading to the maximum energy deposition possible [Akasofu81, (Akasofu
1981)], $\epsilon=v_{w}B_{w}^{2}R_{M}^{2}$.

Heating resulting from these potential/conductivity combinations was
calculated and incorporated into the MESA models. The resulting energy
deposition profiles for two typical models are shown in Figure 1a. Total
heating in both cases is limited by the power available from the wind rather
than by the wind-induced field strength. Figure 1b illustrates planetary
interior models resulting from the two cases; note the growth of the 1-bar
planetary radius from $\approx 1.05R_{J}$ at 1 AU to $\approx 1.58R_{J}$ at
0.014 AU. The latter corresponds to a stellar irradiance of $F=7\times
10^{9}\rm~{}erg~{}cm^{-2}~{}s^{-1}$ and an orbital period of 0.6 days.

(a)

(b)

Figure 1: The left panel shows internal electric heating input due to GEC as a
function of internal pressure for models with semimajor axes $a=1\rm~{}AU$
(blue) and $a=0.014\rm~{}AU$ (red). Maximum heating occurs at $r/R_{PL}=0.973$
at $1\rm~{}AU$ and at $r/R_{PL}=0.945$ at $0.014\rm~{}AU$. The right panel
illustrates MESA models incorporating GEC heating for the same models, both
with solar composition and $B=10\rm~{}G$, and incorporating stellar
irradiation via a gray atmosphere.

##  3 Discussion

Successful models which account for the observed radius excess in hot Jupiters
by additional heating must be capable of supplying
$>10^{27}\rm~{}erg~{}s^{-1}$ to the convective portion of the planetary
interior. The proposed GEC model is capable of such heating, and produces
planetary radii broadly in accord with observations (Figure 2). Variations in
planetary mass, planetary composition (leading to changes in conductivity),
planetary magnetic field, and stellar magnetic field may account for the range
of radii observed in the sample. In addition, we note that it is likely that
multiple heating models coexist, including tidal heating (for noncircular
orbits) and Ohmic heating ([Batygin & Stevenson (2010), Batygin & Stevenson
2010], [Batygin, Stevenson, & Bodenheimer (2011), Batygin et al. 2011]).

Observational testing of the model is possible. In particular, predictions
include

  1. 1.

The correlation of radius inflation with stellar magnetic field, potentially
derivable from accurate stellar activity proxies such as Ca II h+k and/or
starspot coverage.

  2. 2.

Planetary temperatures in excess of those possible based solely on radiative
equilibrium calculations. Precise photometric transit observations and
comparisons of atmospheric spectra to models should enable such testing.

(a)

(b)

Figure 2: Left panel: planetary radii at $P=1\rm~{}bar$ for models at varying
distances from the Sun. Heating leads to significant radius inflation for
models with semimajor axes less than $\sim 0.1\rm~{}AU$. All models were
evolved to $t=5\rm~{}Gyr$. Right panel: model radii plotted against stellar
irradiation and compared to all known hot Jupiters ($R_{PL}>0.8R_{J}$). All
models assume $10\rm~{}G$ planetary magnetic fields.

Figure 2a summarizes model results over a range of semimajor axes, and shows
that the effect of GEC heating on planetary radius becomes important for
$a>0.1\rm~{}AU$, and is capable of inflating planets to radii consistent with
those observed. Note that variations in (a) stellar magnetic field, (b)
planetary magnetic field, and c) planetary composition will have potentially
significant impacts which are not explored in detail here (though see [Buzasi
(2013), Buzasi 2013] for limited discussion). Further improvements to the
model are planned, and include examining the effects of variations in
composition and magnetic field, as well as GEC model interactions with other
heating mechanisms, such as Ohmic ([Batygin, Stevenson, & Bodenheimer (2011),
Batygin et al. 2011]) and tidal heating.

##  4 Acknowledgments

I am grateful to the Whitaker Foundation for its support of the Whitaker Chair
at Florida Gulf Coast University. Funding for the Kepler mission is provided
by the NASA Science Mission Directorate, and this work was in part funded by
the Kepler Participating Science Program grant NNH09CE70C.

## References

  * [Akasofu (1981)] Akasofu, S.I. 1981, Space Sci. Revs, 28, 121
  * [Batygin & Stevenson (2010)] Batygin, K. & Stevenson, D.J. 2010, ApJ, 714, 238
  * [Batygin, Stevenson, & Bodenheimer (2011)] Batygin, K., Stevenson, D.J., & Bodenheimer, P.H. 2011, ApJ, 738, 1
  * [Buzasi (2013)] Buzasi, D.L. 2013, ApJ, 765, 25
  * [Demory & Seager (2011)] Demory, B.O. & Seager, S. 2011, ApJS, 197, 12
  * [Fortney & Nettlemann (2010)] Fortney, J.J. & Nettlemann, N. 2010, Space Sci. Revs, 152, 423
  * [Gu, Bodenheimer, & Lin (2004)] Gu, P.-G., Bodenheimer, P.H., & Lin, D.N.C. 2004, ApJ, 608, 1076
  * [Menou (2012)] Menou, K. 2012, ApJ, 754, 9
  * [Paxton et al. (2012)] Paxton, B. et al. 2012, ApJS, 192, 3
  * [Suzuki (2006)] Suzuki, T.K. 2006, ApJ, 640, 75
  * [Tinsley, Burns, & Zhou (2007)] Tinsley, B.A., Burns, G.B., & Zhou, L. 2007, Adv. Sp. Res., 40, 1126


# Knot concordance invariants from Seiberg–Witten theory and slice genus
bounds in $4$-manifolds

David Baraglia  School of Mathematical Sciences, The University of Adelaide,
Adelaide SA 5005, Australia<EMAIL_ADDRESS>
###### Abstract.

We construct a new family of knot concordance invariants $\theta^{(q)}(K)$,
where $q$ is a prime number. Our invariants are obtained from the equivariant
Seiberg–Witten–Floer cohomology, constructed by the author and Hekmati,
applied to the degree $q$ cyclic cover of $S^{3}$ branched over $K$. In the
case $q=2$, our invariant $\theta^{(2)}(K)$ shares many similarities with the
knot Floer homology invariant $\nu^{+}(K)$ defined by Hom and Wu. Our
invariants $\theta^{(q)}(K)$ give lower bounds on the genus of any smooth,
properly embedded, homologically trivial surface bounding $K$ in a definite
$4$-manifold with boundary $S^{3}$.

##  1\. Introduction

In this paper we introduce a series of new knot concordance invariants. Our
invariants are obtained from the equivariant Seiberg–Witten–Floer cohomology,
constructed by the author and Hekmati [1], applied to cyclic branched covers
of knots. Our invariants can be used to bound the slice genus of knots, or
more generally the genus of any smooth, properly embedded, homologically
trivial surface in a definite $4$-manifold with $S^{3}$ boundary which bounds
the knot.

For each prime $q$, we obtain a knot concordance invariant $\theta^{(q)}(K)$.
For simplicity we will restrict attention to the $q=2$ invariant in the
introduction and write $\theta(K)=\theta^{(2)}(K)$.

######  Theorem 1.1.

The invariant $\theta(K)$ is a knot concordance invariant valued in the non-
negative integers. Moreover $\theta$ satisfies the following properties:

  * (1)

$-\sigma(K)/2\leq\theta(K)\leq g_{4}(K)$.

  * (2)

$\theta(K_{1}+K_{2})\leq\theta(K_{1})+\theta(K_{2})$.

  * (3)

Let $K_{+},K_{-}$ be knots where $K_{-}$ is obtained from $K_{+}$ by changing
a positive crossing into a negative crossing. Then

$0\leq\theta(K_{+})-\theta(K_{-})\leq 1.$

  * (4)

If $K$ is quasi-alternating, then

$\theta(K)=\begin{cases}-\sigma(K)/2,&\sigma(K)\leq 0,\\\
0,&\sigma(K)>0.\end{cases}$

  * (5)

If $\delta(K)<-\sigma(K)/2$ and $\sigma(K)\leq 0$, then $\theta(K)\geq
1+\sigma(K)/2$. Here $\delta(K)$ is the Manolescu–Owens knot concordance
invariant [12].

Note that property (1) above means that the slice genus bound
$g_{4}(K)\geq\theta(K)$ can be thought of as a refinement of the bound
$g_{4}(K)\geq-\sigma(K)/2$. A similar situation occurs for the Ozsváth–Szabó
tau-invariant $\tau(K)$, namely there exists a non-negative, integer-valued
concordance invariant $\nu^{+}(K)$ which refines the slice genus bound
$g_{4}(K)\geq\tau(K)$ in the sense that $\tau(K)\leq\nu^{+}(K)\leq g_{4}(K)$
[7]. There are multiple similarities between $\theta$ and $\nu^{+}$. From [7]
and [2], we have that $\nu^{+}(K)$ satisfies properties (2)-(5) of Theorem
1.1. Another similarity is that $\nu^{+}(K)$ arises as the smallest $j$ such
that $V_{j}(K)=0$, where $V_{j}(K)$ is a sequence of decreasing knot
concordance invariants originally defined in [15] (in a different notation),
while $\theta(K)$ is similarly defined as the smallest $j$ for which a certain
sequence of concordance invariants vanish.

For prime knots with $9$ or fewer crossings we have found that in all but one
case we have $\theta(K)=\nu^{+}(K)$ and $\theta(-K)=\nu^{+}(-K)$. The
exception is the knot $K=9_{42}$ which has $\theta(K)=\nu^{+}(K)=0$, but
$\theta(-K)=1$ and $\nu^{+}(-K)=0$. It is interesting to note that $9_{42}$ is
the only prime knot with $9$ or fewer crossings which is neither
quasialternating nor quasipositive (or the mirror of a quasipositive knot).

The invariant $\theta(K)$ sometimes gives a better slice genus bound than can
be obtained from the $\tau,\nu^{+}$ or $s$ invariants, or from the
Levine–Tristram signature function.

######  Example 1.2.

Let $K=-9_{42}\\#Wh(T_{2,2a+1})$, where $Wh(T_{2,2a+1})$ is the untwisted
Whitehead double of the torus knot $T_{2,2a+1}$ with positive clasp and $a\geq
1$. Then $\tau(K)=1$, $s(K)/2=1$, where $s$ is the Rasmussen $s$-invariant and
$|\sigma_{K}(\omega)/2|\leq 1$ for all $\omega$, where $\sigma_{K}(\omega)$ is
the Levine–Tristram signature function. We also have $\nu^{+}(K)\leq 1$ and
$\nu^{+}(-K)=0$. Thus all these invariants give a slice genus bound
$g_{4}(K)\geq 1$. However we will see that $\theta(K)=2$, giving the better
bound $g_{4}(K)\geq 2$. In fact, since $9_{42}$ and $Wh(T_{2,2a+1})$ both have
slice genus $1$ and unknotting number $1$ we get that $g_{4}(K)=u(K)=2$.

While it is not always easy to compute $\theta(K)$, we can obtain a lower
bound for it in some circumstances. For instance, we have the following:

######  Theorem 1.3.

Let $\Sigma_{2}(K)$ denote the double cover of $S^{3}$ branched over $K$ and
let $\mathfrak{s}_{0}$ denote the unique spinc-structure which arises from a
spin structure. Let $\ell(K)$ denote the lowest $i$ for which
$HF^{+}_{i}(\Sigma_{2}(K),\mathfrak{s}_{0})$ is non-zero (where $HF^{+}$ is
taken with coefficients in $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$). Then

$\theta(K)\geq\ell(-K)-\frac{3\sigma(K)}{4}.$

In particular this implies the following bound on the slice genus

$g_{4}(K)\geq\ell(-K)-\frac{3\sigma(K)}{4}.$

Conversely, if the value of $g_{4}(K)$ is known then we can use this result to
give an upper bound for $\ell(-K)$:

$\ell(-K)\leq g_{4}(K)+\frac{3\sigma(K)}{4}.$

The invariant $\theta(K)$ is not only a lower bound for the slice genus
$g_{4}(K)$, but also for the genus of any null-homologous surface bounding $K$
in a negative definite $4$-manifold:

######  Theorem 1.4.

Let $X$ be a smooth, compact, oriented $4$-manifold with boundary $S^{3}$ and
with negative definite intersection form. Suppose also that
$H_{1}(X;\mathbb{Z})=0$. Let $K\subset S^{3}$ be a knot. Then for any, smooth,
oriented, properly embedded, null-homologous surface $\Sigma\subset X$
bounding $K$, we have

$g(\Sigma)\geq\theta(K),$

where $g(\Sigma)$ is the genus of $\Sigma$.

For a $4$-manifold $X$ with boundary $S^{3}$, a knot $K\subset S^{3}$ is said
to be $H$-slice in $X$ if $K$ bounds a smooth, oriented, properly embedded,
null-homologous disc $\Sigma\subset X$. More generally, we define the
$H$-slice genus of $K$ in $X$ to be the minimum genus of such a null-
homologous surface $\Sigma\subset X$ bounding $K$ and we denote it by
$g_{H}(K,X)$. Thus Theorem 1.4 says that $\theta$ provides a lower bound for
the $H$-slice genus of any knot in a negative definite $4$-manifold $X$ with
boundary $S^{3}$ and with $H_{1}(X;\mathbb{Z})=0$.

Similar bounds on the $H$-slice genus of $K$ in negative definite manifolds
have been obtained by different methods. From [14], we have that
$g_{H}(K,X)\geq\tau(K)$, where $\tau$ is the Ozsváth–Szabó $\tau$-invariant.
In the special case that $X=\\#^{t}\overline{\mathbb{CP}^{2}}\setminus B^{4}$,
there is a similar inequality $g_{H}(K,X)\geq s(K)/2$ [11], where $s$ is the
Rasmussen $s$-invariant [16]. Yet another bound is given by the
Levine–Tristram signature [3], which in the case of a negative defininte
$4$-manifold with $H_{1}(X;\mathbb{Z})=0$ gives

$g_{H}(K,X)\geq-\sigma_{K}(\omega)/2$

for all $\omega\in S^{1}_{!}$. Here $\sigma_{K}(\omega)$ is the
Levine–Tristram signature of $K$ and $S^{1}_{!}$ is the set of unit complex
numbers that are not zeros of any integral coefficient Laurent polynomial $p$
with $p(1)=1$. Note that $S^{1}_{!}$ contains all roots of unity of prime
power order. In particular, since the signature of $K$ is given by
$\sigma(K)=\sigma_{K}(-1)$, we have $g_{H}(K,X)\geq-\sigma(K)/2$.

We have examples where the genus bound given by $\theta(K)$ is better than
that given by $\tau$, $s$ or $\sigma_{K}(\omega)$. Indeed the knots
$K=-9_{42}\\#Wh(T_{2,2a+1})$ are an example of this. In fact, since these
knots satisfy $\theta(K)=2=g_{4}(K)$, we see that the minimal genus of any
null-homologous, properly embedded surface bounding $K$ in any negative
definite $4$-manifold with zero integral first homology is exactly $2$.

###  1.1. Structure of the paper

In Section 2 we collect various results concerning branched covers and spinc-
structures that will needed for the main results. In Section 3 we recall
briefly the details of the equivariant Seiberg–Witten–Floer cohomology groups
needed for this paper. In Section 4 we define the concordance invariants
$\theta^{(q)}$ and prove various properties of them. In Section 5 we give some
examples of computations of the invariant $\theta$. Finally in Section 6 we
prove the slice genus bounds for surfaces in definite $4$-manifolds bounding a
knot $K$.

