# The $(2,1)$-cable of the figure-eight knot is not smoothly slice

Irving Dai  Department of Mathematics, Stanford University<EMAIL_ADDRESS>,  Sungkyung Kang  Center for Geometry and Physics, Institute for Basic
Science (IBS), Pohang 37673, Korea<EMAIL_ADDRESS>,  Abhishek Mallick
Max-Planck-Institut für Mathematik<EMAIL_ADDRESS>,  JungHwan Park
Department of Mathematical Sciences, Korea Advanced Institute for Science and
Technology<EMAIL_ADDRESS>and  Matthew Stoffregen  Department of
Mathematics, Michigan State University<EMAIL_ADDRESS>
###### Abstract.

We prove that the $(2,1)$-cable of the figure-eight knot is not smoothly slice
by showing that its branched double cover bounds no equivariant homology ball.

ID was supported by NSF grant DMS-1902746. SK was supported by the Institute
for Basic Science (IBS-R003-D1). AM was supported by the Max-Planck-Institut
für Mathematik. JP was supported by Samsung Science and Technology Foundation
(SSTF-BA2102-02) and the POSCO TJ Park Science Fellowship. MS was supported by
NSF grant DMS-1952755.

##  1\. Introduction

In this paper, we prove that the $(2,1)$-cable of the figure-eight knot is not
smoothly slice, answering a forty-year-old question posed by Kawauchi
[Kaw80a]. This problem has attracted considerable attention (see for example
[BIRS16, AIM19]) due to its connection to the slice-ribbon conjecture of Fox
[Fox62, Problem 25]. Explicitly, in [Miy94, Example 2] Miyazaki showed that if
$K$ is a fibered, negative amphicheiral knot with irreducible Alexander
polynomial, then the $(2n,1)$-cable of $K$ is not (homotopy) ribbon for any
$n\neq 0$. On the other hand, these knots are known to be algebraically slice
[Kaw80b, Proposition 4.1] and rationally slice [Kaw80a, Cha07,
KW18].111Actually, [Kaw80a, Cha07, KW18] establish that such examples are
strongly rationally slice; see [KW18, Definition 2]. While such cables are
generally believed not to be slice, the fact that no argument has appeared in
the literature has left open the possibility that these generate
counterexamples to the slice-ribbon conjecture.

Prior to the current article, there has been no proof of nonsliceness for any
of these knots, even in the simplest case of the $(2,1)$-cable of the figure-
eight knot. For all of Miyazaki’s examples, the usual concordance invariants
defined using knot Floer homology [OS04a, Ras03] vanish, as these are rational
concordance invariants (see [Hom17] for a survey). One can additionally prove
that for the $(2,1)$-cable of the figure-eight knot, the concordance
invariants coming from involutive knot Floer homology [HM17] vanish (see
[HKPS20, Remark 1.4]) and the $s$-invariant [Ras10] from Khovanov homology
[Kho00] is zero.222This follows (for example) from [MMSW19, Theorem 1.8] and
the fact that the $(2,1)$-cable of the figure-eight knot bounds nullhomologous
slice disks in both $\mathbb{CP}^{2}$ and
$\smash{\mathbb{\overline{CP}}^{2}}$.

One general strategy for obstructing the sliceness of a knot is to show that
its branched double cover bounds no $\mathbb{Z}/2\mathbb{Z}$-homology ball.
This idea is well-known; see for example [CG88, End95, Lis07, MO07, GJ11,
HLR12, HPC21]. The new ingredient we use in this paper is to remember the data
of the branching involution, which will allow us to obtain a more refined
obstruction by asking for the existence of an equivariant homology ball. This
approach was employed successfully in [AKS20, DHM22] using tools from Heegaard
Floer homology [OS04c, OS04b]. Building on recent results from [DMS22, Mal22],
we extend these methods to prove:

######  Theorem 1.1.

Let $K=4_{1}$ be the figure-eight knot. Then the $(2,1)$-cable $K_{2,1}$ is
not smoothly slice. In fact, $K_{2,1}$ has infinite order in the smooth
concordance group.

More generally, our proof shows that $K_{2,1}$ is not slice in any
$\mathbb{Z}/2\mathbb{Z}$-homology ball; see Remark 5.4.

Our approach applies to many knots other than $K=4_{1}$. The precise
conditions of our theorem are slightly technical and have to do with the form
of the knot Floer homology of $K$:

######  Theorem 1.2.

Let $K$ be a Floer-thin knot with $\mathrm{Arf}(K)=1$. Then the $(2,1)$-cable
$K_{2,1}$ is not smoothly slice. In fact, $K_{2,1}$ has infinite order in the
smooth concordance group.

The definition of Floer-thin may be found in [MO08]; all alternating or quasi-
alternating knots are Floer-thin. Theorem 1.2 shows that the next few of
Miyazaki’s examples after $4_{1}$ also have $(2,1)$-cables which are not
slice: these are $6_{3},8_{12}$, and $8_{17}$. The smallest fibered, negative
amphicheiral knot with irreducible Alexander polynomial which is not subsumed
by Theorem 1.2 is $10_{17}$, which has Arf invariant zero.

## Acknowledgements

The authors would like to thank Jae Choon Cha, Matthew Hedden, and Ian Zemke
for helpful discussions.

##  2\. General Overview

The topological outline of our strategy is quite straightforward. If $K$ is a
knot with slice disk $D$, then the branched double cover $\Sigma_{2}(K)$
bounds a $\mathbb{Z}/2\mathbb{Z}$-homology ball given by the branched double
cover $\Sigma_{2}(D)$. One can thus obstruct the sliceness of $K$ by showing
that $\Sigma_{2}(K)$ does not bound any $\mathbb{Z}/2\mathbb{Z}$-homology
ball, essentially moving the question into the setting of homology cobordism.
In recent years, a refinement of this strategy has emerged: in addition to
considering the 3-manifold $\Sigma_{2}(K)$, we may also remember the action of
the branching involution $\tau\colon\Sigma_{2}(K)\rightarrow\Sigma_{2}(K)$. We
can then ask if $\Sigma_{2}(K)$ bounds an equivariant
$\mathbb{Z}/2\mathbb{Z}$-homology ball; that is, a homology ball $W$ over
which $\tau$ extends as a smooth involution $\tau_{W}\colon W\rightarrow W$.
If $K$ has slice disk $D$, then such an extension is obtained by taking
$\tau_{W}$ to be the branching action over $D$. Thus, asking for the existence
of an equivariant homology ball also provides an obstruction to the sliceness
of $K$, which is (in principle) a strictly stronger refinement of the
previous.333For example, it is easy to see that $\Sigma_{2}((4_{1})_{2,1})$
indeed bounds a $\mathbb{Z}/2\mathbb{Z}$-homology ball; see Remark 2.2. Here,
we show that it bounds no equivariant ball. This strategy was employed
successfully in several examples by the second author (in joint work with
Alfieri and Stipsicz [AKS20]) and the first and third authors (in joint work
with Hedden [DHM22]); see also [BH21, KMT21].