###  1.2. Acknowledgements

We thank Pedram Hekmati and Hokuto Konno for comments on a draft of this
paper.

##  2\. Branched covers and spinc-structures

In this section $q$ is allowed to be a prime power. In later sections we will
require $q$ to be prime. Let $K\subset S^{3}$ be a knot in $S^{3}$. We denote
by $Y=\Sigma_{q}(K)$ the degree $q$ cyclic cover of $S^{3}$ branched over $K$.
Since $q$ is a prime power it follows that $Y$ is a rational homology
$3$-sphere [10, Corollary 3.2]. Let $\pi:Y\to S^{3}$ denote the covering map
and let $\sigma:Y\to Y$ denote the generator of the natural
$\mathbb{Z}_{q}$-action. From [8, Corollary 2.2], we have that any spinc-
structure on $Y\setminus\pi^{-1}(K)$ uniquely extends to $Y$. Now since
$H^{2}(S^{3}\setminus K;\mathbb{Z})=0$, there is a unique spinc-structure on
$S^{3}\setminus K$. The pullback of this spinc-structure under $\pi$ extends
uniquely to a spinc-structure on $Y$. Following [8], we denote this spinc-
structure by $\mathfrak{s}_{0}=\mathfrak{s}_{0}(K,q)$ and we call
$\mathfrak{s}_{0}$ the distinguished spinc-structure on $Y$. We note here that
$\mathfrak{s}_{0}$ is $\sigma$-invariant. To see this first note that the
restriction of $\mathfrak{s}_{0}$ to $Y\setminus\pi^{-1}(K)$ is
$\sigma$-invariant because it is a pullback from $S^{3}\setminus K$. Then
since spinc-structures extend uniquely over $\pi^{-1}(K)$, we get that
$\mathfrak{s}_{0}$ is $\sigma$-invariant.

Now we consider branched covers of $4$-manifolds with boundary. We will be
concerned specifically with the case that $X$ is a compact, oriented, smooth
$4$-manifold whose boundary is either $S^{3}$ or two copies of $S^{3}$. In the
latter case, we view one copy of $S^{3}$ as an ingoing boundary and the other
as an outgoing boundary so that $X$ is a cobordism from $S^{3}$ to $S^{3}$.
Suppose $\Sigma\subset X$ is a smooth, connected, oriented, properly embedded
surface which meets the outgoing boundary in a knot $K_{1}$ and meets the
ingoing boundary (if there is one) in a knot $K_{0}$.

We assume that $H_{1}(X;\mathbb{Z})=0$. Then $H_{2}(X,\partial
X;\mathbb{Z})\cong H^{2}(X;\mathbb{Z})$ is torsion-free.

######  Lemma 2.1.

Suppose that $H_{1}(X;\mathbb{Z})=0$. If $[\Sigma]\in H_{2}(X,\partial
X;\mathbb{Z})$ is trivial, then
$H_{1}(X\setminus\Sigma;\mathbb{Z})\cong\mathbb{Z}$. If $[\Sigma]\in
H_{2}(X,\partial X;\mathbb{Z})$ is $k$ times a primitive element, where $k\geq
1$, then $H_{2}(X\setminus\Sigma;\mathbb{Z})\cong\mathbb{Z}/k\mathbb{Z}$. In
either case $H_{2}(X\setminus\Sigma;\mathbb{Z})$ is generated by a meridian
around $\Sigma$.

###### Proof.

From the long exact sequence of the pair $(X,X\setminus\Sigma)$ we obtain an
exact sequence

$H_{2}(X;\mathbb{Z})\to
H_{2}(X,X\setminus\Sigma;\mathbb{Z})\buildrel\partial\over{\longrightarrow}H_{1}(X\setminus\Sigma;\mathbb{Z})\to
H_{1}(X;\mathbb{Z})=0.$

Let $N$ be a tubular neighbourhood of $\Sigma$, which we identify with the
open unit disc bundle in the normal bundle of $\Sigma$. By excision and the
Thom isomorphism, $H_{2}(X,X\setminus\Sigma;\mathbb{Z})\cong
H_{2}(N,N\setminus\Sigma;\mathbb{Z})\cong\mathbb{Z}[D]$, where $D$ denotes the
unit disc in a fibre of the normal bundle. The boundary of $[D]$ is a
meridian, so we see that $H_{1}(X\setminus\Sigma;\mathbb{Z})$ is generated by
a meridian. It remains to determine the image of the map
$H_{2}(X;\mathbb{Z})\to H_{2}(X,X\setminus\Sigma;\mathbb{Z})$. Any class $u\in
H_{2}(X;\mathbb{Z})$ can be represented by a compact oriented embedded surface
$S$ in the interior of $X$, meeting $\Sigma$ transversally. The image of
$u=[S]$ in $H_{2}(X,X\setminus\Sigma;\mathbb{Z})$ under the excision
isomorphism is easily seen to be $\langle u,[\Sigma]\rangle[D]$, where
$\langle u,[\Sigma]\rangle$ denotes the intersection pairing between
$H_{2}(X;\mathbb{Z})$ and $H_{2}(X,\partial X;\mathbb{Z})$. This is a dual
pairing, hence the image of $H_{2}(X;\mathbb{Z})\to
H_{2}(X,X\setminus\Sigma;\mathbb{Z})$ is exactly $k[D]$ if $[\Sigma]$ is $k$
times a primitive element and is zero is $[\Sigma]=0$. ∎

Assume that $H_{1}(X;\mathbb{Z})=0$ and that $[\Sigma]$ is divisible by $q$.
By Lemma 2.1, it follows that there is a unique homomorphism
$H_{1}(X\setminus\Sigma;\mathbb{Z})\to\mathbb{Z}_{q}$ sending the meridian to
$1\;({\rm mod}\;q)$. Hence there is a uniquely determined cyclic branched
cover $\pi:W\to X$ of degree $q$ and with branching set $\Sigma$. We let
$\sigma:W\to W$ denote the generator of the natural $\mathbb{Z}_{q}$-action.
If $\partial X=S^{3}$ and $\Sigma$ meets $\partial S^{3}$ in the knot $K_{1}$,
then $\partial W=\Sigma_{q}(K_{1})$ and the $\mathbb{Z}_{q}$-action on $W$
extends the $\mathbb{Z}_{q}$-action on $\Sigma_{q}(K_{1})$. If $\partial
X=S^{3}\cup S^{3}$ and $\Sigma$ meets the ingoing and outgoing boundaries in
$K_{0}$ and $K_{1}$, then $W$ is a $\mathbb{Z}_{q}$-equivariant cobordism from
$\Sigma_{q}(K_{0})$ to $\Sigma_{q}(K_{1})$.

Given a knot $K$, let $\sigma_{K}(\omega)$ denote the Levine–Tristram
signature of $K$ and define

$\sigma^{(q)}(K)=\sum_{j=1}^{q-1}\sigma_{K}(e^{2\pi ij/q}).$

Suppose that $\Sigma\subset D^{4}$ is a smooth, oriented, properly embedded
surface bounding $K$ and let $W\to D^{4}$ be the $q$-fold cyclic branched
cover of $D^{4}$ branched over $\Sigma$. Then the signature of $W$ equals
$\sigma^{(q)}(K)$ [9]. We note here that when $q$ is odd, $\sigma^{(q)}(K)$ is
divisible by $4$ because of the symmetry
$\sigma_{K}(\omega)=\sigma_{K}(\omega^{-1})$ of the Levine–Tristram signature.

######  Lemma 2.2.

Let $X$ be a compact, oriented, smooth $4$-manifold with
$H_{1}(X;\mathbb{Z})=0$ and whose boundary is either $S^{3}$ or two copies of
$S^{3}$. Let $\Sigma\subset X$ be a smooth, connected, oriented, properly
embedded surface of genus $g$ which meets the outgoing boundary in a knot
$K_{1}$ and meets the ingoing boundary (if there is one) in a knot $K_{0}$.
Suppose that $[\Sigma]\in H_{2}(X,\partial X;\mathbb{Z})$ is a multiple of a
prime power $q$. Let $\pi:W\to X$ be the degree $q$ cyclic cover of $X$
branched over $\Sigma$. Then $b_{1}(W)=0$. If $\partial X=S^{3}$, then

$\displaystyle b_{2}(W)$ $\displaystyle=qb_{2}(X)+(q-1)(2g),$
$\displaystyle\sigma(W)$
$\displaystyle=q\sigma(X)-\frac{(q^{2}-1)}{3q}[\Sigma]^{2}+\sigma^{(q)}(K_{1}),$
$\displaystyle b_{+}(W)$
$\displaystyle=qb_{+}(X)+(q-1)g-\frac{(q^{2}-1)}{6q}[\Sigma]^{2}+\frac{\sigma^{(q)}(K_{1})}{2},$
$\displaystyle\quad b_{-}(W)$
$\displaystyle=qb_{-}(X)+(q-1)g+\frac{(q^{2}-1)}{6q}[\Sigma]^{2}-\frac{\sigma^{(q)}(K_{1})}{2}.$

and if $\partial X$ is two copies of $S^{3}$, then

$\displaystyle b_{2}(W)$ $\displaystyle=qb_{2}(X)+(q-1)(2g),$
$\displaystyle\sigma(W)$
$\displaystyle=q\sigma(X)-\frac{(q^{2}-1)}{3q}[\Sigma]^{2}+\sigma^{(q)}(K_{1})-\sigma^{(q)}(K_{0}),$
$\displaystyle b_{+}(W)$
$\displaystyle=qb_{+}(X)+(q-1)g-\frac{(q^{2}-1)}{6q}[\Sigma]^{2}+\frac{\sigma^{(q)}(K_{1})}{2}-\frac{\sigma^{(q)}(K_{0})}{2},$
$\displaystyle\quad b_{-}(W)$
$\displaystyle=qb_{-}(X)+(q-1)g+\frac{(q^{2}-1)}{6q}[\Sigma]^{2}-\frac{\sigma^{(q)}(K_{1})}{2}+\frac{\sigma^{(q)}(K_{0})}{2}.$

###### Proof.

We give the proofs in the case that $\partial X=S^{3}$. The case that
$\partial X$ has two components is similar.

Choose a surface $\Sigma_{1}\subset D^{4}$ bounding $K_{1}$. Let $X^{\prime}$
be the closed $4$-manifold $X^{\prime}=X\cup_{S^{3}}\overline{D^{4}}$, and let
$\Sigma^{\prime}\subset X^{\prime}$ be the closed embedded surface given by
$\Sigma^{\prime}=\Sigma\cup_{K_{1}}\overline{\Sigma_{1}}$. Let $W_{1}\to
D^{4}$ be the degree $q$ cyclic covering of $D^{4}$ branched over
$\Sigma_{1}$. Recall that $\sigma(W_{1})$ is given by $\sigma^{(q)}(K_{1})$.
Let $\pi:W^{\prime}\to X^{\prime}$ be the degree $q$ cyclic cover of
$X^{\prime}$ branched over $\Sigma^{\prime}$. Thus
$W^{\prime}=W\cup_{Y}\overline{W_{1}}$, where $Y=\Sigma_{q}(K_{1})$. Since $Y$
is a rational homology $3$-sphere, the Mayer–Vietoris sequence implies that
$H_{1}(W;\mathbb{Q})\to H_{1}(W^{\prime};\mathbb{Q})$ is injective. But
$b_{1}(W^{\prime})=0$ by [17], hence $b_{1}(W)=0$ as well.

Next we compute the second Betti number of $W$. Since $\pi:W\to X$ is a degree
$q$ covering branched over $\Sigma$, we clearly have

$\chi(W)=q\chi(X)-(q-1)\chi(\Sigma).$

But since $b_{1}(X)=b_{1}(W)=0$ and $\chi(\Sigma)=1-2g$, we get

$b_{2}(W)=qb_{2}(X)+(q-1)(2g).$

Now consider the signature of $W$. Novikov additivity gives
$\sigma(W^{\prime})=\sigma(W)-\sigma(W_{1})=\sigma(W)-\sigma^{(q)}(K_{1})$. On
the other hand the $G$-signature theorem for $G=\mathbb{Z}_{q}$ acting on
$W^{\prime}$ gives

$\sigma(X)=\sigma(X^{\prime})=\frac{\sigma(W^{\prime})}{q}+\frac{q^{2}-1}{3q}[\widetilde{\Sigma^{\prime}}]^{2}$

where $\widetilde{\Sigma^{\prime}}$ is the preimage of $\Sigma^{\prime}$ under
$\pi:W^{\prime}\to X^{\prime}$. The degree of the normal bundle of
$\Sigma^{\prime}$ in $X^{\prime}$ is $q$ times the degree of the normal bundle
of $\widetilde{\Sigma^{\prime}}$ in $W^{\prime}$, so
$[\widetilde{\Sigma^{\prime}}]^{2}=[\Sigma^{\prime}]^{2}/q=[\Sigma]^{2}/q$ and
so

$\sigma(W^{\prime})=q\sigma(X)-\frac{(q^{2}-1)}{3q}[\Sigma]^{2}.$

Thus

$\sigma(W)=q\sigma(X)-\frac{(q^{2}-1)}{3q}[\Sigma]^{2}+\sigma^{(q)}(K_{1}).$

Using $b_{\pm}(W)=(b_{2}(W)\pm\sigma(W))/2$, we get

$b_{+}(W)=qb_{+}(X)+(q-1)g-\frac{(q^{2}-1)}{6q}[\Sigma]^{2}+\frac{\sigma^{(q)}(K_{1})}{2}$

and

$b_{-}(W)=qb_{-}(X)+(q-1)g+\frac{(q^{2}-1)}{6q}[\Sigma]^{2}-\frac{\sigma^{(q)}(K_{1})}{2}.$

∎

Next we turn our attention to spinc-structures on $W$. Let $X$ and $\pi:W\to
X$ be as in Lemma 2.2.

######  Proposition 2.3.

For any spinc-structure $\mathfrak{s}_{X}$ on $X$, there exists a
$\sigma$-invariant spinc-structure $\mathfrak{s}$ on $W$ such that the
restriction of $\mathfrak{s}$ to $W\setminus\pi^{-1}(\Sigma)$ is the pullback
of $\mathfrak{s}_{X}$ restricted to $X\setminus\Sigma$. Moreover, the
restriction of $\mathfrak{s}$ to any boundary component $\Sigma_{q}(K_{i})$ of
$W$ coincides with the distinguished spinc-structure $\mathfrak{s}_{0}$.

###### Proof.

We will construct $\mathfrak{s}$ as an equivariant spinc-structure, by which
we mean that we will construct the corresponding spinor bundles and equip them
with a $\mathbb{Z}_{q}$-action lifting $\sigma$. Let
$\widetilde{\Sigma}=\pi^{-1}(\Sigma)$ be the fixed point set of $\sigma$. Let
$S_{X}^{\pm}\to X$ denote the spinor bundles on $X$ corresponding to the
spinc-structure $\mathfrak{s}_{X}$. Let $S_{0}^{\pm}$ denote the restriction
of $S_{X}^{\pm}$ to $X\setminus\Sigma$ and let $\widetilde{S_{0}}^{\pm}$
denote the pullbacks of $S_{0}^{\pm}$ to $W\setminus\widetilde{\Sigma}$ under
$\pi$. Since $\pi:W\setminus\widetilde{\Sigma}\to X\setminus\Sigma$ is an
unbranched cover, $\widetilde{S_{0}}^{\pm}$ define a spinc-structure
$\mathfrak{s}^{\prime}$ on $W\setminus\widetilde{\Sigma}$. Furthermore, since
$\widetilde{S_{0}}^{\pm}$ are obtained by pullback, they carry a natural
action of $\mathbb{Z}_{q}$ which makes $\mathfrak{s}^{\prime}$ into an
equivariant spinc-structure.