In both [AKS20] and [DHM22], the obstruction to the existence of such a
homology ball was established through the use of Heegaard Floer homology. We
recall the essential features of this formalism here. Let $Y$ be a 3-manifold
equipped with a ${\operatorname{spin^{c}}}$-structure $\mathfrak{s}$. In
[OS04c, OS04b], Ozsváth and Szabó construct a free, finitely-generated chain
complex $\mathit{CF}^{-}(Y,\mathfrak{s})$ over the polynomial ring
$\mathbb{F}[U]$ (where $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$) whose homotopy
type is a diffeomorphism invariant of the pair $(Y,\mathfrak{s})$. (For a
discussion and proof of naturality, see [JTZ21].) As explained in [OS06],
$\mathit{CF}^{-}(Y,\mathfrak{s})$ has a relative $\mathbb{Z}$-grading with
$\mathrm{gr}(U)=-2$; if $\mathfrak{s}$ is torsion then this can be lifted to
an absolute $\mathbb{Q}$-grading. We briefly review the important aspects of
Heegaard Floer homology which will be relevant for the present article.

Firstly, we have the $\mathrm{TQFT}$ structure of Heegaard Floer homology
[OS06]. Let $W$ be a cobordism from $Y_{1}$ to $Y_{2}$ and let $\mathfrak{s}$
be a ${\operatorname{spin^{c}}}$-structure on $W$ restricting to
$\mathfrak{s}_{i}=\mathfrak{s}|_{Y_{i}}$. Then Ozsváth and Szabó construct a
Heegaard Floer cobordism map:

$F_{W,\mathfrak{s}}\colon\mathit{CF}^{-}(Y_{1},\mathfrak{s}_{1})\rightarrow\mathit{CF}^{-}(Y_{2},\mathfrak{s}_{2}).$

Secondly, by work of Juhász, Thurston, and Zemke, the mapping class group
$\mathrm{MCG}(Y)$ acts naturally on $\mathit{CF}^{-}(Y)$ [JTZ21]. More
precisely, let $\tau$ be a self-diffeomorphism of $Y$, and let $\mathfrak{s}$
be a ${\operatorname{spin^{c}}}$-structure on $Y$ which is invariant under
pullback by $\tau$. Then $\tau$ induces a grading-preserving,
$\mathbb{F}[U]$-equivariant map

$\tau\colon\mathit{CF}^{-}(Y,\mathfrak{s})\rightarrow\mathit{CF}^{-}(Y,\mathfrak{s}),$

which by abuse of notation we also denote by $\tau$. There is a slight
subtlety in that generally speaking, only the based mapping class group acts
naturally on $\mathit{CF}^{-}(Y,\mathfrak{s})$. However, if $Y$ is a rational
homology sphere (as will be the case in our setting), it follows from work of
Zemke [Zem15] that this subtlety can be disregarded; see [DHM22, Lemma 4.1].
Finally, we also make essential use of the involutive Heegaard Floer package
of Hendricks and Manolescu [HM17]. This is an enhancement of the usual
Heegaard Floer formalism which utilizes the conjugation symmetry present in a
Heegaard diagram. If $\mathfrak{s}$ is a self-conjugate
${\operatorname{spin^{c}}}$-structure on $Y$, then Hendricks and Manolescu
define a grading-preserving, $\mathbb{F}[U]$-equivariant homotopy involution

$\iota\colon\mathit{CF}^{-}(Y,\mathfrak{s})\rightarrow\mathit{CF}^{-}(Y,\mathfrak{s}).$

The homotopy type of the pair $(\mathit{CF}^{-}(Y,\mathfrak{s}),\iota)$ is a
diffeomorphism invariant of $(Y,\mathfrak{s})$; $\iota$ should be thought of
as an additional internal symmetry of $\mathit{CF}^{-}(Y,\mathfrak{s})$. The
importance of considering $\iota$ in addition to $\tau$ was observed in
[DHM22]; in the current article, the usage of $\iota$ will likewise be
essential.

The crucial point is the following:

######  Theorem 2.1.

[DHM22, Proof of Theorem 1.2] Let $W$ be a rational homology cobordism between
rational homology spheres $Y_{1}$ and $Y_{2}$. Let $\mathfrak{s}$ be a
${\operatorname{spin^{c}}}$-structure on $W$ restricting to
$\mathfrak{s}_{i}=\mathfrak{s}|_{Y_{i}}$. Then the Heegaard Floer cobordism
map

$F_{W,\mathfrak{s}}\colon\mathit{CF}^{-}(Y_{1},\mathfrak{s}_{1})\rightarrow\mathit{CF}^{-}(Y_{2},\mathfrak{s}_{2})$

satisfies the following properties:

  1. (1)

The Heegaard Floer grading shift of $F_{W,\mathfrak{s}}$ is zero.

  2. (2)

The induced map on the localized homology

$H_{*}(U^{-1}\mathit{CF}^{-}{(Y_{1},\mathfrak{s}_{1})})\xrightarrow{\cong}H_{*}(U^{-1}\mathit{CF}^{-}{(Y_{2},\mathfrak{s}_{2})})$

is an isomorphism.444It is a standard structural result that
$H_{*}(U^{-1}\mathit{CF}^{-}{(Y,\mathfrak{s})})=\mathit{HF}^{\infty}(Y,\mathfrak{s})=\mathbb{F}[U,U^{-1}]$
for any rational homology sphere $Y$.

Suppose moreover that $\mathfrak{s}$ is self-conjugate. Then:

  1. (3)

Up to homotopy, $F_{W,\mathfrak{s}}$ intertwines the $\iota$-action on
$\mathit{CF}^{-}(Y,\mathfrak{s}_{1})$ with the $\iota$-action on
$\mathit{CF}^{-}(Y,\mathfrak{s}_{2})$:

$F_{W,\mathfrak{s}}\circ\iota_{1}\simeq\iota_{2}\circ F_{W,\mathfrak{s}}.$

Finally, let $\tau$ be a self-diffeomorphism of $W$ restricting to
$\tau_{i}=\tau|_{Y_{i}}$. Suppose $\mathfrak{s}$ is invariant under $\tau$.
Then:

  1. (4)

Up to homotopy, $F_{W,\mathfrak{s}}$ intertwines the action of $\tau_{1}$ on
$\mathit{CF}^{-}(Y,\mathfrak{s}_{1})$ with the action of $\tau_{2}$ on
$\mathit{CF}^{-}(Y,\mathfrak{s}_{2})$:

$F_{W,\mathfrak{s}}\circ\tau_{1}\simeq\tau_{2}\circ F_{W,\mathfrak{s}}.$

The first two parts of Theorem 2.1 are standard results in the literature,
while the third is work of Hendricks and Manolescu [HM17]. The only subtlety
in establishing the remaining claim is to analyze the choice of path needed to
define $F_{W,\mathfrak{s}}$ (see [Zem15]), since this may not be invariant
under $\tau$; see [DHM22, Section 6]. This is done in the proof of [DHM22,
Theorem 1.2] (in fact, for a much wider class of cobordisms). We call a map
satisfying the conditions of Theorem 2.1 a local map, or sometimes a
$(\tau,\iota)$-local map for emphasis. Given two complexes $C_{1}$ and $C_{2}$
as above with actions of $\iota$ and $\tau$, we say that $C_{1}$ and $C_{2}$
are locally equivalent if there are local maps $f\colon C_{1}\rightarrow
C_{2}$ and $g\colon C_{2}\rightarrow C_{1}$ in both directions. Occasionally,
we will use this terminology but relax the grading shift condition when our
meaning is clear.

To see the relevance of Theorem 2.1, let $Y$ be a
$\mathbb{Z}/2\mathbb{Z}$-homology ball with involution $\tau$ and suppose
there exists an equivariant $\mathbb{Z}/2\mathbb{Z}$-homology ball $W$ with
boundary $Y$. Note that $W$ has a unique self-conjugate
${\operatorname{spin^{c}}}$-structure $\mathfrak{s}$ which is necessarily
preserved by the extension $\tau_{W}$; write
$\mathfrak{s}|_{Y}=\mathfrak{s}_{0}$. Puncturing $W$ gives a
$(\tau,\iota)$-local equivalence between the trivial complex
$\mathit{CF}^{-}(S^{3})=\mathbb{F}[U]$ and
$\mathit{CF}^{-}(Y,\mathfrak{s}_{0})$, with the actions of $\iota$ and $\tau$
on the former being the identity. Here and throughout, we normalize gradings
so that the uppermost generator of $\mathit{CF}^{-}(S^{3})$ lies in Maslov
grading zero; this differs slightly from the usual convention in which
$\mathit{CF}^{-}(S^{3})$ starts in Maslov grading $-2$.

Our strategy will hence be to compute the actions of $\iota$ and $\tau$ on the
Heegaard Floer complex of $\Sigma_{2}(K_{2,1})$ and then algebraically rule
out the existence of a local map from the trivial complex. Techniques for
establishing the nonexistence of maps as in Theorem 2.1 have been developed by
many authors and have led to the formalism of the local equivalence group; see
especially the work of Hendricks, Manolescu, and Zemke [HMZ18]. Although it
will be helpful for the reader to have a broad familiarity with these ideas,
for the main claim of Theorem 1.1 this will not be needed. See [AKS20, Section
3] or [DHM22, Section 3] for a more comprehensive discussion of local
equivalence in this setting.

The main crux of this paper will thus be the computation of $\iota$ and
$\tau$. In general, however, determining the induced action of $\tau$ is
difficult. Here, we make essential use of the following topological
observation: for any knot $K$, the Montesinos trick [Mon75] shows that the
branched double cover $\Sigma_{2}(K_{2,1})$ is homeomorphic to
$S^{3}_{+1}(K\\#K^{r})$; see for example the algorithm of [Sav02, Section
1.1.12]. Moreover, this homeomorphism identifies the branching action on
$\Sigma_{2}(K_{2,1})$ with the involution on $S^{3}_{+1}(K\\#K^{r})$ induced
from the obvious strong inversion on $K\\#K^{r}$ interchanging the two
factors. (See Figure 2.1.) We refer to this strong inversion as the swapping
involution.

Figure 2.1. The swapping involution on $K\\#K^{r}$. This is constructed as
follows: fix an axis of rotation far away from $K$, and rotate $K$ about this
axis by $180^{\circ}$ degrees. Reversing orientation on the resulting image
and summing along the obvious band gives $K\\#K^{r}$.

In recent work [DMS22], the first, third, and fifth authors carried out a
program for studying involutions on knots using the knot Floer homology
package. (For background on knot Floer homology, see [OS04a, Ras03].)
Explicitly, in [DMS22, Section 3] it is shown that a strong inversion on a
knot induces an automorphism of its knot Floer complex (see also [Mal22,
Section 2.2]). Building on work of Juhász and Zemke [JZ18, Theorem 8.1], in
[DMS22, Theorem 4.3] the authors computed the induced action for such examples
$K\\#K^{r}$ as in Figure 2.1. Combining this with a large surgery formula due
to the third author [Mal22, Theorem 1.1], this will allow us to compute the
induced action of $\tau$ on $\smash{S^{3}_{n}(K\\#K^{r})}$ for $n$ large. We
then use an auxiliary argument involving an equivariant, negative definite
cobordism from $n$-surgery to $(+1)$-surgery to constrain the induced action
of $\tau$ on $\smash{S^{3}_{+1}(K\\#K^{r})}$. By the above topological
observation, this is the same as the branching action on
$\Sigma_{2}(K_{2,1})$.

######  Remark 2.2.

The discussion of this section can be summarized succinctly as follows. Taking
double branched covers yields a homomorphism from the smooth concordance group
to the $(\mathbb{Z}/2\mathbb{Z})$-homology cobordism group
$\smash{\Theta^{3}_{\mathbb{Z}/2\mathbb{Z}}}$. In [DHM22, Section 2], a
refinement of $\smash{\Theta^{3}_{\mathbb{Z}/2\mathbb{Z}}}$ called the
$(\mathbb{Z}/2\mathbb{Z})$-homology bordism group of involutions was defined.
Denoted by $\smash{\Theta^{\tau}_{\mathbb{Z}/2\mathbb{Z}}}$, this additionally
takes into account an involution on each homology sphere. Remembering the
branching action gives a homomorphism from $\mathcal{C}$ to
$\smash{\Theta^{\tau}_{\mathbb{Z}/2\mathbb{Z}}}$, through which the previous
homomorphism factors:

$\Sigma_{2}\colon\mathcal{C}\xrightarrow{2\text{x
cover}}\Theta^{\tau}_{\mathbb{Z}/2\mathbb{Z}}\xrightarrow{\text{forgetful
map}}\Theta^{3}_{\mathbb{Z}/2\mathbb{Z}}.$

The approach of the present article is to show that $K_{2,1}$ is not slice by
showing that its image is nonzero in
$\smash{\Theta^{\tau}_{\mathbb{Z}/2\mathbb{Z}}}$. Note that (for example)
$(4_{1})_{2,1}$ does have trivial image in
$\smash{\Theta^{3}_{\mathbb{Z}/2\mathbb{Z}}}$: since $4_{1}\\#4_{1}^{r}$ is
slice, $S^{3}_{+1}(4_{1}\\#4_{1}^{r})$ bounds a homology ball (in fact, a
contractible manifold). However, $4_{1}\\#4_{1}^{r}$ is not equivariantly
slice; while $4_{1}^{r}$ is isotopic to $-4_{1}$, this isotopy does not
preserve the swapping involution. Readers familiar with the work of Lin,
Ruberman, and Saveliev [LRS18] will recognize that we show
$S^{3}_{+1}(4_{1}\\#4_{1}^{r})$ (with the branching involution) is a
$(\mathbb{Z}/2\mathbb{Z})$-strong cork. See [LRS18, Section 1.2.2] or [DHM22,
Definition 1.8] for further discussion.