We will show that $\mathfrak{s}^{\prime}$ extends equivariantly over
$\widetilde{\Sigma}$. Choose a $\sigma$-invariant tubular neighbourhood
$\widetilde{N}$ of $\widetilde{\Sigma}$, which we may identify with the normal
bundle of $\widetilde{\Sigma}$. Let $N=\pi(\widetilde{N})$ be the image of
$\widetilde{N}$ under $\pi$. Then $N$ is a tubular neighbourhood of $\Sigma$
which can be identified with the normal bundle of $\Sigma$. Then we have an
unbranched covering $\widetilde{N}\setminus\widetilde{\Sigma}\to
N\setminus\Sigma$.

Choose a $\sigma$-invariant metric $g$ on $W$. Then we obtain a
$\sigma$-equivariant isomorphism $T(\widetilde{N})\cong
T\widetilde{\Sigma}\oplus\widetilde{N}$. This is a direct sum of
$\sigma$-equivariant complex line bundles, so has a $\sigma$-equivariant
complex structure. Associated to this complex structure is
$\sigma$-equivariant spinc-structure which we denote by
$\mathfrak{s}_{\widetilde{N}}$. If we can show that the restriction of
$\mathfrak{s}_{\widetilde{N}}$ to $\widetilde{N}\setminus\widetilde{\Sigma}$
is equivariantly isomorphic to the restriction of $\mathfrak{s}^{\prime}$ to
$\widetilde{N}\setminus\widetilde{\Sigma}$, then it follows that
$\mathfrak{s}^{\prime}$ extends equivariantly over $\widetilde{\Sigma}$, since
we can glue $\mathfrak{s}^{\prime}$ and $\mathfrak{s}_{\widetilde{N}}$
together over $\widetilde{N}\setminus\widetilde{\Sigma}$ using this
isomorphism.

Now since $\sigma$ acts freely on $\widetilde{N}\setminus\widetilde{\Sigma}$
with quotient space $N\setminus\Sigma$, we have that any equivariant spinc-
structure on $\widetilde{N}\setminus\widetilde{\Sigma}$ descends to a spinc-
structure on $N\setminus\Sigma$. Moreover any two equivariant spinc-structures
on $\widetilde{N}\setminus\widetilde{\Sigma}$ are equivariantly isomorphic if
and only if the corresponding descended spinc-structures on $N\setminus\Sigma$
are isomorphic.

Consider the restriction of $\mathfrak{s}^{\prime}$ to
$\widetilde{N}\setminus\widetilde{\Sigma}$. This is the pullback of
$\mathfrak{s}_{X}|_{N\setminus\Sigma}$, hence it descends to
$\mathfrak{s}_{X}|_{N\setminus\Sigma}$. Note that this spinc-structure extends
to $N$ as $\mathfrak{s}_{X}|_{N}$. Next consider the restriction of
$\mathfrak{s}_{\widetilde{N}}$ to $\widetilde{N}\setminus\widetilde{\Sigma}$.
Since $N=\widetilde{N}/\sigma$, it is easily seen that
$\mathfrak{s}_{\widetilde{N}}$ descends to the spinc-structure on
$N\setminus\Sigma$ associated to the complex structure on
$T(N\setminus\Sigma)\cong T\Sigma\oplus N$. Since this isomorphism extends
over $\Sigma$ to an isomorphism $T(N)\cong T\Sigma\oplus N$, it follows that
the spinc-structure obtained by descending $\mathfrak{s}_{\widetilde{N}}$ to
$N\setminus\Sigma$ extends to a spinc-structure on $N$.

Lastly, since $N$ deformation retracts to $\Sigma$ and
$H^{2}(\Sigma;\mathbb{Z})=0$ (because $\Sigma$ has non-empty boundary), it
follows that there is a unique spinc-structure on $N$. Hence the spinc-
structures on $N\setminus\Sigma$ obtained by descending
$\mathfrak{s}^{\prime}$ and $\mathfrak{s}_{\widetilde{N}}$ to
$N\setminus\Sigma$ are isomorphic, since they both extend to the unique spinc-
structure on $N$. This proves the claim that $\mathfrak{s}^{\prime}$ extends
equivariantly to a spinc-structure on $W$, which we denote by $\mathfrak{s}$.
Since $\mathfrak{s}$ is an equivariant spinc-structure, its isomorphism class
is $\sigma$-invariant.

It remains to show that the restriction of $\mathfrak{s}$ to any boundary
component $\Sigma_{q}(K_{i})$ of $W$ coincides with the distinguished spinc-
structure $\mathfrak{s}_{0}$. From the construction of $\mathfrak{s}$, it
follows that the restriction of $\mathfrak{s}$ to
$\Sigma_{q}(K_{i})\setminus\pi^{-1}(K_{i})$ is the pullback of the unique
spinc-structure on $S^{3}\setminus K_{i}$. But this property uniquely
characterises the distinguished spinc-structure $\mathfrak{s}_{0}$, so the
restriction of $\mathfrak{s}$ to $\Sigma_{q}(K_{i})$ is the distinguished
spinc-structure. ∎

######  Lemma 2.4.

The pullback $\pi^{*}:H^{2}(X;\mathbb{Z})\to H^{2}(W;\mathbb{Z})$ is injective
and the image is precisely the $\sigma$-invariant elements of
$H^{2}(W;\mathbb{Z})$.

###### Proof.

For $a\in H^{2}(X;\mathbb{Z})$, $\pi_{*}(\pi^{*}a)=qa$. Injectivity of
$\pi^{*}$ follows, since $H^{2}(X;\mathbb{Z})$ is torsion free.

To prove that the image of $\pi^{*}$ is precisely the $\sigma$-invariant
elements of $H^{2}(W;\mathbb{Z})$ we will compute the degree $2$ equivariant
cohomology of $W$ in two ways. First using the Leray–Serre spectral sequence
for the Borel fibration. This spectral sequence has
$E_{2}^{p,q}=H^{p}(BG;H^{q}(W;\mathbb{Z}))$ where $G=\mathbb{Z}_{q}$ and abuts
to the equivariant cohomology $H^{*}_{G}(W;\mathbb{Z})$. Since $b_{1}(W)=0$,
we have $H^{1}(W;\mathbb{Z})=0$ and so the only non-zero terms in
$E_{2}^{p,q}$ with $p+q=2$ are $E_{2}^{0,2}=H^{2}(W;\mathbb{Z})^{G}$ and
$E_{2}^{2,0}=H^{2}(BG;\mathbb{Z})=H^{2}_{G}(pt;\mathbb{Z})\cong\mathbb{Z}_{q}$.
Furthermore $E_{2}^{0,1}=E_{2}^{2,1}=E_{2}^{3,0}=0$, so there are no non-zero
differentials out of $E_{r}^{0,2}$ or in to $E_{r}^{2,0}$ for any $r$. So we
get an exact sequence

$0\to H^{2}_{G}(pt;\mathbb{Z})\to H^{2}_{G}(W;\mathbb{Z})\to
H^{2}(W;\mathbb{Z})^{G}\to 0.$

Furthermore, the action of $G$ on $W$ has fixed points. Taking a fixed point
$w\in W$ gives a splitting

$H^{2}_{G}(W;\mathbb{Z})\cong H^{2}(W;\mathbb{Z})^{G}\oplus
H^{2}_{G}(w;\mathbb{Z})$

and it follows that we have an isomorphism

$H^{2}_{G}(W,w;\mathbb{Z})\cong H^{2}(W;\mathbb{Z})^{G}.$

Next, we compute the degree $2$ equivariant cohomology using the
Mayer–Vietoris sequence. Let $G$ act trivially on $X$ so that $\pi:W\to X$ may
be regarded as an equivariant map. Consider the open cover of $X$ by
$X\setminus\Sigma$, $N$, where $N$ is a tubular neighbourhood of $\Sigma$. Let
$\widetilde{\Sigma}=\pi^{-1}(\Sigma)$ and $\widetilde{N}=\pi^{-1}(N)$. Then
$W\setminus\widetilde{\Sigma}$, $\widetilde{N}$ is an open $G$-invariant cover
of $W$. Since the cover of $W$ is obtained by pulling back the cover of $X$ we
get a commutative diagram between Mayer–Vietors sequences (all cohomology
groups having coefficients in $\mathbb{Z}$):

$\textstyle{H^{1}_{G}(\widetilde{\Sigma})\\!\oplus\\!H^{1}_{G}(W\\!\setminus\\!\widetilde{\Sigma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\\!}$$\scriptstyle{\tilde{i}}$$\textstyle{\\!H^{1}_{G}(\widetilde{N}\\!\setminus\\!\widetilde{\Sigma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\\!}$$\textstyle{\\!H^{2}_{G}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\\!}$$\textstyle{\\!H^{2}_{G}(\widetilde{\Sigma})\\!\oplus\\!H^{2}_{G}(W\\!\setminus\\!\widetilde{\Sigma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\\!}$$\scriptstyle{\tilde{j}}$$\textstyle{\\!H^{2}_{G}(\widetilde{N}\\!\setminus\\!\widetilde{\Sigma})}$$\textstyle{H^{1}_{G}(\Sigma)\\!\oplus\\!H^{1}_{G}(X\\!\setminus\\!\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\\!}$$\scriptstyle{i}$$\scriptstyle{\pi^{*}}$$\textstyle{\\!H^{1}_{G}(N\\!\setminus\\!\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\\!}$$\scriptstyle{\pi^{*}}$$\textstyle{\\!H^{2}_{G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\\!}$$\scriptstyle{\pi^{*}}$$\textstyle{\\!H^{2}_{G}(\Sigma)\\!\oplus\\!H^{2}_{G}(X\\!\setminus\\!\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\\!}$$\scriptstyle{j}$$\scriptstyle{\pi^{*}}$$\textstyle{\\!H^{2}_{G}(N\\!\setminus\\!\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$

We claim that $\pi^{*}$ induces isomorphisms $coker(\tilde{i})\cong coker(i)$
and $ker(\tilde{j})\cong ker(j)$. These claims imply that
$\pi^{*}:H^{2}_{G}(X)\to H^{2}_{G}(W)$ is an isomorphism. Consider first $i$
and $\tilde{i}$. We have that $\pi:\widetilde{\Sigma}\to\Sigma$ is a
homeomorphism. Also $G$ acts freely on
$\widetilde{W}\setminus\widetilde{\Sigma}$ and
$\widetilde{N}\setminus\widetilde{\Sigma}$ with quotient spaces
$X\setminus\Sigma$ and $N\setminus\Sigma$. Furthermore
$H^{1}_{G}(pt;\mathbb{Z})=0$ so it follows that the first two vertical maps in
the above diagram are isomorphisms. This shows that $\pi^{*}:coker(i)\to
coker(\tilde{i})$ is an isomorphism. Similarly the last two vertical maps fit
into a commutative diagram with exact columns:

$\textstyle{0}$$\textstyle{0}$$\textstyle{H^{2}_{G}(\widetilde{\Sigma})\oplus
H^{2}_{G}(W\setminus\widetilde{\Sigma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{j}}$$\textstyle{H^{2}_{G}(\widetilde{N}\setminus\widetilde{\Sigma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}_{G}(\Sigma)\oplus
H^{2}_{G}(X\setminus\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\scriptstyle{j}$$\textstyle{H^{2}_{G}(N\setminus\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\textstyle{H^{2}_{G}(pt)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(0,\iota)}$$\scriptstyle{\cong}$$\textstyle{H^{2}_{G}(pt)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $\iota:H^{2}_{G}(pt)\to H^{2}_{G}(X\setminus\Sigma)$ and
$\iota:H^{2}_{G}(pt)\to H^{2}(N\setminus\Sigma)$ are the maps induced by
$X\setminus\Sigma\to pt$ and $N\setminus\Sigma\to pt$. It follows that
$\pi^{*}:ker(j)\to ker(\tilde{j})$ is an isomorphism.

We have proven that $\pi^{*}:H^{2}_{G}(X;\mathbb{Z})\to
H^{2}_{G}(W;\mathbb{Z})$ is an isomorphism. Let $w\in W$ be a fixed point and
$x=\pi(w)$. Then we get an isomorphism $\pi^{*}:H^{2}(X,x;\mathbb{Z})\to
H^{2}(W,w;\mathbb{Z})$. But $H^{2}_{G}(X,x;\mathbb{Z})\cong
H^{2}(X;\mathbb{Z})$ via the forgetful map from equivariant to non-equivariant
cohomology and similarly $H^{2}_{G}(W,w;\mathbb{Z})\cong H^{2}(W)^{G}$ via the
forgetful map (corresponding to the $p=0$ column of the spectral sequence
$E_{r}^{p,q}$). Hence $\pi^{*}:H^{2}(X;\mathbb{Z})\to H^{2}(W;\mathbb{Z})^{G}$
is an isomorphism. ∎

Since each boundary component of $X$ is a $3$-sphere, we have
$H^{2}(X;\mathbb{Z})$ $\cong$ $H^{2}(X,\partial X;\mathbb{Z})$ and that the
intersection form on $H^{2}(X,\partial X;\mathbb{Z})$ is unimodular.
Furthermore $H_{1}(X;\mathbb{Z})=0$ implies that $H^{2}(X;\mathbb{Z})$ and
$H^{2}(X,\partial X;\mathbb{Z})$ are torsion-free. So spinc-structures on $X$
are in bijection with characteristics $c\in H^{2}(X;\mathbb{Z})$, that is,
elements such that $\langle c,x\rangle=\langle x,x\rangle\;({\rm mod}\;2)$ for
every $x\in H^{2}(X,\partial X;\mathbb{Z})$. Now we can give an improved
version of Proposition 2.3.

######  Proposition 2.5.

If $q$ is odd, then for any characteristic $c\in H^{2}(X;\mathbb{Z})$, there
is a $\sigma$-invariant spinc-structure $\mathfrak{s}$ on $W$ such that
$c_{1}(\mathfrak{s})=\pi^{*}(c)$ in $H^{2}(W;\mathbb{Q})$ and such that
$\mathfrak{s}$ restricts to the distinguished spinc-structure on each
component of $\partial W$.

If $q=2$, then for any characteristic $c\in H^{2}(X;\mathbb{Z})$, there is a
$\sigma$-invariant spinc-structure $\mathfrak{s}$ on $W$ such that
$c_{1}(\mathfrak{s})=\pi^{*}(c+[\Sigma]/2)$ in $H^{2}(W;\mathbb{Q})$ and such
that $\mathfrak{s}$ restricts to the distinguished spinc-structure on each
component of $\partial W$.