##  3\. Analysis of large surgery

We now describe how to compute the action of the swapping involution on
$\mathit{CFK}(K\\#K^{r})$. Throughout, we use Zemke’s conventions for knot
Floer homology [Zem19c, Zem19b]. This assigns to a (doubly-based, oriented)
knot $K$ a bigraded complex $\mathit{CFK}(K)$ over the polynomial ring
$\mathbb{F}[\mathcal{U},\mathcal{V}]$. Denote the two gradings by
$\mathrm{gr}=(\mathrm{gr}_{U},\mathrm{gr}_{V})$; we have
$\mathrm{gr}(\mathcal{U})=(-2,0)$, $\mathrm{gr}(\mathcal{V})=(0,-2)$, and
$\mathrm{gr}(\partial)=(-1,-1)$. See [Zem19b, Section 1.5] for further
discussion.

Building on the naturality results of [Zem19c], in [DHM22, Section 3] it is
shown that a strong inversion $\tau$ on $K$ induces a well-defined homotopy
involution $\tau_{K}$ on $\mathit{CFK}(K)$. This is a skew-graded, skew-
equivariant automorphism of $\mathit{CFK}(K)$ in the sense that $\tau_{K}$
interchanges $\mathrm{gr}_{U}$ and $\mathrm{gr}_{V}$ and satisfies
$\tau_{K}(\mathcal{U}^{i}\mathcal{V}^{j}x)=\mathcal{V}^{i}\mathcal{U}^{j}\tau_{K}(x)$.
It will also be important for us to consider the involutive knot Floer map
defined by Hendricks and Manolescu in [HM17]. This is similarly a skew-graded,
skew-equivariant automorphism of $\mathit{CFK}(K)$ which we denote by
$\iota_{K}$. See [HM17, Section 6] for further discussion.

###  3.1. The swapping involution

Fix any knot $K$. By a slight abuse of notation, throughout this subsection we
write $\tau$ for the $180^{\circ}$-degree rotation about an axis far away from
$K$. We form the connected sum $K\\#\tau K^{r}$ by reversing orientation on
$\tau K$ and summing along an obvious band as in Figure 2.1. While $\tau K$ is
of course isotopic to $K$, in order to discuss the action of the swapping
involution it will be helpful for us to clearly distinguish between the two.
We thus follow the notation of [DMS22, Theorem 4.3] and temporarily write
$K\\#\tau K^{r}$, rather than $K\\#K^{r}$. Denote the swapping involution on
$K\\#\tau K^{r}$ by $\tau_{\mathrm{sw}}$.

The following tensor product formula allows us to simultaneously calculate the
action of $\tau_{\mathrm{sw}}$ and the involutive knot Floer map on
$\mathit{CFK}(K\\#\tau K^{r})$:

######  Theorem 3.1.

[DMS22, Theorem 4.3] The triples

$(\mathit{CFK}(K\\#\tau K^{r}),\tau_{\mathrm{sw}},\iota_{K\\#\tau
K^{r}})\quad\text{and}\quad(\mathit{CFK}(K)\otimes\mathit{CFK}(\tau
K^{r}),\tau_{\otimes},\iota_{\otimes})$

are homotopy equivalent, where

$\tau_{\otimes}=(\mathrm{id}\otimes\mathrm{id}+\Psi\otimes\Phi)\circ\tau_{\mathrm{exch}}$

and

$\iota_{\otimes}=\varsigma_{\otimes}\circ(\mathrm{id}\otimes\mathrm{id}+\Psi\otimes\Phi)\circ(\iota_{K}\otimes\iota_{\tau
K^{r}}).$

For convenience of the reader, we briefly describe the terms appearing in
Theorem 3.1. The most important map to explain is $\tau_{\mathrm{exch}}$,
which should be thought of as the induced action of $\tau$ on the Floer
homology of $K\sqcup\tau K^{r}$. Note that $\tau$ interchanges the two
components of $K\sqcup\tau K^{r}$ and reverses orientation on both; this
parallels the fact that a strong inversion on a knot is orientation-reversing.
We first give an actual definition of $\tau_{\mathrm{exch}}$ for the reader
more familiar with knot Floer homology, followed by an explanation of how to
compute $\tau_{\mathrm{exch}}$ in practice.

Let $w$ and $z$ be an ordered pair of basepoints on $K$. Fixing Heegaard data
$\mathcal{H}$ compatible with $w$, $z$, and the orientation on $K$ defines the
knot Floer complex $\mathit{CFK}(K,w,z)$. There are three other closely-
related sets of Heegaard data which we will need to consider:

  1. (1)

Keeping the same Heegaard splitting (including the designation of the
$\alpha$\- and $\beta$-curves and the orientation of the Heegaard surface) but
interchanging the roles of $w$ and $z$ gives a set of Heegaard data for the
reverse of $K$. Denote the corresponding knot Floer complex by
$\mathit{CFK}(K^{r},z,w)$.

  2. (2)

The pushforward of $\mathcal{H}$ under $\tau$ forms a set of Heegaard data for
$(\tau K,\tau w,\tau z)$. Denote the corresponding knot Floer complex by
$\mathit{CFK}(\tau K,\tau w,\tau z)$.

  3. (3)

Taking the pushforward of $\mathcal{H}$ under $\tau$ and then interchanging
the roles of $\tau w$ and $\tau z$ gives a set of Heegaard data for $(\tau
K^{r},\tau z,\tau w)$. Denote the corresponding knot Floer complex by
$\mathit{CFK}(\tau K^{r},\tau z,\tau w)$.

These complexes are all isomorphic to each other in different ways. Firstly,
note that $\tau$ induces a tautological isomorphism from $\mathit{CFK}(K,w,z)$
to $\mathit{CFK}(\tau K,\tau w,\tau z)$, and likewise an isomorphism from
$\mathit{CFK}(\tau K^{r},\tau z,\tau w)$ to $\mathit{CFK}(K^{r},z,w)$. Using
this, define a map

$t\colon\mathit{CFK}(K,w,z)\otimes\mathit{CFK}(\tau K^{r},\tau z,\tau
w)\rightarrow\mathit{CFK}(K^{r},z,w)\otimes\mathit{CFK}(\tau K,\tau w,\tau z)$

which takes the first factor on the left isomorphically onto the second factor
on the right, and the second factor on the left isomorphically onto the first
factor on the right.

Secondly, observe that we have a tautological map
$\mathrm{sw}\colon\mathit{CFK}(K^{r},z,w)\rightarrow\mathit{CFK}(K,w,z)$ given
by mapping intersection-points to intersection-points. Unlike the previous
isomorphisms, this is skew-graded and skew-equivariant. We call $\mathrm{sw}$
the switch map; see [DHM22, Definition 3.1] for further discussion. Similarly,
there is a switch map from $\mathit{CFK}(\tau K,\tau w,\tau z)$ to
$\mathit{CFK}(\tau K^{r},\tau z,\tau w)$. Define

$\mathrm{sw}\otimes\mathrm{sw}\colon\mathit{CFK}(K^{r},z,w)\otimes\mathit{CFK}(\tau
K,\tau w,\tau z)\rightarrow\mathit{CFK}(K,w,z)\otimes\mathit{CFK}(\tau
K^{r},\tau z,\tau w)$

by taking the switch map in both factors. We then define
$\tau_{\mathrm{exch}}$ to be the composition

$\tau_{\mathrm{exch}}=(\mathrm{sw}\otimes\mathrm{sw})\circ t.$

Note that $\tau_{\mathrm{exch}}$ is skew-graded and skew-equivariant. Roughly
speaking, $t$ exchanges the complexes of $K$ and $\tau K^{r}$ using the
tautological pushforward under $\tau$, but we must apply
$\mathrm{sw}\otimes\mathrm{sw}$ to account for the fact that $\tau$ is
orientation-reversing on each component.

In practice, computing the action of $\tau_{\mathrm{exch}}$ is quite
straightforward. Start with a knot Floer complex
$\mathit{CFK}(K)=\mathit{CFK}(K,w,z)$ for $K$, as schematically depicted on
the left in Figure 3.1. Reflecting this complex over a diagonal line gives the
complex $\mathit{CFK}(K^{r})=\mathit{CFK}(K^{r},z,w)$ for $K^{r}$. This is the
same (via the tautological pushforward) as the complex $\mathit{CFK}(\tau
K^{r})=\mathit{CFK}(\tau K^{r},\tau z,\tau w)$ for $\tau K^{r}$. Note that the
switch map isomorphism from $\mathit{CFK}(K)$ to $\mathit{CFK}(K^{r})$ given
by reflection over the diagonal line is indeed a skew-equivariant, skew-graded
chain map. The action of $\tau_{\mathrm{exch}}$ on
$\mathit{CFK}(K)\otimes\mathit{CFK}(\tau K^{r})$ is then given by the
interchange-and-reflect operation, as in the right of Figure 3.1.

Figure 3.1. A schematic depiction of $\tau_{\mathrm{exch}}$. Here, we have
deliberately chosen an asymmetric complex for $\mathit{CFK}(K)$ to better
illustrate the switch map from $\mathit{CFK}(K)$ to $\mathit{CFK}(K^{r})$.

The second subtlety in Theorem 3.1 is the determination of the knot Floer
involution $\iota_{\tau K^{r}}$ on $\mathit{CFK}(\tau K^{r})=\mathit{CFK}(\tau
K^{r},\tau z,\tau w)$. In Figure 3.1 we have identified $\mathit{CFK}(\tau
K^{r},\tau z,\tau w)$ with the reflection of $\mathit{CFK}(K,w,z)$. However,
the map $\iota_{\tau K^{r}}$ should not be identified with the reflection of
the map $\iota_{K}$ on $\mathit{CFK}(K,w,z)$. This is because the skew
isomorphism

$\mathrm{sw}\colon\mathit{CFK}(K,w,z)\rightarrow\mathit{CFK}(K^{r},z,w)$

does not intertwine $\iota_{K}$ on the left with $\iota_{K^{r}}$ on the right.
Instead, it follows from arguments as in [DHM22, Lemma 3.4] or [DHM22, Lemma
3.8] that $\mathrm{sw}$ intertwines $\iota_{K}$ on the left with
$\varsigma_{K^{r}}\circ\iota_{K^{r}}$ on the right, where $\varsigma_{K^{r}}$
is the Sarkar map (see below). Indeed, for any oriented knot $K$, let
$\rho_{K}$ be the pushforward isomorphism on the knot Floer complex induced by
the half-Dehn twist along the orientation of $K$; denote the half-Dehn twist
in the opposite direction by $\bar{\rho}_{K}$. Recall from [HM17, Section 6]
that $\iota_{K}$ is defined by composing the conjugation map $\eta$ with
$\rho_{K}$ (followed by a naturality map returning to the original Heegaard
data). By [DHM22, Lemma 3.4],

$\iota_{K^{r}}\circ\mathrm{sw}=(\rho_{K^{r}}\circ\eta)\circ\mathrm{sw}=\mathrm{sw}\circ(\bar{\rho}_{K}\circ\eta)\simeq\mathrm{sw}\circ(\bar{\rho}_{K}^{2})\circ(\rho_{K}\circ\eta)=\mathrm{sw}\circ(\varsigma_{K}^{-1}\circ\iota_{K}).$

Roughly speaking, the point is that $\iota_{K}$ is defined using a half-Dehn
twist following the orientation of $K$. However, from the point of view of
$K^{r}$, this half-Dehn twist goes against the orientation of the knot in
question; see [DHM22, Section 3.1] for further discussion. Using the fact that
$\varsigma_{K}$ is its own (homotopy) inverse, this shows
$\iota_{K^{r}}\simeq\mathrm{sw}\circ(\varsigma_{K}\circ\iota_{K})\circ\mathrm{sw}$.

Finally, we briefly remind the reader of the meaning of the maps $\Psi$,
$\Phi$, and $\varsigma$. The first two of these were introduced by Zemke in
[Zem17, Section 3] and are the formal derivatives of $\partial$ with respect
to $\mathcal{U}$ and $\mathcal{V}$:

$\Phi=\left(\dfrac{d}{d\mathcal{U}}\partial\right)\quad\text{and}\quad\Psi=\left(\dfrac{d}{d\mathcal{V}}\partial\right).$

The map $\varsigma$ is the Sarkar basepoint-moving map [Sar15]. In [Zem17,
Theorem B], it is shown that

$\varsigma\simeq\mathrm{id}+\Phi\circ\Psi.$

In the statement of Theorem 3.1, $\Psi$ acts on $\mathit{CFK}(K)$ while $\Phi$
acts on $\mathit{CFK}(\tau K^{r})$; the map $\varsigma_{\otimes}$ is the
Sarkar map on the entire tensor product complex
$\mathit{CFK}(K)\otimes\mathit{CFK}(\tau K^{r})$.

We are now finally ready to put everything together. Let $K=4_{1}$. As shown
in [HM17, Section 8.2], the knot Floer complex $\mathit{CFK}(K)$ for $K$
(together with the action of $\iota_{K}$) is homotopy equivalent to the
complex displayed on the left in Figure 3.2. Following the above discussion,
we reflect this complex across a diagonal line to obtain $\mathit{CFK}(\tau
K^{r})$. The action of $\iota_{\tau K^{r}}$ is displayed on the right in
Figure 3.2.

Figure 3.2. The knot Floer complex $\mathit{CFK}(K)$ (left) and
$\mathit{CFK}(\tau K^{r})$ (right). Horizontal and vertical arrows denote
$\mathcal{U}$ and $\mathcal{V}$ terms in the differential. We label the
generators by $\\{x,a,b,c,d\\}$ in both cases, so that
$\tau_{\mathrm{exch}}(p\otimes q)=q\otimes p$. On the left, the gradings of
these are as follows: $\mathrm{gr}(x)=\mathrm{gr}(a)=\mathrm{gr}(d)=(0,0)$,
$\mathrm{gr}(b)=(1,-1)$, and $\mathrm{gr}(c)=(-1,1)$. The gradings on the
right are the same, except $\mathrm{gr}(b)=(-1,1)$ and
$\mathrm{gr}(c)=(1,-1)$.

We may then take the tensor product of $\mathit{CFK}(K)$ and
$\mathit{CFK}(\tau K^{r})$ and perform the algebra of Theorem 3.1 to compute
$\tau_{\mathrm{sw}}$ and $\iota_{K\\#\tau K^{r}}$. In Figure 3.3, we have
displayed an especially convenient basis for
$\mathit{CFK}(K)\otimes\mathit{CFK}(\tau K^{r})$, which will be useful
presently. However, since we will not actually need to know the actions of
$\tau_{\mathrm{sw}}$ and $\iota_{K\\#\tau K^{r}}$ on the entire complex, we
suppress the full computation. Instead, it will suffice for us to discuss the
action on the large surgery subcomplex, which we do below.