###### Proof.

Let $q$ be odd. Let $\mathfrak{s}_{X}$ be any spinc-structure on $X$. Then
from Proposition 2.3, there exists a $\sigma$-invariant spinc-structure
$\mathfrak{s}$ on $W$ which restricts to $\mathfrak{s}_{0}$ on each component
of $\partial W$. Then $c_{1}(\mathfrak{s})$ is $\sigma$-invariant, so by Lemma
2.4 we have $c_{1}(\mathfrak{s})=\pi^{*}(c)$ for some $c\in
H^{2}(X;\mathbb{Z})$. We claim that $c$ is a characteristic for
$H^{2}(X;\mathbb{Z})$. To see this, let $a\in H^{2}(X;\mathbb{Z})$. Then using
the fact that $c_{1}(\mathfrak{s})$ is a characteristic, we get

$\displaystyle q\langle c,a\rangle$
$\displaystyle=\langle\pi^{*}(c),\pi^{*}(a)\rangle$ $\displaystyle=\langle
c_{1}(\mathfrak{s}),\pi^{*}(a)\rangle$
$\displaystyle=\langle\pi^{*}(a),\pi^{*}(a)\rangle\;({\rm mod}\;2)$
$\displaystyle=q\langle a,a\rangle\;({\rm mod}\;2).$

Since $q$ is odd, we deduce that $\langle c,a\rangle=\langle a,a\rangle\;({\rm
mod}\;2)$, so $c$ is a characteristic. Now given any other characteristic
$d\in H^{2}(X;\mathbb{Z})$, we have that $d=c+2u$ for some $u\in
H^{2}(X;\mathbb{Z})$. Let $L\to X$ be a line bundle with $c_{1}(L)=u$. Then
$\mathfrak{s}_{L}=L\otimes\mathfrak{s}$ is a $\sigma$-invariant spinc-
structure on $W$ with
$c_{1}(\mathfrak{s}_{L})=c_{1}(\mathfrak{s})+2\pi^{*}(c_{1}(L))=\pi^{*}(c+2u)=\pi^{*}(d)$.
Also, $L$ restricted to the boundary of $X$ is trivial (since
$H^{2}(S^{3};\mathbb{Z})=0$), so $\mathfrak{s}_{L}$ is isomorphic to
$\mathfrak{s}$ on the boundary of $W$ and hence $\mathfrak{s}_{L}$ restricts
to the distinguished spinc-structure on each component of $\partial W$.

Now consider the case $q=2$. As in the odd case, let $\mathfrak{s}_{X}$ be any
spinc-structure on $X$. Then from Proposition 2.3, there exists a
$\sigma$-invariant spinc-structure $\mathfrak{s}$ on $W$ which restricts to
$\mathfrak{s}_{0}$ on each component of $\partial W$. Then
$c_{1}(\mathfrak{s})$ is $\sigma$-invariant, so by Lemma 2.4 we have that
$c_{1}(\mathfrak{s})=\pi^{*}(y)$ for some $y\in H^{2}(X;\mathbb{Z})$. Set
$c=y-[\Sigma]/2$ so that $c_{1}(\mathfrak{s})=\pi^{*}(c+[\Sigma]/2)$.

We claim that $c$ is a characteristic for $H^{2}(X;\mathbb{Z})$. From [13,
Lemma 3.4] we have
$w_{2}(W)=\pi^{*}(w_{2}(X))+[\widetilde{\Sigma}]=\pi^{*}(w_{2}(X)+[\Sigma]/2)$.
Let $c^{\prime}\in H^{2}(X;\mathbb{Z})$ be a characteristic. Then
$c^{\prime}=w_{2}(X)\;({\rm mod}\;2)$ and hence
$\pi^{*}(c^{\prime}+[\Sigma]/2)=w_{2}(W)\;({\rm mod}\;2)$. So
$\pi^{*}(c^{\prime}+[\Sigma]/2)$ is a characteristic for
$H^{2}(W;\mathbb{Z})$. But $\pi^{*}(c+[\Sigma]/2)$ is also a characteristic,
so it follows that $\pi^{*}(c)=\pi^{*}(c^{\prime})\;({\rm mod}\;2)$. So
$\pi^{*}(c)=\pi^{*}(c^{\prime})+2w$ for some $w\in H^{2}(W;\mathbb{Z})$.
Applying $\pi_{*}$ we get $2c=2c^{\prime}+2\pi_{*}(w)$. But
$H^{2}(X;\mathbb{Z})$ is torsion free, so $c=c^{\prime}+\pi_{*}(w)$. Applying
$\pi^{*}$ and using $\pi^{*}(\pi_{*}(w))=w+\sigma^{*}(w)$, we see that
$\pi^{*}(c)=\pi^{*}(c^{\prime})+w+\sigma^{*}(w)$. Comparing this with
$\pi^{*}(c)=\pi^{*}(c^{\prime})+2w$, we see that $\sigma^{*}(w)=w$. So by
Lemma 2.4, we have that $w=\pi^{*}(v)$ for some $v\in H^{2}(X;\mathbb{Z}$.
Hence $\pi^{*}(c)=\pi^{*}(c^{\prime}+2v)$. By Lemma 2.4, $\pi^{*}$ is
injective and hence $c=c^{\prime}+2v$. This proves the claim that $c$ is a
characteristic for $H^{2}(X;\mathbb{Z})$, because $c^{\prime}$ is a
characteristic for $H^{2}(X;\mathbb{Z})$.

The rest of the argument works the same as the odd case. Given any other
characteristic $d\in H^{2}(X;\mathbb{Z})$, we have that $d=c+2u$ for some
$u\in H^{2}(X;\mathbb{Z})$. Let $L\to X$ be a line bundle with $c_{1}(L)=u$.
Then $\mathfrak{s}_{L}=L\otimes\mathfrak{s}$ is a $\sigma$-invariant spinc-
structure on $W$ with
$c_{1}(\mathfrak{s}_{L})=c_{1}(\mathfrak{s})+2\pi^{*}(c_{1}(L))=\pi^{*}(c+2u+[\Sigma]/2)=\pi^{*}(d+[\Sigma]/2)$.
Also, $L$ restricted to the boundary of $X$ is trivial, so $\mathfrak{s}_{L}$
restricts to the distinguished spinc-structure on each component of $\partial
W$. ∎

##  3\. Equivariant Seiberg–Witten–Floer cohomology and equivariant
$d$-invariants

Let $q$ be a prime and let $G$ be the cyclic group $G=\mathbb{Z}_{q}$. Let
$\sigma$ denote a generator of $G$. Let $Y$ be a rational homology $3$-sphere
and $\mathfrak{s}$ a spinc-structure on $Y$. Suppose that $\mathbb{Z}_{q}$
acts on $Y$ by orientation preserving diffeomorphisms and that $\mathfrak{s}$
is invariant under this action. In [1], the author and Hekmati constructed the
equivariant Seiberg–Witten–Floer cohomology groups
$HSW^{*}_{G}(Y,\mathfrak{s})$ (the construction works more generally for any
finite group action, but we will only need the case that $G$ is cyclic of
prime order). Up to a degree shift, $HSW^{*}_{G}(Y,\mathfrak{s})$ is equal to
the $S^{1}\times G$-equivariant cohomology of the Conley index
$I(Y,\mathfrak{s})$ of a finite dimensional approximation of the
Chern–Simons–Dirac gradient flow. We take cohomology with respect to the
coefficient group $\mathbb{F}=\mathbb{Z}_{q}$. Thus
$HSW^{*}_{G}(Y,\mathfrak{s})$ is a module over the ring $R=H^{*}_{S^{1}\times
G}(pt;\mathbb{F})$. If $q=2$, then $R\cong\mathbb{F}[U,Q]$, where $deg(U)=2$,
$deg(Q)=1$. If $q$ is odd, then $R\cong\mathbb{F}[U,R,S]/(R^{2})$, where
$deg(U)=2$, $deg(R)=1$, $deg(S)=2$.

The localisation theorem in equivariant cohomology implies that the
localisation $U^{-1}HSW^{*}_{G}(Y,\mathfrak{s})$ is a free $U^{-1}R$-module of
rank $1$. Thus we have an isomorphism of the form

$\iota:U^{-1}HSW^{*}_{G}(Y,\mathfrak{s})\to U^{-1}R\tau$

for some element $\tau$. We then define a sequence of invariants, which are to
be thought of as an equivariant analogue of the Ozsváth–Szabó $d$-invariant
$d(Y,\mathfrak{s})$. These are defined as follows. If $q=2$, we set
$d_{G,Q^{j}}(Y,\mathfrak{s})=i-j$, where $i$ is the least degree for which
there exists an element $x\in HSW^{i}_{G}(Y,\mathfrak{s})$ and a
$k\in\mathbb{Z}$ such that

$\iota x=Q^{j}U^{k}\tau\;({\rm mod}\;Q^{j+1})$

(cf [1, §3.6]). If $q$ is odd, we set $d_{G,S^{j}}(Y,\mathfrak{s})=i-2j$,
where $i$ is the least degree for which there exists an element $x\in
HSW^{i}_{G}(Y,\mathfrak{s})$ and a $k\in\mathbb{Z}$ such that

$\iota x=S^{j}U^{k}\tau\;({\rm mod}\;S^{j+1})$

For convenience, we also define equivariant $\delta$-invariants which are
related to the $d$-invariants by

$\delta_{G,Q^{j}}(Y,\mathfrak{s})=\frac{1}{2}d_{G,Q^{j}}(Y,\mathfrak{s})$

if $q=2$ and

$\delta_{G,S^{j}}(Y,\mathfrak{s})=\frac{1}{2}d_{G,S^{j}}(Y,\mathfrak{s})$

if $q$ is odd.

Most of the properties of the $d$-invariants that we need are proven in [1],
however we also need a result concerning equivariant connected sums.

Let $Y_{1},Y_{2}$ be rational homology $3$-spheres. Suppose that
$G=\mathbb{Z}_{q}$ acts smoothly and orientation preservingly on $Y_{1},Y_{2}$
and that the action has non-empty fixed point sets. Since $G$ acts orientation
preservingly, the fixed points sets $F_{1},F_{2}$ must be $1$-dimensional. For
$i=1,2$, let $N_{i}$ denote the normal bundle of $F_{i}$ in $Y_{i}$. Let
$y_{1}\in Y_{1}$, $y_{2}\in Y_{2}$ be fixed points. Assume that the action of
$G$ on the normal spaces $(N_{1})_{y_{1}}$, $(N_{2})_{y_{2}}$ are isomorphic
as representations of $G$ by an orientation preserving isomorphism $\varphi$.
Then a neighbourhood of $y_{i}$ in $Y_{i}$ takes the form
$(-1,1)\times(N_{i})_{y_{i}}$ where $G$ acts trivially on the first factor.
Then $(t,x)\mapsto(-t,\varphi(x))$ defines a $G$-equivariant orientation
reversing diffeomorphism from $(-1,1)\times(N_{1})_{y_{1}}$ to
$(-1,1)\times(N_{2})_{y_{2}}$. Thus by removing $G$-invariant neighbourhoods
of $y_{1},y_{2}$ and identifying their boundaries via this map, we can form
the $G$-equivariant connected sum $Y=Y_{1}\\#Y_{2}$.

Let $\mathfrak{s}_{1},\mathfrak{s}_{2}$ be $G$-invariant spinc-structures on
$Y_{1},Y_{2}$ and set $\mathfrak{s}=\mathfrak{s}_{1}\\#\mathfrak{s}_{2}$.

######  Proposition 3.1.

For all $i,j\geq 0$, we have

$d_{G,Q^{i+j}}(Y,\mathfrak{s})\leq
d_{G,Q^{i}}(Y_{1},\mathfrak{s}_{1})+d_{G,Q^{j}}(Y_{2},\mathfrak{s}_{2})$

if $q=2$, and

$d_{G,S^{i+j}}(Y,\mathfrak{s})\leq
d_{G,S^{i}}(Y_{1},\mathfrak{s}_{1})+d_{G,S^{j}}(Y_{2},\mathfrak{s}_{2}).$

if $q$ is odd.

###### Proof.

We give the proof in the $q=2$ case. The case where $q$ is odd is similar. In
the $q=2$ case we have $R\cong\mathbb{F}[U,Q]$. For an equivariant connected
sum, one has that the Conley indices are related by $I(Y,\mathfrak{s})\cong
I(Y_{1},\mathfrak{s}_{1})\wedge I(Y_{2},\mathfrak{s}_{2})$ and this defines a
product map on cohomology groups

(3.1)
$\mu:HSW^{i}_{G}(Y_{1},\mathfrak{s}_{1})\otimes_{\mathbb{F}[U,Q]}HSW^{j}_{G}(Y_{2},\mathfrak{s}_{2})\to
HSW^{i+j}_{\mathbb{Z}_{2}}(Y,\mathfrak{s}).$

Recall that we have localisation isomorphisms

$\iota_{1}:U^{-1}HSW^{*}_{G}(Y_{1},\mathfrak{s}_{1})\to\mathbb{F}[U,U^{-1},Q]\tau_{1}$

and

$\iota_{2}:U^{-1}HSW^{*}_{G}(Y_{2},\mathfrak{s}_{2})\to\mathbb{F}[U,U^{-1},Q]\tau_{2}$

for some $\tau_{1},\tau_{2}$. One can show that
$I(Y,\mathfrak{s})^{S^{1}}\cong I(Y_{1},\mathfrak{s}_{1})^{S^{1}}\wedge
I(Y_{2},\mathfrak{s}_{2})^{S^{1}}$ and from this it follows that the
localisation of the product map (3.1) is an isomorphism. So letting
$\iota:U^{-1}HSW^{*}_{G}(Y,\mathfrak{s})\to\mathbb{F}[U,U^{-1},Q]\tau$ denote
the localisation map corresponding to $(Y,\mathfrak{s})$, it follows that we
must have

$\iota(\mu(\iota_{1}^{-1}(\tau_{1})\otimes\iota_{2}^{-1}(\tau_{2})))=cU^{k}\tau\;({\rm
mod}\;Q)$

for some $c\in\mathbb{F}\setminus\\{0\\}$ and some $k\in\mathbb{Z}$.

Now set $a_{1}=d_{G,Q^{i}}(Y_{1},\mathfrak{s}_{1})+i$,
$a_{2}=d_{G,Q^{j}}(Y_{2},\mathfrak{s}_{2})+j$. Then by definition of the
$d$-invariants there exists $x_{1}\in HSW^{a_{1}}_{G}(Y_{1},\mathfrak{s}_{1})$
such that $\iota_{1}x_{1}=Q^{i}U^{k_{1}}\tau_{1}\;({\rm mod}\;Q^{i+1})$ and
$\iota_{2}x_{2}=Q^{j}U^{k_{2}}\tau_{2}\;({\rm mod}\;Q^{j+1})$ for some
$k_{1},k_{2}$. It follows that

$\iota(\mu(c^{-1}x_{1},x_{2}))=Q^{i+j}U^{k_{1}+k_{2}+k}\tau\;({\rm
mod}\;Q^{i+j+1}).$

But $\mu(c^{-1}x_{1},x_{2})\in HSW^{a_{1}+a_{2}}_{G}(Y,\mathfrak{s})$, so from
the definition of $d_{G,Q^{i+j}}(Y,\mathfrak{s})$, we get

$d_{G,Q^{i+j}}(Y,\mathfrak{s})\leq
a_{1}+a_{2}-i-j=d_{G,Q^{i}}(Y_{1},\mathfrak{s}_{1})+d_{G,Q^{j}}(Y_{2},\mathfrak{s}_{2}).$

∎

##  4\. Concordance invariants from equivariant Seberg–Witten–Floer cohomology

Let $K\subset S^{3}$ be a knot. Let $\sigma(K)$ and $g_{4}(K)$ denote the
signature and smooth $4$-genus of $K$. In [1], the author and Hekmati
constructed a sequence of integer-valued knot concordance invariants
$\\{\delta_{j}(K)\\}_{j\geq 0}$ with the following properties (see [1, §6]):

  * (1)

$\delta_{0}(K)\geq\delta(K)$, where $\delta(K)$ is the Manolescu–Owens
invariant [12].

  * (2)

$\delta_{j+1}(K)\leq\delta_{j}(K)$ for all $j\geq 0$.

  * (3)

$\delta_{j}(K)\geq-\sigma(K)/2$ for all $j\geq 0$ and
$\delta_{j}(K)=-\sigma(K)/2$ for $j\geq g_{4}(K)-\sigma(K)/2$.

  * (4)

$\delta_{j}(K)+\delta_{j}(-K)\geq 0$ for all $j\geq 0$.

  * (5)

$\delta_{j}(K)=-\sigma(K)/2\;({\rm mod}\;4)$ for all $j\geq 0$.

  * (6)

If $K$ is quasi-alternating, then $\delta_{j}(K)=-\sigma(K)/2$ for all $j\geq
0$.

The invariant $\delta_{j}(K)$ is defined by

$\delta_{j}(K)=2d_{\mathbb{Z}_{2},Q^{j}}(\Sigma_{2}(K),\mathfrak{s}_{0})=4\delta_{\mathbb{Z}_{2},Q^{j}}(\Sigma_{2}(K),\mathfrak{s}_{0}).$

For convenience we define a new set of concordance invariants $\xi_{j}(K)$ by
setting

$\xi_{j}(K)=\frac{1}{4}\delta_{j}(K)+\frac{1}{8}\sigma(K).$

Then the above properties of $\delta_{j}(K)$ imply that $\xi_{j}(K)$ is
integer-valued, decreasing and is zero for $j\geq g_{4}(K)-\sigma(K)/2$.

######  Proposition 4.1.

Let $K_{+},K_{-}$ be knots where $K_{-}$ is obtained from $K_{+}$ by changing
a positive crossing into a negative crossing. Then
$\sigma(K_{-})-\sigma(K_{+})=0$ or $2$, and:

  * (1)

If $\sigma(K_{-})=\sigma(K_{+})$, then
$\xi_{j+1}(K_{-})\leq\xi_{j}(K_{+})\leq\xi_{j}(K_{-})$ for all $j\geq 0$.

  * (2)

If $\sigma(K_{-})=\sigma(K_{+})+2$, then
$\xi_{j+1}(K_{+})\leq\xi_{j}(K_{-})\leq\xi_{j}(K_{+})$ for all $j\geq 0$.

###### Proof.

Resolving the singularity of a regular homotopy associated to the crossing
change gives a genus $1$ surface $\Sigma$ properly embedded in $X=[0,1]\times
S^{3}$ which meets the ingoing boundary of $X$ in $K_{-}$ and the outgoing
boundary in $K_{+}$. Let $W\to X$ be the double cover of $X$ branched over
$\Sigma$. This gives a $\mathbb{Z}_{2}$-equivariant cobordism from
$\Sigma_{2}(K_{-})$ to $\Sigma_{2}(K_{+})$ where $\mathbb{Z}_{2}$ acts by
covering transformation. By Proposition 2.5, we can choose a
$\mathbb{Z}_{2}$-invariant spinc-structure $\mathfrak{s}$ on $W$ which
restricts to the distinguished spinc-structure on either boundary and has
$c_{1}(\mathfrak{s})=0$. By Lemma 2.2, we find
$\sigma(W)=\sigma(K_{+})-\sigma(K_{-})$ and $b_{2}(W)=2$. Thus

$b_{+}(W)=1+\frac{\sigma(K_{+})-\sigma(K_{-})}{2}.$

Recall that for a single crossing change one has
$\sigma(K_{+})-\sigma(K_{-})\in\\{0,-2\\}$. If $\sigma(K_{+})=\sigma(K_{-})$,
then $b_{+}(W)=b_{-}(W)=1$. In this case the equivariant Froyshov inequality
[1, Theorem 5.3] applied to $(W,\mathfrak{s})$ gives

$-\frac{\sigma(W)}{2}+\delta_{j+1}(K_{-})\leq\delta_{j}(K_{+}).$

Since $\sigma(W)=\sigma(K_{+})-\sigma(K_{-})$, this can be rewritten as

$\xi_{j+1}(K_{-})\leq\xi_{j}(K_{+}).$

Next, consider the case $\sigma(K_{+})=\sigma(K_{-})-2$. Then $b_{+}(W)=0$ and
$b_{-}(W)=2$. In this case the equivariant Froyshov inequalities for $W$ and
$\overline{W}$ give

$\xi_{j}(K_{-})\leq\xi_{j}(K_{+}).$

Consider again a regular homotopy associated to the crossing change. This
gives an immersed surface in $[0,1]\times S^{3}$ which meets the ingoing
boundary in $K_{-}$, the outgoing boundary in $K_{+}$ and has a single
transverse point of self-intersection with self-intersection $-1$. Instead of
resolving the crossing, we take a blow-up. This gives a genus zero surface
$\Sigma$ properly embedded in $X$, where $X$ is $\mathbb{CP}^{2}$ with two
balls removed and $\Sigma$ meets the ingoing boundary of $X$ in $K_{-}$ and
the outgoing boundary in $K_{+}$. Let $\pi:W\to X$ be the double cover of $X$
branched over $\Sigma$. We again obtain a $\mathbb{Z}_{2}$-equivariant
cobordism from $\Sigma_{2}(K_{-})$ to $\Sigma_{2}(K_{+})$. By Proposition 2.5,
we can choose a $\mathbb{Z}_{2}$-invariant spinc-structure $\mathfrak{s}$ on
$W$ such that $\mathfrak{s}$ restricts to the distinguished spinc-structure on
each boundary and $c_{1}(\mathfrak{s})^{2}=2$ (take $c$ in Proposition 2.5 to
be a characteristic satisfying $c^{2}=1$). Lemma 2.2 gives
$\sigma(W)=\sigma(K_{+})-\sigma(K_{-})+2$ and $b_{2}(W)=2$.

Note that $H^{+}(\overline{W})^{\mathbb{Z}_{2}}\cong
H^{-}(W)^{\mathbb{Z}_{2}}\cong H^{-}(X)=0$. Hence reversing orientation on $W$
and swapping ingoing and outgoing boundaries, we may apply the Froyshov
inequality. If $\sigma(K_{+})=\sigma(K_{-})$, then $b_{+}(W)=2$, $b_{-}(W)=0$
and the equivariant Froyshov inequality applied to $\overline{W}$ gives:

$\frac{\sigma(W)-c_{1}(\mathfrak{s})^{2}}{2}+\delta_{j}(K_{+})\leq\delta_{j}(K_{-}).$

Since $\sigma(W)=\sigma(K_{+})-\sigma(K_{-})-2$ and
$c_{1}(\mathfrak{s})^{2}=-2$, this simplifies to

$\frac{\sigma(K_{+})}{2}-\frac{\sigma(K_{-})}{2}+\delta_{j}(K_{+})\leq\delta_{j}(K_{-})$

which can be re-written as

$\xi_{j}(K_{+})\leq\xi_{j}(K_{-}).$

Similarly if $\sigma(K_{+})=\sigma(K_{-})-2$, then applying the equivariant
Froyshov inequality to $\overline{W}$ gives:

$\xi_{j+1}(K_{+})\leq\xi_{j}(K_{-}).$

Putting together all these inequalities, we see that if
$\sigma(K_{-})=\sigma(K_{+})$, then
$\xi_{j+1}(K_{-})\leq\xi_{j}(K_{+})\leq\xi_{j}(K_{-})$ for all $j\geq 0$ and
if $\sigma(K_{-})=\sigma(K_{+})+2$, then
$\xi_{j+1}(K_{+})\leq\xi_{j}(K_{-})\leq\xi_{j}(K_{+})$ for all $j\geq 0$. ∎

Extend the definition of $\xi_{j}(K)$ to negative $j$ by setting
$\xi_{j}(K)=\xi_{0}(K)$ for all $j<0$. Consider the shifted invariants
$\rho_{j}(K)=\xi_{j-\sigma(K)/2}(K)$. Then $\\{\rho_{j}(K)\\}$ is integer-
valued, decreasing and $\rho_{j}(K)=0$ for $j\geq g_{4}(K)$ by property (3) of
the $\delta_{j}$-invariants. We define $\theta(K)$ to be the smallest $j\geq
0$ for which $\rho_{j}(-K)=0$. Equivalently,
$\theta(K)=\max\\{0,j(-K)-\sigma(K)/2\\}$ where $j(-K)$ is the smallest $j$
such that $\xi_{j}(-K)=0$. Since $\rho_{j}$ are knot concordance invariants,
it follows that $\theta$ is also a concordance invariant.

######  Theorem 4.2.

The invariant $\theta(K)$ satisfies the following properties:

  * (1)

$-\sigma(K)/2\leq\theta(K)\leq g_{4}(K)$.

  * (2)

$\theta(K_{1}+K_{2})\leq\theta(K_{1})+\theta(K_{2})$.

  * (3)

Let $K_{+},K_{-}$ be knots where $K_{-}$ is obtained from $K_{+}$ by changing
a positive crossing into a negative crossing. Then

$0\leq\theta(K_{+})-\theta(K_{-})\leq 1.$

  * (4)

If $K$ is quasi-alternating, then

$\theta(K)=\begin{cases}-\sigma(K)/2,&\sigma(K)\leq 0,\\\
0,&\sigma(K)>0.\end{cases}$

  * (5)

If $\delta(K)<-\sigma(K)/2$ and $\sigma(K)\leq 0$, then $\theta(K)\geq
1-\sigma(K)/2$.

###### Proof.

The inequality $\theta(K)\geq-\sigma(K)/2$ follows from the definition
$\theta(K)=\max\\{0,j(-K)-\sigma(K)/2\\}$ and the fact that $j(-K)\geq 0$.

The inequality $\theta(K)\leq g_{4}(K)$ follows from the definition of
$\theta(K)$ and the fact that $j(-K)\leq g_{4}(K)+\sigma(K)/2$ (which in turn
follows from the fact that $\xi_{j}(-K)=0$ for $j\geq g_{4}(K)+\sigma(K)/2$).

For (2), we note that $Y=\Sigma_{2}(K_{1}+K_{2})$ is (equivariantly)
diffeomorphic to the connected sum of $Y_{1}=\Sigma_{2}(K_{1})$ and
$Y_{2}=\Sigma_{2}(K_{2})$. Furthermore
$\mathfrak{s}_{0}(K_{1}\\#K_{2})=\mathfrak{s}_{0}(K_{1})\\#\mathfrak{s}_{0}(K_{2})$
[8, Theorem 2.4]. Then from Proposition 3.1, we have

$d_{\mathbb{Z}_{2},Q^{i+j}}(\Sigma_{2}(K_{1}+K_{2}),\mathfrak{s}_{0})\leq
d_{\mathbb{Z}_{2},Q^{i}}(\Sigma_{2}(K_{1}),\mathfrak{s}_{0})+d_{\mathbb{Z}_{2},Q^{j}}(\Sigma_{2}(K_{2}),\mathfrak{s}_{0})$

for all $i,j\geq 0$. It follows that
$\delta_{i+j}(K_{1}+K_{2})\leq\delta_{i}(K_{1})+\delta_{j}(K_{2})$ for all
$i,j\geq 0$. Hence $j(K_{1}+K_{2})\leq j(K_{1})+j(K_{2})$. Similarly
$j(-K_{1}-K_{2})\leq j(-K_{1})+j(-K_{2})$. From this and additivity of the
signature, we get $\theta(K_{1}+K_{2})\leq\theta(K_{1})+\theta(K_{2})$.

For (3), we consider separately the cases $\sigma(K_{+})=\sigma(K_{-})$ and
$\sigma(K_{+})=\sigma(K_{-})-2$. If $\sigma(K_{+})=\sigma(K_{-})$, then by
Proposition 4.1 (1), we have

$\xi_{j+1}(K_{-})\leq\xi_{j}(K_{+})\leq\xi_{j}(K_{-}).$

Setting $j=j(K_{-})$, we have $\xi_{j}(K_{-})=\xi_{j+1}(K_{-})=0$, hence
$\xi_{j}(K_{+})=0$. So $j(K_{+})\leq j(K_{-})$. If $j(K_{-})>1$, then
$\xi_{j-1}(K_{-})>0$ and hence $\xi_{j-2}(K_{+})\geq\xi_{j-1}(K_{-})>0$, so
$j(K_{+})\geq j(K_{-})-1$. Thus we have $j(K_{-})-1\leq j(K_{+})\leq
j(K_{-})$. Consider the mirrors $-K_{+}$ and $-K_{-}$. Then $-K_{+}$ is
obtained from $-K_{-}$ by changing a positive crossing to a negative crossing
and $\sigma(-K_{-})=\sigma(-K_{+})$, so we get $j(-K_{+})-1\leq j(-K_{-})\leq
j(-K_{+})$. Hence

(4.1) $j(-K_{+})-\frac{\sigma(K_{+})}{2}-1\leq
j(-K_{-})-\frac{\sigma(K_{-})}{2}\leq j(-K_{+})-\frac{\sigma(K_{+})}{2}$

from which is follows easily that $0\leq\theta(K_{+})-\theta(K_{-})\leq 1$. In
the case $\sigma(K_{+})-\sigma(K_{-})=-2$, Proposition 4.1 (2) gives

$\xi_{j+1}(K_{+})\leq\xi_{j}(K_{-})\leq\xi_{j}(K_{+}).$

Arguing as above this gives, $j(K_{+})-1\leq j(K_{-})\leq j(K_{+})$. A similar
argument applied to the mirrors of $K_{+}$ and $K_{-}$ gives $j(-K_{-})-1\leq
j(-K_{+})\leq j(-K_{-})$. Using this and $\sigma(K_{-})=\sigma(K_{+})+2$, we
again find that the inequality (4.1) holds and hence
$0\leq\theta(K_{+})-\theta(K_{-})\leq 1$.

For (4), if $K$ is quasi-alternating then $\delta_{j}(-K)=-\sigma(-K)/2$ for
all $j\geq 0$. Hence $j(-K)=0$ and $\theta(K)=\max\\{-\sigma(K)/2,0\\}$.

For (5), if $\delta(K)<-\sigma(K)/2$, then
$\delta_{0}(-K)\geq\delta(-K)>-\sigma(-K)/2$. Hence $\xi_{0}(-K)>0$ and
$j(-K)\geq 1$. Hence $\theta(K)\geq j(-K)-\sigma(K)/2\geq 1-\sigma(K)/2$. ∎

######  Corollary 4.3.

For any knot $K$ we have $\theta(K)+\theta(-K)\leq u(K)$ where $u(K)$ is the
unknotting number of $K$.