Figure 3.3. The knot Floer complex of $K\\#K^{r}$. Horizontal and vertical
arrows denote $\mathcal{U}$ and $\mathcal{V}$ terms in the differential.

######  Remark 3.2.

The reader familiar with involutive knot Floer homology may note that the
connected sum involution $\iota_{\otimes}$ in Theorem 3.1 appears to be
different from the connected sum formula of [Zem19a, Theorem 1.1]. This is not
actually material: it follows from [DMS22, Lemma 2.21] that defining

$\tau_{\otimes}^{\prime}=\varsigma_{\otimes}\circ(\mathrm{id}\otimes\mathrm{id}+\Psi\otimes\Phi)\circ\tau_{\mathrm{exch}}$

and

$\iota_{\otimes}^{\prime}=(\mathrm{id}\otimes\mathrm{id}+\Psi\otimes\Phi)\circ(\iota_{K}\otimes\iota_{\tau
K^{r}}).$

results in a homotopy equivalent triple, bringing us into alignment with
[Zem19a, Theorem 1.1]. Likewise, the reader may wonder why the formula for
$\tau_{\otimes}$ is not simply $\tau_{\mathrm{exch}}$. As in [Zem19a, Theorem
1.1], this is due to a subtlety involving the definition of $\tau_{K}$ and the
choice of decoration for the usual pair-of-pants cobordism from $K\sqcup\tau
K^{r}$ to $K\\#\tau K^{r}$; see [DMS22, Section 4.2]. In fact, one can show
that taking $\tau_{\otimes}=\tau_{\mathrm{exch}}$ does not satisfy the
commutation relation with $\iota_{\otimes}$ required by [DMS22, Theorem 1.7],
unlike the expression for $\tau_{\otimes}$ in Theorem 3.1.

###  3.2. The large surgery formula

For any knot $K$, let $A_{0}(K)$ denote the subset of $\mathit{CFK}(K)$ lying
in Alexander grading $A=(\mathrm{gr}_{U}-\mathrm{gr}_{V})/2=0$. We think of
this as a singly-graded chain complex over $\mathbb{F}[U]$, where
$U=\mathcal{U}\mathcal{V}$ and the Maslov grading is given by
$\mathrm{gr}_{U}=\mathrm{gr}_{V}$. It is straightforward to check that
$\iota_{K}$ and $\tau_{K}$ both preserve $A_{0}(K)$. We may thus view both of
these as $\mathbb{F}[U]$-equivariant, grading-preserving chain automorphisms
on $A_{0}(K)$, which (by abuse of notation) we continue to denote by
$\iota_{K}$ and $\tau_{K}$.

The large surgery formula of Ozsváth and Szabó [OS04a] (see also [Ras03])
states that if $n\geq g_{3}(K)$, then we have a homotopy equivalence of
$\mathbb{F}[U]$-complexes

$\Gamma_{n,0}\colon\mathit{CF}^{-}(S^{3}_{n}(K),\mathfrak{s}_{0})\xrightarrow{\simeq}A_{0}(K).$

Here, $\mathfrak{s}_{0}$ denotes the (self-conjuate)
${\operatorname{spin^{c}}}$-structure on $S^{3}_{n}(K)$ which corresponds to
the element $0\in\mathbb{Z}/n\mathbb{Z}$ under the identification defined in
[OS08, Lemma 2.2]. This homotopy equivalence is relatively graded with grading
shift $-(n-1)/4$.555That is, an element of
$\mathit{CF}^{-}(S^{3}_{n}(K),\mathfrak{s}_{0})$ with grading $(n-1)/4$ has
Maslov grading zero in $A_{0}(K)$. Moreover, by [HM17, Theorem 6.8] and
[Mal22, Theorem 1.1], $\Gamma_{n,0}$ intertwines $\iota$ with $\iota_{K}$ and
$\tau$ with $\tau_{K}$ (up to homotopy), so that we get a homotopy equivalence
of triples

$\Gamma_{n,0}\colon(\mathit{CF}^{-}(S^{3}_{n}(K),\mathfrak{s}_{0}),\tau,\iota)\xrightarrow{\simeq}(A_{0}(K),\tau_{K},\iota_{K}).$

See the proof of [Mal22, Corollary 1.3].

We now return to the case of $K=4_{1}$. In light of the large surgery formula,
we restrict our attention to the large surgery subcomplex $A_{0}(K\\#K^{r})$.
In fact, it turns out that for our purposes it suffices to understand the
actions of $\tau_{\mathrm{sw}}$ and $\iota_{K\\#K^{r}}$ on the homology of
this complex. From Figure 3.3, it is straightforward to verify that
$H_{*}(A_{0}(K\\#K^{r}))$ is generated by the classes

$[x|x],\quad[x|d],\quad[d|x],\quad[a|d+d|a+b|b+c|c],\quad\text{and}\quad[d|d],$

all lying in Maslov grading zero. (We temporarily use $|$ in place of
$\otimes$ for brevity.) The first of these is $U$-nontorsion, while the others
have $U$-torsion order one. In Table 1, we have computed the actions of
$\iota$ and $\tau$ on these using Theorem 3.1. To illustrate the calculation,
we do the most complicated case of $a|d+d|a+b|b+c|c$ and leave the other
entries to the reader. We have:

$\displaystyle(\iota_{K}|\iota_{\tau K^{r}})(a|d+d|a+b|b+c|c)$
$\displaystyle=(a+x)|d+d|(a+x+d)+c|c+b|b$
$\displaystyle=a|d+d|a+b|b+c|c+x|d+d|x+d|d.$

Among these terms, the action of $\Psi|\Phi$ vanishes outside of
$(\Psi|\Phi)(b|b)=d|d$. The action of $\varsigma_{\otimes}$ is then easily
checked to be the identity; hence
$\iota_{\otimes}=\varsigma_{\otimes}\circ(\mathrm{id}|\mathrm{id}+\Psi|\Phi)\circ(\iota_{K}|\iota_{\tau
K^{r}})$ is as displayed on the fourth row of Table 1. Similarly,

$\displaystyle\tau_{\mathrm{exch}}(a|d+d|a+b|b+c|c)$
$\displaystyle=d|a+a|d+b|b+c|c.$

The same computation of $\Psi|\Phi$ then gives the action of
$\tau_{\otimes}=(\mathrm{id}|\mathrm{id}+\Phi|\Psi)\circ\tau_{\mathrm{exch}}$
in the fourth row of Table 1.