###### Proof.

By Theorem 4.2 (3), each crossing change can increase $\theta(K)+\theta(-K)$
by at most one. Hence by considering a sequence of $u(K)$ crossing changes
starting at the unknot and finishing with $K$, we get
$\theta(K)+\theta(-K)\leq u(K)$. ∎

######  Proposition 4.4.

For prime knots with $9$ or fewer crossings we have that
$\theta(K)=\nu^{+}(K)$ and $\theta(K)=\nu^{+}(-K)$, except for $K=9_{42}$. For
$K=9_{42}$ we have $\theta(K)=\nu^{+}(K)=\nu^{+}(-K)=0$ and $\theta(-K)=1$.

###### Proof.

All prime knots with $9$ or fewer crossings are quasi-alternating except for
$8_{19},9_{42}$ and $9_{46}$, so we only need to check these three cases. The
knot $K=8_{19}$ has $-\sigma(K)/2=\tau(K)=g_{4}(K)=u(K)=3$. Hence
$\theta(K)=\nu^{+}(K)=3$. Then since $\theta(K)+\theta(-K)=3+\theta(-K)\leq
u(K)=3$, we must have $\theta(-K)=0$. Similarly, $\nu^{+}(K)+\nu^{+}(-K)\leq
3$, so $\nu^{+}(-K)=0$.

The knot $K=9_{42}$ has $\sigma(K)/2=g_{4}(K)=u(K)=1$. Hence
$1=\sigma(K)/2\leq\omega(-K)\leq g_{4}(K)=1$, giving $\theta(-K)=1$. Then
since $\theta(K)+\theta(-K)\leq u(K)=1$, we have $\theta(K)=0$. On the other
hand, $\tau(K)=0$. Then from [18, Theorem 1.1], it follows that
$\nu^{+}(K)=\nu^{+}(-K)=0$.

Lastly, the knot $K=9_{46}$ is slice, so
$\theta(K)=\theta(-K)=\nu(K)=\nu^{+}(-K)=0$. ∎

So far we have obtained concordance invariants related to the branched double
cover $\Sigma_{2}(K)$. We now generalise the above results by considering
analogous concordance invariants associated to cyclic branched covers of any
prime order.

Given a knot $K$ in $S^{3}$ and a prime number $q$, we let $Y=\Sigma_{q}(K)$
denote the degree $q$ cyclic cover of $S^{3}$ branched over $K$. As discussed
in Section 2, there is a distinguished spinc-structure $\mathfrak{s}_{0}$ on
$Y$. In [8], Jabuka defined a series of knot concordance invariants by
definining

$\delta^{(q)}(K)=4\delta(\Sigma_{q}(K),\mathfrak{s}_{0}).$

Jabuka’s invariants are defined more generally for prime powers but we
restrict here to the prime case because this is necessary in order to apply
the equivariant Froyshov inequality [1, Theorem 5.3]. Note also that Jabuka
used the notation $\delta_{q}(K)$ for what we have denoted $\delta^{(q)}(K)$.

For any odd prime $q$ and any $j\geq 0$, we define a knot concordance
invariant $\delta_{j}^{(q)}(K)$ by setting

$\delta_{j}^{(q)}(K)=4\delta_{\mathbb{Z}_{q},S^{j}}(\Sigma_{q}(K),\mathfrak{s}_{0}),$

where $\delta_{\mathbb{Z}_{q},S^{j}}$ are the equivariant $\delta$-invariants,
as defined in [1]. We also write $\delta_{j}^{(2)}(K)=\delta_{j}(K)$ for the
case $q=2$.

######  Proposition 4.5.

Let $q$ be an odd prime. The concordance invariants $\delta_{j}^{(q)}(K)$
satisfy the following properties:

  * (1)

$\delta^{(q)}_{0}(K)\geq\delta^{(q)}(K)$, where $\delta^{(q)}(K)$ is the
Jabuka invariant [8].

  * (2)

$\delta^{(q)}_{j+1}(K)\leq\delta^{(q)}_{j}(K)$ for all $j\geq 0$.

  * (3)

$\delta^{(q)}_{j}(K)\geq-\sigma^{(q)}(K)/2$ for all $j\geq 0$ and
$\delta^{(q)}_{j}(K)=-\sigma^{(q)}(K)/2$ for
$2j\geq(q-1)g_{4}(K)-\sigma^{(q)}(K)/2$.

  * (4)

$\delta^{(q)}_{j}(K)+\delta^{(q)}_{j}(-K)\geq 0$ for all $j\geq 0$.

  * (5)

$\delta^{(q)}_{j}(K)=-\sigma^{(q)}(K)/2\;({\rm mod}\;4)$ for all $j\geq 0$.

###### Proof.

The proof is a straightforward extension of the $q=2$ case [1, Proposition
6.2] to an arbitrary prime, so we omit the details. ∎

For convenience we set

$\xi_{j}^{(q)}(K)=\frac{1}{4}\delta^{(q)}_{j}(K)+\frac{1}{8}\sigma^{(q)}(K).$

Then $\xi^{(q)}_{j}(K)$ is integer-valued, decreasing as a function of $j$ and
is zero for $2j\geq(q-1)g_{4}(K)-\sigma^{(q)}(K)/2$.

As in the $q=2$ case, we have the following properties:

######  Proposition 4.6.

Let $K_{+},K_{-}$ be knots where $K_{-}$ is obtained from $K_{+}$ by changing
a positive crossing into a negative crossing. Then
$-(q-1)\leq\sigma^{(q)}(K_{+})-\sigma^{(q)}(K_{-})\leq 0$ or $2$ and for all
$j\geq 0$, we have

$\xi^{(q)}_{j+\alpha}(K_{-})\leq\xi_{j}^{(q)}(K_{+})\text{ and
}\xi^{(q)}_{j+\beta}\leq\xi^{(q)}_{j}(K_{-}),$

where $\alpha=(q-1)/2+\sigma^{(q)}(K_{+})/4-\sigma^{(q)}(K_{-})/4$ and
$\beta=\sigma^{(q)}(K_{-})/4-\sigma^{(q)}(K_{+})/4$.

###### Proof.

The proof is a straightforward extension of the proof of Proposition 4.1, so
we give only a sketch. Resolving the singularity of a regular homotopy
associated to the crossing change gives a genus $1$ surface $\Sigma$ properly
embedded in $X=[0,1]\times S^{3}\times$ which meets the ingoing boundary of
$X$ in $K_{-}$ and the outgoing boundary in $K_{+}$. Let $W\to X$ be the
$q$-fold cover of $X$ branched over $\Sigma$. This gives a
$\mathbb{Z}_{q}$-equivariant cobordism from $\Sigma_{q}(K_{-})$ to
$\Sigma_{q}(K_{+})$. One finds that $b_{2}(W)=2(q-1)$ and that
$\sigma(W)=\sigma^{(q)}(K_{+})-\sigma^{(q)}(K_{-})$, hence
$b_{+}(W)=(q-1)+\sigma^{(q)}(K_{+})/2-\sigma^{(q)}(K_{-})/2=2\alpha$. Note
that $2\alpha=b_{+}(W)\geq 0$, hence
$\sigma^{(q)}(K_{+})-\sigma^{(q)}(K_{-})\geq-(q-1)$. Applying the equivariant
Froyshov inequality to $W$ gives
$\xi^{(q)}_{j+\alpha}(K_{-})\leq\xi^{(q)}_{j}(K_{+})$.

Consider again a regular homotopy associated to the crossing change. Instead
of resolving the crossing, we take a blow-up. This gives a genus zero surface
$\Sigma$ properly embedded in $X$, where $X$ is $\mathbb{CP}^{2}$ with two
balls removed and $\Sigma$ meets the ingoing boundary of $X$ in $K_{-}$ and
the outgoing boundary in $K_{+}$. Let $\pi:W\to X$ be the $q$-fold cover of
$X$ branched over $\Sigma$. We find $b_{2}(W)=q$ and
$\sigma(W)=q+\sigma^{(q)}(K_{+})-\sigma^{(q)}(K_{-})$, hence
$b_{-}(W)=\sigma^{(q)}(K_{-})/2-\sigma^{(q)}(K_{+})/2=2\beta$. Since
$2\beta=b_{-}(W)\geq 0$, we have $\sigma^{(q)}(K_{+})-\sigma^{(q)}(K_{-})\leq
0$. Applying the Froyshov inequality to $\overline{W}$ (with
$\Sigma_{q}(K_{+})$ regarded as an ingoing boundary and $\Sigma_{q}(K_{-})$ an
outgoing boundary gives $\xi^{(q)}_{j+\beta}(K_{+})\leq\xi^{(q)}_{j}(K_{-})$.
∎

For an odd prime $q$, we define

$\theta^{(q)}(K)=\max\left\\{0,\dfrac{2j^{(q)}(-K)}{q-1}-\dfrac{\sigma^{(q)}(K)}{2(q-1)}\right\\},$

where $j^{(q)}(-K)$ is the smallest $j$ such that $\xi^{(q)}_{j}(-K)=0$. Note
that by this definition $\theta^{(q)}(K)$ is valued in
$\frac{1}{(q-1)}\mathbb{Z}$.

######  Theorem 4.7.

The invariant $\theta^{(q)}(K)$ satisfies the following properties:

  * (1)

$-\dfrac{\sigma^{(q)}(K)}{2(q-1)}\leq\theta^{(q)}(K)\leq g_{4}(K)$.

  * (2)

$\theta^{(q)}(K_{1}+K_{2})\leq\theta^{(q)}(K_{1})+\theta^{(q)}(K_{2})$.

  * (3)

Let $K_{+},K_{-}$ be knots where $K_{-}$ is obtained from $K_{+}$ by changing
a positive crossing into a negative crossing. Then

$0\leq\theta^{(q)}(K_{+})-\theta^{(q)}(K_{-})\leq 1.$

  * (4)

If $\Sigma_{q}(K)$ is an $L$-space, then

$\theta^{(q)}(K)=\begin{cases}-\dfrac{\sigma^{(q)}(K)}{2(q-1)},&\sigma^{(q)}(K)\leq
0,\\\ 0,&\sigma^{(q)}(K)>0.\end{cases}$

  * (5)

If $\delta^{(q)}(K)<-(q-1)\sigma^{(q)}(K)/(2(q-1))$ and $\sigma^{(q)}(K)\leq
0$, then $\theta^{(q)}(K)\geq 1/(q-1)-\sigma^{(q)}(K)/(2(q-1))$.

###### Proof.

We prove only part (3), since the other parts are similar to the $q=2$ case.
First note that for any knot $K$, we have

(4.2) $(q-1)\theta^{(q)}(-K)=\max\\{0,2j^{(q)}(K)+\sigma^{(q)}(K)/2\\}.$

Now let $K_{-}$ be obtained from $K_{+}$ by changing a positive crossing into
a negative crossing. Then from Proposition 4.6 we have

$\xi^{(q)}_{j+\alpha}(K_{-})\leq\xi_{j}^{(q)}(K_{+})\text{ and
}\xi^{(q)}_{j+\beta}\leq\xi^{(q)}_{j}(K_{-}),$

where $2\alpha=(q-1)+\sigma^{(q)}(K_{+})/2-\sigma^{(q)}(K_{-})/2$ and
$2\beta=\sigma^{(q)}(K_{-})/2-\sigma^{(q)}(K_{+})/2$. Now if
$\xi^{(q)}_{j}(K_{+})=0$, then $\xi^{(q)}_{j+\alpha}(K_{-})\leq 0$, hence
$\xi^{(q)}_{j+\alpha}(K_{-})=0$. It follows that $j^{(q)}(K_{-})\leq
j^{(q)}(K_{+})+\alpha$. Similarly, if $\xi^{(q)}_{j}(K_{-})=0$, then
$\xi^{(q)}_{j+\beta}(K_{+})\leq 0$, so we deduce that $j^{(q)}(K_{+})\leq
j^{(q)}(K_{-})+\beta$. Putting these together, we have

$j^{(q)}(K_{+})-\beta\leq j^{(q)}(K_{-})\leq j^{(q)}(K_{+})+\alpha.$

From the definitions of $\alpha$ and $\beta$, this is equivalent to

(4.3) $2j^{(q)}(K_{+})+\frac{\sigma^{(q)}(K_{+})}{2}\leq
2j^{(q)}(K_{-})+\frac{\sigma^{(q)}(K_{-})}{2}\leq
2j^{(q)}(K_{+})+\frac{\sigma^{(q)}(K_{+})}{2}+(q-1).$

If $2j^{(q)}(K_{+})+\sigma^{(q)}(K_{+})/2\geq 0$, then we also have
$2j^{(q)}(K_{-})+\sigma^{(q)}(K_{-})/2\geq 0$. So from (4.2), we see that
$(q-1)\theta^{(q)}(-K_{+})=2j^{(q)}(K_{+})+\sigma^{(q)}(K_{+})/2$ and
$\theta^{(q)}(-K_{-})=2j^{(q)}(K_{-})+\sigma^{(q)}(K_{-})/2$. Then from (4.3),
we get

$(q-1)\theta^{(q)}(-K_{+})\leq(q-1)\theta^{(q)}(-K_{-})\leq(q-1)\theta^{(q)}(-K_{+})+(q-1).$

If $j^{(q)}(K_{+})+\sigma^{(q)}(K_{+})/2\leq 0$, then $\theta^{(q)}(-K_{+})=0$
and from (4.3) we get

$(q-1)\theta^{(q)}(-K_{-})\leq
2j^{(q)}(K_{-})+\frac{\sigma^{(q)}(K_{-})}{2}\leq
2j^{(q)}(K_{+})+\frac{\sigma^{(q)}(K_{+})}{2}+(q-1)\leq(q-1),$

hence

$0\leq(q-1)\theta^{(q)}(-K_{-})\leq(q-1).$

So in either case, we get that

(4.4)
$\theta^{(q)}(-K_{+})\leq\theta^{(q)}(-K_{-})\leq\theta^{(q)}(-K_{+})+1.$

Lastly, we note that if $K_{-}$ is obtained from $K_{+}$ by changing a
positive crossing into a negative crossing, then $-K_{+}$ is obtained from
$-K_{-}$ by changing a positive crossing to a negative crossing. Replacing
$K_{+}$ by $-K_{-}$ and $K_{-}$ by $-K_{+}$ in (4.4), we get

$\theta^{(q)}(K_{-})\leq\theta^{(q)}(K_{+})\leq\theta^{(q)}(K_{-})+1$

or equivalently $0\leq\theta^{(q)}(K_{+})-\theta^{(q)}(K_{-})\leq 1$. ∎

Using the same argument as the $q=2$ case, we have:

######  Corollary 4.8.

For any knot $K$ we have $\theta^{(q)}(K)+\theta^{(q)}(-K)\leq u(K)$.

We consider a more general concordance invariant $\theta^{(q)}(K,m)$ depending
also on a non-negative integer $m$.

######  Definition 4.9.

Let $m$ be a non-negative integer and $q$ a prime. For any knot $K$, define
$j^{(q)}(K,m)$ to be the least value of $j$ such that $\delta_{j}^{(q)}(K)\leq
m-\sigma^{(q)}(K)/2$. Further, define
$\omega^{(q)}(K,m)\in\frac{1}{(q-1)}\mathbb{Z}$ by

$\theta^{(2)}(K,m)=\max\left\\{0,j^{(2)}(-K,m)-\frac{\sigma(K)}{2}\right\\}$

for $q=2$, and

$\theta^{(q)}(K,m)=\max\left\\{0,\frac{2j^{(q)}(-K,m)}{q-1}-\frac{\sigma^{(q)}(K)}{2(q-1)}\right\\}$

for odd $q$.

######  Proposition 4.10.

Let $\ell^{(q)}(K)$ denote the lowest $i$ for which
$HF^{+}_{i}(\Sigma_{q}(K),\mathfrak{s}_{0})$ is non-zero (where $HF^{+}$ is
taken with coefficients in $\mathbb{F}=\mathbb{Z}/q\mathbb{Z}$). Then

$j^{(2)}(K,m)\geq\ell^{(2)}(K)-\frac{m}{2}+\frac{\sigma(K)}{4}$

and

$2j^{(q)}(K,m)\geq\ell^{(q)}(K)-\frac{m}{2}+\frac{\sigma^{(q)}(K)}{4}$

for odd $q$. Hence for all $q$ we have

$\theta^{(q)}(K,m)\geq\frac{\ell^{(q)}(-K)}{(q-1)}-\frac{m}{2(q-1)}-\frac{3\sigma^{(q)}(K)}{4(q-1)}.$

In particular, setting $m=0$, we have

$\theta^{(q)}(K)\geq\frac{\ell^{(q)}(-K)}{(q-1)}-\frac{3\sigma^{(q)}(K)}{4(q-1)}.$

###### Proof.

We consider the case where $q$ is odd. The case $q=2$ is similar. Let
$Y=\Sigma_{q}(K)$ and let $\mathfrak{s}_{0}$ be the distinguished spinc-
structure. Then
$\delta^{(q)}_{j}(K)/2=d_{\mathbb{Z}_{q},S^{j}}(Y,\mathfrak{s}_{0})$ is
defined to be $k-2j$, where $k$ is the minimum degree of an element of the
equivariant Seiberg–Witten–Floer cohomology
$HSW^{*}_{\mathbb{Z}_{q}}(Y,\mathfrak{s}_{0})$ satisfying the condition given
in Section 3. From this definition it follows that $\delta^{(q)}_{j}(K)\geq
2(k_{0}-2j)$ where $k_{0}$ is the lowest degree in which
$HSW^{*}_{\mathbb{Z}_{q}}(Y,\mathfrak{s}_{0})$ is non-zero. From [1, Theorem
3.2], there is a spectral sequence $E_{r}^{p,q}$ abutting to
$HSW^{*}_{\mathbb{Z}_{q}}(Y,\mathfrak{s}_{0})$ which has
$E_{2}^{p,q}=H^{p}(B\mathbb{Z}_{q};HSW^{q}(Y,\mathfrak{s}_{0}))$. Hence
$k_{0}$ is at least the lowest degree in which $HSW^{q}(Y,\mathfrak{s}_{0})$
is non-zero. Now recall that the Seiberg–Witten–Floer cohomology
$HSW^{*}(Y,\mathfrak{s}_{0})$ is isomorphic to the Heegaard–Floer cohomology
$HF_{+}^{*}(Y,\mathfrak{s}_{0})$. Further, since we are working over a field
this is also isomorphic to the Heegaard–Floer homology
$HF^{+}_{*}(Y,\mathfrak{s}_{0})$. So the lowest degree in which
$HSW^{*}(Y,\mathfrak{s}_{0})$ is non-zero is precisely $\ell^{(q)}(K)$. So we
have

$\delta^{(q)}_{j}(K)\geq 2(k_{0}-2j)\geq 2\ell^{(q)}(K)-4j.$

Now let $j=j^{(q)}(K,m)$, so $\delta^{(q)}_{j}(K)\leq m-\sigma^{(q)}(K)/2$,
giving

$2j^{(q)}(K,m)\geq\ell^{(q)}(K)-\frac{m}{2}+\frac{\sigma^{(q)}(K)}{4}$

and thus

$2\frac{j^{(q)}(K,m)}{q-1}+\frac{\sigma^{(q)}(K)}{2}\geq\frac{\ell^{(q)}(K)}{q-1}-\frac{m}{2(q-1)}+\frac{3\sigma^{(q)}(K)}{4(q-1)}.$

Replacing $K$ by $-K$, we thus get

$\theta^{(q)}(K,m)\geq\frac{\ell^{(q)}(-K)}{(q-1)}-\frac{m}{2(q-1)}-\frac{3\sigma^{(q)}(K)}{4(q-1)}.$

∎

##  5\. Examples

###  5.1. Torus knots

Let $a,b$ be positive coprime integers and let $T_{a,b}$ denote the $(a,b)$
torus knot.

######  Proposition 5.1.

Let $q$ be coprime to $a$ and $b$. Then $\theta^{(q)}(-T_{a,b})=0$.

###### Proof.

We have that $\sigma^{(q)}(T_{a,b})\leq 0$ as $T_{a,b}$ is braid positive.
From [1, Proposition 7.2], we have
$\delta_{j}^{(q)}(T_{a,b})=-\sigma^{(q)}(T_{a,b})/2$ for all $j\geq 0$. So
$j^{(q)}(T_{a,b})=0$ and
$\theta^{(q)}(-T_{a,b})=\max\\{0,\sigma^{(q)}(T_{a,b})/(2(q-1))\\}=0$. ∎

######  Proposition 5.2.

For each $n\geq 1$, we have $\theta(T_{3,6n-1})=g_{4}(T_{3,6n-1})=6n-2$ and
$\theta(T_{3,6n+1})=g_{4}(T_{3,6n+1})=6n$.

###### Proof.

From [1, Proposition 7.3], we have that $j(T_{3,6n-1})=2n-2$. Also
$\sigma(T_{3,6n-1})=-8n$, hence
$\theta(T_{3,6n-1})=2n-2+4n=6n-2=g_{4}(T_{3,6n-1})$. Similarly from [1,
Proposition 7.4], we have that $j(T_{3,6n+1})=2n$ and we also have
$\sigma(T_{3,6n+1})=-8n$. Hence
$\theta(T_{3,6n+1})=2n+4n=6n=g_{4}(T_{3,6n+1})$. ∎

######  Proposition 5.3.

For each $n\geq 1$, we have $\theta^{(3)}(T_{2,6n-1})=g_{4}(T_{2,6n-1})=3n-1$
and $\theta^{(3)}(T_{2,6n+1})=g_{4}(T_{2,6n+1})=3n$.

###### Proof.

From [1, Proposition 7.2], we have that $\sigma^{(3)}(T_{2,6n\pm
1})=\sigma^{(2)}(T_{3,6n\pm 1})=\sigma(T_{3,6n\pm 1})$, which equals $-8n$.
Also from [1, Proposition 7.3, 7.4], we find that $\ell^{(3)}(-T_{2,6n-1})=-2$
and $\ell^{(3)}(-T_{2,6n+1})=0$. Putting this into Proposition 4.10 gives
$\theta^{(3)}(T_{2,6n-1})\geq 3n-1$ and $\theta^{(3)}(T_{2,6n+1})\geq 3n$. But
we also find that $g_{4}(T_{2,6n-1})=3n-1$, $g_{4}(T_{2,6n+1})=3n$, so we must
have equalities $\theta^{(3)}(T_{2,6n-1})=3n-1$,
$\theta^{(3)}(T_{2,6n+1})=3n$. ∎

In light of the above propositions we suspect that
$\theta^{(q)}(T_{a,b})=g_{4}(T_{a,b})$ for all primes $q$ and all positive
coprime integers $a,b$.

######  Proposition 5.4.

For each $n\geq 1$ and $m\geq 0$, we have
$\theta(T_{3,6n-1},m)=\max\\{4n,6n-2-2\lfloor\frac{m}{4}\rfloor\\}$ and
$\theta(T_{3,6n+1},m)=\max\\{4n,6n-2\lfloor\frac{m}{4}\rfloor\\}$.

###### Proof.

Recall that $\sigma(T_{3,6n\pm 1})=-8n$. Then $\theta(T_{3,6n\pm
1},m)=j(-T_{3,6n\pm 1},m)+4n$. From [1, Proposition 7.3], we have that

$\delta_{j}(-T_{3,6n-1})=\begin{cases}-4\left(\left\lfloor\frac{j}{2}\right\rfloor+1\right)&0\leq
j\leq 2n-3,\\\ -4n&j\geq 2n-2.\end{cases}$

It follows that $j(-T_{3,6n-1},m)=\max\\{0,2n-2-2\lfloor\frac{m}{4}\rfloor\\}$
and thus $\theta(T_{3,6n-1},m)=\max\\{4n,6n-2-2\lfloor\frac{m}{4}\rfloor\\}$.
The case of $T_{3,6n+1}$ is similar. From [1, Proposition 7.4], we have that

$\delta_{j}(-T_{3,6n+1})=\begin{cases}-4\left\lfloor\frac{j}{2}\right\rfloor&0\leq
j\leq 2n-1,\\\ -4n&j\geq 2n.\end{cases}$

Hence $j(-T_{3,6n+1},m)=\max\\{0,2n-2\lfloor\frac{m}{4}\rfloor\\}$ and thus
$\theta(T_{3,6n+1},m)=\max\\{4n,6n-2\lfloor\frac{m}{4}\rfloor\\}$. ∎

###  5.2. Examples involving Whitehead doubles

For any knot $K$, let $Wh(K)$ denote the untwisted Whitehead double of $K$
with a positive clasp. Whitehead doubles are known to have trivial Alexander
polynomial and are therefore topologically slice [4]. This also implies that
the Levine–Tristram signature is identically zero. A Whitehead double can be
unknotted by a single crossing change, so they have $g_{4}(Wh(K))\leq 1$ and
$u(Wh(K))\leq 1$.

######  Proposition 5.5.

If $\tau(K)>0$, then $\theta(Wh(K))=1$, $\theta(-Wh(K))=0$.

###### Proof.

By [12, Theorem 1.5], we have that $\delta(Wh(K))<0$ whenever $\tau(K)>0$.
Then since $\sigma(Wh(K))=0$, Theorem 4.2 (5) gives $\theta(Wh(K))\geq 1$. On
the other hand, Corollary 4.3 gives $\theta(Wh(K))+\theta(-Wh(K))\leq
u(Wh(K))\leq 1$. Hence $\theta(Wh(K))=1$ and $\theta(-Wh(K))=0$. ∎

Note that from [6, Theorem 1.4], we have that $\tau(Wh(K))=1$ whenever
$\tau(K)>0$. Hence for such knots we also have $\nu^{+}(Wh(K))=1$,
$\nu^{+}(-Wh(K))=0$.

######  Example 5.6.

The Whitehead double $Wh(T_{2,2a+1})$ of the torus knot $T_{2,2a+1}$ has
$\tau(Wh(T_{2,2a+1}))=1$, $s(Wh(T_{2,2a+1}))=2$ and
$\delta(Wh(T_{2,2a+1}))=-4a$ [12]. Consider the knot
$K=Wh(T_{2,2a+1})\\#(-Wh(T_{2,2b+1}))$. Then $\tau(K)=s(K)=0$ and the
Levine–Tristram signature of $K$ is identically zero. However
$\delta(K)=4(b-a)$. So if $a>b$ then $\theta(K)\geq 1$. On the other hand
$\theta(K)\leq\theta(Wh(T_{2,2a+1}))+\theta(-Wh(T_{2,2b+1}))=1$, so we have
$\theta(K)=1$.

######  Example 5.7.

Let $K=-9_{42}\\#Wh(T_{2,2a+1})$ where $a$ is any positive integer. Then
$\tau(K)=1$, $s(K)=2$, $\sigma(K)=-2$ and $\delta(K)=1-4a<-\sigma(K)/2$.
Therefore $\theta(K)\geq 2$. But $9_{42}$ and $Wh(T_{2,2a+1})$ both have slice
genus $1$ and unknotting number $1$, hence $g_{4}(K),u(K)\leq 2$. It follows
that $\theta(K)=g_{4}(K)=u(K)=2$. On the other hand $\tau$, $s$ and $\sigma$
(or more generally the Levine–Tristram signature) of $K$ only give the bounds
$g_{4}(K)\geq 1$ and $u(K)\geq 1$. We also have
$\nu^{+}(K)\leq\nu^{+}(-9_{42})+\nu^{+}(Wh(T_{2,2a+1}))=1$ and
$\nu^{+}(-K)\leq\nu+(9_{42})+\nu^{+}(-Wh(T_{2,2a+1}))=0$, so $\theta(K)$ also
gives a better genus bound than $\nu^{+}(K)$ and $\nu^{+}(-K)$.

######  Example 5.8.

Consider a connected sum such as

$K=-Wh(T_{2,2a+1})\\#Wh(T_{2,2b+1})\\#Wh(T_{2,2c+1})$

with $a>b+c$. Then $\tau(K)=1$, $\sigma(K)=0$ and $\delta(K)=4(a-b-c)>0$,
hence $\theta(-K)\geq 1$. Also
$\theta(-K)\leq\theta(Wh(T_{2,2a+1}))+\theta(-Wh(T_{2,2b+1}))+\theta(-Wh(T_{2,2c+1}))=1$,
hence $\theta(-K)=1$. Since $\tau(K)>0$, we see that $K$ is not smoothly
$H$-slice in any negative definite $4$-manifold (with zero integral first
homology) and since $\theta(-K)>0$, we see that $K$ is not smoothly $H$-slice
in any positive definite $4$-manifold either. Moreover, $K$ is topologically
slice since it is a connected sum of Whitehead doubles.

######  Example 5.9.

Most of the examples we have seen have either $\theta(K)=0$ or $\theta(-K)=0$.
However it is certainly possible to have $\theta(K)$ and $\theta(-K)$ both
non-zero. Consider for example $K=T_{2,5}\\#(-Wh(T_{2,3}))$. Then $T_{2,5}$ is
concordant to $K\\#Wh(T_{2,3})$, hence

$2=\theta(T_{2,5})\leq\theta(K)+\theta(Wh(T_{2,3}))=\theta(K)+1$

which gives $\theta(K)\geq 1$. On the other hand $\sigma(K)=-4$ and
$\delta(K)=6>-\sigma(K)/2$, so $\theta(-K)\geq 1$. Furthermore, $Wh(T_{2,3})$
can be unknotted by changing a positive crossing in the clasp to a negative
crossing. So $T_{2,3}$ is obtained from $-K$ by changing a positive crossing
to negative. Hence

$0\leq\theta(-K)-\theta(T_{2,5})\leq 1,$

hence $\theta(-K)\geq 2$. Further, as $u(T_{2,5})=2$ and $u(Wh(T_{2,3}))=1$,
we get $\theta(K)+\theta(-K)\leq u(K)\leq 3$. So $\theta(K)=1$, $\theta(-K)=2$
and $u(K)=3$.

##  6\. Genus bounds for definite $4$-manifolds

Let $X$ be a compact, oriented, smooth $4$-manifold with boundary $\partial
X=S^{3}$. Note that $H^{2}(X,\partial X;\mathbb{Z})$ and $H^{2}(X;\mathbb{Z})$
are canonically isomorphic.