Cycles | Homology class | Image under $\iota$ | Image under $\tau$
---|---|---|---
$x|x$ | free | $x|x+x|d+d|x+d|d$ | $x|x$
$x|d$ |  $U$-torsion | $x|d+d|d$ | $d|x$
$d|x$ |  $U$-torsion | $d|x+d|d$ | $x|d$
$a|d+d|a+b|b+c|c$ |  $U$-torsion | $a|d+d|a+b|b+c|c+x|d+d|x$ | $a|d+d|a+b|b+c|c+d|d$
$d|d$ |  $U$-torsion | $d|d$ | $d|d$
Table 1. Actions of $\iota$ and $\tau$ on $H_{*}(A_{0}(4_{1}\\#4_{1}^{r}))$.

The crucial observation is now the following:

######  Lemma 3.3.

There is no $U$-nontorsion class in $H_{*}(A_{0}(4_{1}\\#4_{1}^{r}))$ which
has Maslov grading zero and is invariant under both $\iota$ and $\tau$.

###### Proof.

With respect to the ordered basis
$\\{[x|x],[x|d],[d|x],[a|d+d|a+b|b+c|c],[d|d]\\}$, the action of
$\mathrm{id}+\iota$ is given by the matrix

$\mathrm{id}+\iota=\begin{pmatrix}0&0&0&0&0\\\ 1&0&0&1&0\\\ 1&0&0&1&0\\\
0&0&0&0&0\\\ 1&1&1&0&0\end{pmatrix}.$

Hence we see that the $\iota$-invariant subspace of
$H_{*}(A_{0}(4_{1}\\#4_{1}^{r}))$ is generated by

$[x|x]+[x|d]+[a|d+d|a+b|b+c|c],\quad[x|d]+[d|x],\quad\text{and}\quad[d|d].$

Among these, the homology classes $[x|d]+[d|x]$ and $[d|d]$ are
$\tau$-invariant, while we have

$(\mathrm{id}+\tau)([x|x]+[x|d]+[a|d+d|a+b|b+c|c])=[x|d]+[d|x]+[d|d].$

Hence the $(\tau,\iota)$-invariant subspace of
$H_{\ast}(A_{0}(4_{1}\\#4_{1}^{r}))$ is generated by $[x|d]+[d|x]$ and
$[d|d]$, both of which are annihilated by $U$. ∎

##  4\. Reduction to small surgery

We now present the construction which will allow us to pass from large surgery
to small surgery. Let $K$ be any strongly invertible knot. For any $n>0$,
define a cobordism $W_{1,n}$ whose incoming end is $S^{3}_{+1}(K)$ by
attaching 2-handles along $(-1)$-framed meridians of $K$, as indicated in
Figure 4.1. We denote these meridians by $x_{1},\ldots,x_{n-1}$. Clearly,
blowing down all of the $x_{i}$ shows that the outgoing end of the cobordism
is diffeomorphic to $S^{3}_{n}(K)$. We can always arrange $W_{1,n}$ to be
equivariant: if $n$ is odd, then all of the 2-handles may be divided into
symmetric pairs, whereas if $n$ is even, then there is a single 2-handle which
intersects the axis of symmetry.

Figure 4.1. The cobordism $W_{1,n}$, formed by attaching $(-1)$-framed
$2$-handles along $n-1$ meridians of $K$.

######  Lemma 4.1.

The cobordism $W_{1,n}$ is spin and negative definite.

###### Proof.

We calculate the intersection form of the cobordism. Orient the $x_{i}$ so
that $x_{i}\cdot K=1$. Consider the basis change $x_{i}^{\prime}=x_{i}-K$.
Clearly, $x_{i}^{\prime}\cdot K=0$ for all $i$. We then compute that
$x_{i}^{\prime}\cdot x_{i}^{\prime}=-2$ for all $i$, while
$x_{i}^{\prime}\cdot x_{j}^{\prime}=-1$ for all $i\neq j$. Since both of these
terms are negative, the cobordism is evidently negative definite; since the
intersection form is even, the cobordism is spin. ∎

The existence of $W_{1,n}$ gives us partial information regarding the $\tau$\-
and $\iota$-actions on $S^{3}_{+1}(K)$ in terms of those on $S^{3}_{n}(K)$.
More precisely, we have the following:

######  Lemma 4.2.

Let $K$ be any strongly invertible knot. If $n$ is odd, then there is an
$(\tau,\iota)$-local map

$f_{1,n}\colon\mathit{CF}^{-}(S^{3}_{+1}(K))\rightarrow\mathit{CF}^{-}(S^{3}_{n}(K),\mathfrak{s}_{0})$

with absolute grading shift $(n-1)/4$. Here, $\mathfrak{s}_{0}$ is the unique
self-conjugate ${\operatorname{spin^{c}}}$-structure on $S^{3}_{n}(K)$.

###### Proof.

By Lemma 4.1, we have that $W_{1,n}$ is spin and negative definite. Let
$\mathfrak{s}$ be the unique spin structure on $W_{1,n}$. (Note that the
homology of $W_{1,n}$ has no torsion; hence it has a unique spin structure
which moreover corresponds to a unique self-conjugate
${\operatorname{spin^{c}}}$-structure.) Since $n$ is odd, we necessarily have
$\mathfrak{s}|_{S^{3}_{n}(K)}=\mathfrak{s}_{0}$. As discussed above, $W_{1,n}$
may be taken to be equivariant, and by uniqueness, $\mathfrak{s}$ is fixed by
the involution on $W_{1,n}$. Hence the cobordism map

$F_{W_{1,n},\mathfrak{s}}\colon\mathit{CF}^{-}(S^{3}_{+1}(K))\rightarrow\mathit{CF}^{-}(S^{3}_{n}(K),\mathfrak{s}_{0})$

homotopy commutes with $\tau$ (and $\iota$), as in [DHM22, Section 6]. It
remains to compute the resulting grading shift, which is easily seen to be

$\dfrac{2\sigma(W_{1,n})+3\chi(W_{1,n})}{4}=\dfrac{n-1}{4}$

as desired. ∎

The application of Lemma 4.2 will become clear in the next section.

##  5\. Proof of Theorem 1.1

We now turn to the proof of the main theorem. First, we have the following
obstruction to bounding an equivariant homology ball:

######  Lemma 5.1.

If $Y$ bounds an equivariant $\mathbb{Z}/2\mathbb{Z}$-homology ball, then
$\mathit{HF}^{-}(Y,\mathfrak{s}_{0})$ admits a $U$-nontorsion class in Maslov
grading zero which is invariant under both $\iota$ and $\tau$. Here,
$\mathfrak{s}_{0}$ is the unique self-conjugate
${\operatorname{spin^{c}}}$-structure on $Y$.