Let $K\subset S^{3}$ be a knot in $S^{3}$. Consider a smooth, oriented,
properly embedded surface $\Sigma\to X$ bounding $K$. The surface $\Sigma$ has
a relative homology class in $H_{2}(X,\partial X;\mathbb{Z})\cong
H_{2}(X;\mathbb{Z})$. Given a knot $K$ and a homology class $a\in
H_{2}(X;\mathbb{Z})$, we say that an embedded surface $\Sigma$ of the above
type represents the pair $(K,a)$ if $\partial\Sigma=K$ and $[\Sigma]=a$.
Define the slice genus of $K$ with respect to $(X,a)$, $g_{4}(K,X,a)$ to be
the minimum genus of an embedded surface in $X$ which represents the pair
$(K,a)$. Define the $H$-slice genus of $K$ in $X$, $g_{4}(K,X)$ to be the
minimum genus of a smooth, oriented, properly embedded surface $\Sigma$ in $X$
bounding $K$ and such that $\Sigma$ is homologically trivial. Clearly
$g_{4}(K,X,a)$ and $g_{H}(K,X)$ depend only on the isotopy class of $K$.

Assume that $H_{1}(X;\mathbb{Z})=0$. Let $\Sigma\subseteq X$ be a properly
embedded surface in $X$ bounding a knot $K$ and suppose further that the
homology class $a=[\Sigma]\in H_{2}(X,\partial X;\mathbb{Z})$ is divisible by
a prime number $q$. Then we may construct the degree $q$ cyclic branched cover
$\pi:W\to X$ branched over $\Sigma$.

Assume now that $X$ is negative definite, that is, the intersection form on
$H^{2}(X;\mathbb{Z})$ is negative definite. Then from Lemma 2.2, we have:

$\displaystyle b_{2}(W)$ $\displaystyle=qb_{2}(X)+(q-1)(2g),$
$\displaystyle\sigma(W)$
$\displaystyle=-qb_{2}(X)-\frac{(q^{2}-1)}{3q}[\Sigma]^{2}+\sigma^{(q)}(K),$
$\displaystyle b_{+}(W)$
$\displaystyle=(q-1)g-\frac{(q^{2}-1)}{6q}[\Sigma]^{2}+\frac{\sigma^{(q)}(K)}{2},$

where $g$ is the genus of $\Sigma$.

######  Theorem 6.1.

Let $q$ be an odd prime and let $a\in H_{2}(X;\mathbb{Z})$ be divisible by
$q$. Then

$\delta_{j}^{(q)}(-K)\leq-\frac{(q^{2}-1)}{6q}a^{2}+\frac{\sigma^{(q)}(K)}{2}$

where

$2j=(q-1)g_{4}(K,X,a)-\frac{(q^{2}-1)}{6q}a^{2}+\frac{\sigma^{(q)}(K)}{2}.$

Thus, setting $m=-\frac{(q^{2}-1)}{6q}a^{2}$ we have that

$\frac{2j^{(q)}(-K,m)}{q-1}\leq
g_{4}(K,X,a)-\frac{q+1}{6q}a^{2}+\frac{\sigma^{(q)}(K)}{2(q-1)}$

and hence

$g_{4}(K,X,a)\geq\theta^{(q)}\left(K,m\right)+\frac{(q+1)}{6q}a^{2}.$

###### Proof.

Choose an embedded surface $\Sigma$ of genus $g=g_{4}(K,X,a)$ representing
$(K,a)$. Consider the cyclic $q$-fold branched cover $\pi:W\to X$ with branch
locus $\Sigma$. This carries and action of $G=\mathbb{Z}_{p}$ generated by the
covering transformation $\sigma$. The boundary of $W$ is $Y=\Sigma_{q}(K)$.
Let $\mathfrak{s}$ be an invariant spinc-structure on $W$. By Proposition 2.5
we can choose $\mathfrak{s}$ so that
$\widetilde{c}=c_{1}(\mathfrak{s})=\pi^{*}(c)$ where $c$ is a characteristic
on $X$ and such that $\mathfrak{s}|_{Y}=\mathfrak{s}_{0}$ is the distinguished
spinc-structure on $Y$. Since $X$ is negative definite, it has intersection
form $diag(-1,-1,\dots,-1)$ (cap the boundary of $X$ off with a disc and apply
Donaldson’s theorem). Choose $c=(1,1,\dots,1)$, so that $c^{2}=-b_{2}(X)$.

Consider $W$ as a cobordism with ingoing boundary
$\overline{Y}=\Sigma_{q}(\overline{K})$ and with empty outgoing boundary. We
have that $H^{+}(W)^{G}\cong H^{+}(X)=0$, since $X$ is negative definite. Thus
the equivariant Froyshov inequality [1, Theorem 5.3] gives:

$\delta(W,\mathfrak{s})+\delta_{\mathbb{Z}_{q},S^{j}}(\overline{Y},\mathfrak{s}_{0})\leq
0$

where

$2j=b_{+}(W)=(q-1)g_{4}(K,X,a)-\frac{(q^{2}-1)}{6q}[\Sigma]^{2}+\frac{\sigma^{(q)}(K)}{2}.$

Re-arranging and using
$\delta_{j}^{(q)}(-K)=4\delta_{\mathbb{Z}_{q},S^{j}}(\overline{Y},\mathfrak{s}_{0})$,
we have

$\displaystyle\delta_{j}^{(q)}(-K)\leq-4\delta(W,\mathfrak{s})$
$\displaystyle=\frac{\sigma(W)-c_{1}(\mathfrak{s})^{2}}{2}$
$\displaystyle=\frac{q\sigma(X)}{2}-\frac{(q^{2}-1)}{6q}[\Sigma]^{2}+\frac{\sigma^{(q)}(K)}{2}+\frac{qb_{2}(X)}{2}$
$\displaystyle=-\frac{(q^{2}-1)}{6q}[\Sigma]^{2}+\frac{\sigma^{(q)}(K)}{2}.$

This gives the result. ∎

Setting $a=0$, we have:

######  Corollary 6.2.

For every odd prime $q$, we have

$g_{H}(K,X)\geq\theta^{(q)}(K).$

Now consider the case $q=2$. For any $x$, let
$\eta(x)=\min_{c}\\{-(x+c)^{2}-b_{2}(X)\\}$ be the minimum value of
$-(x+c)^{2}-b_{2}(X)$ amongst all characteristics of $H^{2}(X;\mathbb{Z})$.
Since $X$ is negative definite, it has intersection form
$diag(-1,-1,\dots,-1)$. Choosing a characteristic such that each entry of
$x+c$ is $0$ or $1$, we see that $-b_{2}(X)\leq\eta(x)\leq 0$. Then we have

######  Theorem 6.3.

Let $a\in H^{2}(X;\mathbb{Z})$ be divisible by $2$. Then for every
characteristic $c\in H^{2}(X;\mathbb{Z})$, we have

$\delta_{j}^{(2)}(-K)\leq-\frac{a^{2}}{4}+\eta(a/2)+\frac{\sigma(K)}{2},$

where

$j=g_{4}(K,X,a)-\frac{a^{2}}{4}+\frac{\sigma(K)}{2}.$

Thus

$g_{4}(K,X,a)\geq j^{(2)}\left(K,m\right)+\frac{a^{2}}{4}-\frac{\sigma(K)}{2}$

where $m=-\frac{a^{2}}{4}+\eta(a/2)$. Equivalently,

$g_{4}(K,X,a)\geq\theta(K,m)+\frac{1}{4}a^{2}.$

###### Proof.

Set $x=a/2$. The proof is similar to the case that $q$ is an odd prime. The
main difference is that we now have $\widetilde{c}=\pi^{*}(c+x)$, where $c$ is
a characteristic on $X$. The Froyshov inequality gives

$2\delta_{j}^{(2)}(-K)\leq\sigma(W)-2(x+c)^{2}=-2(x+c)^{2}-x^{2}-2b_{+}(X)+\sigma(K)$

where

$j=b_{+}(W)=g_{4}(K,X,a)-x^{2}+\sigma(K)/2.$

Minimising over $c$, we get

$\delta_{j}^{(2)}(-K)\leq-x^{2}+\eta(x)-\sigma(K)/2.$

This gives the result. ∎

If we set $a=0$, then $\eta(x)=\eta(0)=0$, so we get

######  Corollary 6.4.

$g_{H}(K,X)\geq\theta(K).$

######  Corollary 6.5.

Let $q$ be a prime and let $a\in H_{2}(X;\mathbb{Z})$ be divisible by $q$.
Then

$g_{4}(K,X,a)\geq\frac{\ell^{(q)}(-K)}{(q-1)}-\frac{3\sigma^{(q)}(K)}{4(q-1)}+\frac{(q+1)}{4q}a^{2}.$

Moreover, if $q=2$ the inequality can be improved to

$g_{4}(K,X,a)\geq\ell^{(2)}(-K)-\frac{3\sigma(K)}{4}+\frac{3}{8}a^{2}-\frac{1}{2}\eta(a/2).$

###### Proof.

This follows from Theorems 6.1, 6.3 together with the estimate for
$\theta^{(q)}(K,m)$ given by Proposition 4.10. ∎

######  Example 6.6.

We compare our slice genus bound for surfaces bounding a $T_{3,6n+1}$-torus
knot in a negative definite $4$-manifold with other bounds arising from the
$\tau$-invariant and the signature.

Let $X$ be a compact, oriented, smooth negative definite $4$-manifold with
$H_{1}(X;\mathbb{Z})=0$ and with boundary $S^{3}$. Let $\Sigma\subset X$ be a
smooth, oriented, properly embedded surface in $X$ bounding $K=T_{3,6n+1}$ and
representing a class $a\in H_{2}(X,\partial X;\mathbb{Z})\cong
H^{2}(X;\mathbb{Z})$. We assume also that $a$ is divisible by $2$, so $a=2x$
for some $x\in H^{2}(X;\mathbb{Z})$. Choose a basis $e_{1},\dots,e_{r}$ for
$H^{2}(X;\mathbb{Z})$ in which the intersection form is given by
$diag(-1,\dots,-1)$. If $x=x_{1}e_{1}+\cdots+x_{r}e_{r}$ then
$x^{2}=-(x_{1}^{2}+\cdots+x_{r}^{2})$ and $\eta(x)$ is minus the number of $i$
such that $x_{i}$ is odd. It follows that $-x^{2}+\eta(x)$ is divisible by $4$
and so

$\theta(T_{3,6n+1},-x^{2}+\eta(x))=\max\\{4n,6n+x^{2}/2-\eta(x)/2\\}.$

Hence Theorem 6.3 gives the genus bound

$g(\Sigma)\geq\max\\{4n+x^{2},6n+3x^{2}/2-\eta(x)/2\\}.$

In particular, this gives

(6.1) $g(\Sigma)\geq
6n+\frac{3x^{2}}{2}=6n-\frac{3}{2}(x_{1}^{2}+\cdots+x_{r}^{2})+\frac{1}{2}\\#\\{i\;|\;x_{i}\text{
odd}\\}.$

Now we compare this with the genus bound in terms of the $\tau$ invariant
[14]. We get $g(\Sigma)\geq a^{2}/2+||a||/2+\tau(K)$, where
$||a_{1}e_{1}+\cdots+a_{r}e_{r}||=|a_{1}|+\cdots+|a_{r}|$. Since
$\tau(T_{3,6n+1})=6n$, this simplifies to

(6.2) $g(\Sigma)\geq
6n+2x^{2}+||x||=6n-2(x_{1}^{2}+\cdots+x_{r})+(|x_{1}|+\cdots+|x_{r}|).$

One also gets two genus bounds in terms of the signature [5, 19]:

(6.3) $g(\Sigma)\geq 4n+x^{2}=4n-(x_{1}^{2}+\cdots+x_{r}^{2})$

and

(6.4) $g(\Sigma)\geq-4n-x^{2}-r=-4n-r+(x_{1}^{2}+\cdots+x_{r}^{2}).$

We have obtained four genus bounds. Our bound using the $\theta$ invariant is
(6.1). The Ozsváth–Szabó bound using the $\tau$ invariant is (6.2) and the two
bounds (6.3) and (6.4) which use the signature. It is easily seen that (6.1)
is always at least as good as (6.2) and is often an improvement. It is also
easily seen that when $|x^{2}|$ is relatively small, say $|x^{2}|\leq 4n$, our
genus bound (6.1) is a better genus bound than (6.3) or (6.4). On the other
hand, when $|x^{2}|$ sufficiently large, the bound $(\ref{equ:sig2})$ is much
better than all the other bounds. In fact the right hand sides of the bounds
(6.1), (6.2) and (6.3) all become negative when $|x^{2}|$ is sufficiently
large.

## References

  * [1] D. Baraglia, P. Hekmati, Equivariant Seiberg–Witten–Floer cohomology, arXiv:2108.06855 (2021).
  * [2] J. Bodnár, D. Celoria, M. Golla, A note on cobordisms of algebraic knots. Algebr. Geom. Topol. 17 (2017), no. 4, 2543-2564.
  * [3] A. Conway, M. Nagel, Stably slice disks of links. J. Topol. 13 (2020), no. 3, 1261-1301.
  * [4] M. H. Freedman, F. Quinn, Topology of $4$-manifolds. Princeton Mathematical Series, 39. Princeton University Press, Princeton, NJ, (1990). viii+259 pp.
  * [5] P. M. Gilmer, Configurations of surfaces in $4$-manifolds. Trans. Amer. Math. Soc. 264 (1981), no. 2, 353-380.
  * [6] M. Hedden, Knot Floer homology of Whitehead doubles. Geom. Topol. 11 (2007), 2277-2338.
  * [7] J. Hom, Z. Wu, Four-ball genus bounds and a refinement of the Ozsváth–Szabó tau invariant. J. Symplectic Geom. 14 (2016), no. 1, 305-323.
  * [8] S. Jabuka, Concordance invariants from higher order covers. Topology Appl. 159 (2012), no. 10-11, 2694-2710.
  * [9] L. H. Kauffman, On knots. Annals of Mathematics Studies, 115. Princeton University Press, Princeton, NJ, (1987). xvi+481 pp.
  * [10] C. Livingston, Seifert forms and concordance. Geom. Topol. 6 (2002), 403-408.
  * [11] C. Manolescu, M. Marengon, S. Sarkar, M. Willis. A generalization of Rasmussen’s invariant, with applications to surfaces in some four-manifolds. Duke Math. J. (to appear), arXiv:1910.08195 (2019).
  * [12] C. Manolescu, B. Owens, A concordance invariant from the Floer homology of double branched covers. Int. Math. Res. Not. IMRN (2007), no. 20, Art. ID rnm077, 21 pp.
  * [13] S. Nagami, Existence of Spin structures on double branched covering spaces over four-manifolds. Osaka J. Math. 37 (2000), no. 2, 425-440.
  * [14] P. Ozsváth, Z. Szabó, Knot Floer homology and the four-ball genus. Geom. Topol. 7 (2003), 615-639.
  * [15] J. Rasmussen, Lens space surgeries and a conjecture of Goda and Teragaito. Geom. Topol. 8 (2004), 1013-1031.
  * [16] J. Rasmussen, Khovanov homology and the slice genus. Invent. Math. 182 (2010), no. 2, 419-447.
  * [17] V. A. Rokhlin, V. A. Two-dimensional submanifolds of four-dimensional manifolds. Funkcional. Anal. i Prilozen. 5 (1971), no. 1, 48-60.
  * [18] T. Sano, K. Sato, An algorithm for computing the $\Upsilon$-invariant and the $d$-invariants of Dehn surgeries, arXiv:2002.09210 (2020)
  * [19] O. J. Viro, Placements in codimension $2$ with boundary, Uspehi Mat. Nauk 30 (1975), no. 1(181), 231-232.