###### Proof.

Let $W$ be an equivariant $\mathbb{Z}/2\mathbb{Z}$-homology ball with boundary
$Y$. Let $\mathfrak{s}$ be the unique self-conjugate
${\operatorname{spin^{c}}}$-structure on $W$. Puncturing $W$ gives an
equivariant $\mathbb{Z}/2\mathbb{Z}$-homology cobordism whose associated map

$F_{W,\mathfrak{s}}\colon\mathit{CF}^{-}(S^{3})\rightarrow\mathit{CF}^{-}(Y,\mathfrak{s}_{0})$

is local. The image $F_{W,\mathfrak{s}}(1)$ gives the desired class. ∎

This immediately shows that $(4_{1})_{2,1}$ is not slice:

###### Proof of 1.1.

Suppose that $(4_{1})_{2,1}$ were slice. Then
$Y_{1}=S^{3}_{+1}(4_{1}\\#4_{1}^{r})$ bounds an equivariant
$\mathbb{Z}/2\mathbb{Z}$-homology ball. By Lemma 5.1, this gives a
$U$-nontorsion class in $\mathit{HF}^{-}(Y_{1})$ which has Maslov grading zero
and is invariant under both $\iota$ and $\tau$. Denote this class by $x_{1}$.
According to Lemma 4.2, we have a $(\tau,\iota)$-local map from $Y_{1}$ to
$Y_{n}=S^{3}_{n}(4_{1}\\#4_{1}^{r})$:

$(f_{1,n})_{*}\colon\mathit{HF}^{-}(Y_{1})\rightarrow\mathit{HF}^{-}(Y_{n},\mathfrak{s}_{0}).$

This has grading shift $(n-1)/4$. On the other hand, the isomorphism

$(\Gamma_{n,0})_{*}\colon\mathit{HF}^{-}(Y_{n},\mathfrak{s}_{0})\xrightarrow{\simeq}H_{*}(A_{0}(4_{1}\\#4_{1}^{r}))$

has grading shift $-(n-1)/4$ and is both $\tau$\- and $\iota$-equivariant. The
class $(\Gamma_{n,0}\circ f_{1,n})_{*}(x_{1})$ then contradicts Lemma 3.3,
completing the proof.

The argument that $(4_{1})_{2,1}$ is nontorsion is similar. Suppose
$(4_{1})_{2,1}$ had finite order in the concordance group. Then
$\\#_{N}(4_{1})_{2,1}$ is smoothly slice for some $N>0$. Taking branched
double covers shows that

$\Sigma_{2}(\\#_{N}(4_{1})_{2,1})=\\#_{N}\Sigma_{2}((4_{1})_{2,1})=\\#_{N}S^{3}_{+1}(4_{1}\\#4_{1}^{r})$

bounds an equivariant $\mathbb{Z}/2\mathbb{Z}$-homology ball. (Here, the
connected sum $\\#_{N}S^{3}_{+1}(4_{1}\\#4_{1}^{r})$ is done in the obvious
equivariant manner.) It thus suffices to prove that
$\mathit{CF}^{-}(\\#_{N}Y_{1})$ admits no local map from the trivial complex.
For this, we assume the reader has some familiarity with the language of local
equivalence, as discussed in [HMZ18, Section 9]. The desired claim follows
easily from the fact that $\mathit{CF}^{-}(Y_{1})$ admits a local map into the
trivial complex; we prove this below in Lemma 5.2.

Indeed, let $F\colon\mathit{CF}^{-}(Y_{1})\rightarrow\mathit{CF}^{-}(S^{3})$
be such a map. Dualizing gives a local map
$G\colon\mathit{CF}^{-}(S^{3})\rightarrow\mathit{CF}^{-}(Y_{1})^{\vee}$. If we
had a local map from $\mathit{CF}^{-}(S^{3})$ to
$\mathit{CF}^{-}(\\#_{N}Y_{1})$, then tensoring this with $\otimes^{N-1}G$ and
applying the connected sum formulas of [HMZ18, Theorem 1.1] and [DHM22,
Proposition 6.8] would give a local map from $\mathit{CF}^{-}(S^{3})$ to
$\mathit{CF}^{-}(Y_{1})$, which we already know is a contradiction. ∎

######  Lemma 5.2.

There exists a local map from $CF^{-}(S^{3}_{+1}(4_{1}\\#4_{1}^{r}))$ to the
trivial complex.

###### Proof.

By Table 1, the homology $H_{*}(A_{0}(4_{1}\\#4_{1}^{r}))$ is isomorphic to
$\mathbb{F}[U]\oplus\mathbb{F}^{4}$. The structure theorem for modules over a
principal ideal domain implies that $A_{0}(4_{1}\\#4_{1}^{r})$ is homotopy
equivalent to a complex generated over $\mathbb{F}[U]$ by
$\\{x,a,b,c,d,e,f,g,h\\}$, where $\partial a=Ub$, $\partial c=Ud$, $\partial
e=Uf$, and $\partial g=Uh$, and we identify

$x=x|x,\quad b=x|d,\quad d=d|x,\quad f=a|d+d|a+b|b+c|c,\quad\text{and}\quad
h=d|d.$

On this complex, the action of $\iota$ is given by

$\displaystyle x\mapsto x+b+d+h,\quad$ $\displaystyle b\mapsto b+h,\quad
d\mapsto d+h,\quad f\mapsto f+b+d,\quad h\mapsto h$ $\displaystyle a\mapsto
a+g,\quad c\mapsto c+g,\quad e\mapsto e+a+c,\quad g\mapsto g.$

The action of $\tau$ is given by

$\displaystyle x\mapsto x,\quad$ $\displaystyle b\mapsto d,\quad d\mapsto
b,\quad f\mapsto f+h,\quad h\mapsto h$ $\displaystyle a\mapsto c,\quad
c\mapsto a,\quad e\mapsto e+g,\quad g\mapsto g.$

The easiest way to see this is to note that in this case, $\iota$ and $\tau$
are determined by their actions on homology for grading reasons (see for
example the proof of [DM19, Lemma 4.2]) and then use Table 1. Consider the
$U$-equivariant map $F:C\rightarrow\mathbb{F}[U]$, defined by

$x\mapsto 1\quad\text{and}\quad(\text{everything else})\mapsto 0.$

It is clear that $F\iota=\iota F$ and $F\tau=\tau F$. Hence $F$ gives a local
map from $A_{0}(4_{1}\\#4_{1}^{r})$ to the trivial complex. Precomposing this
with the map $f_{1,n}$ in 4.2 and using the $\tau$\- and $\iota$-equivariant
homotopy equivalence $\Gamma_{n,0}$ gives the desired local map. ∎

We now turn to the proof of Theorem 1.2. For this, we again assume the reader
has some familiarity with the machinery of local equivalence. Theorem 1.2 will
follow quickly from the fact if $\mathit{CFK}(K_{1})$ is $\iota_{K}$-locally
equivalent to $\mathit{CFK}(K_{2})$ in the sense of Zemke [Zem19a, Definition
2.4], then $\mathit{CFK}(K_{1}\\#K_{1}^{r})$ and
$\mathit{CFK}(K_{2}\\#K_{2}^{r})$ are $(\tau_{K},\iota_{K})$-locally
equivalent in the sense of [DMS22, Definition 2.13]. More precisely, recall
from the work of Zemke [Zem19a] that we have a homomorphism

$\mathit{CFK}_{UV}:\mathcal{C}\rightarrow\mathfrak{K}_{\iota},\quad[K]\mapsto[(\mathit{CFK}(K),\iota_{K})]$

from the smooth concordance group to the local equivalence group
$\mathfrak{K}_{\iota}$ of $\iota_{K}$-complexes. (We follow the notation of
[DMS22], rather than the usual notation $\mathfrak{I}_{K}$ of [Zem19a].) In
[DMS22], it is shown that we similarly have a homomorphism from Sakuma’s
strongly invertible concordance group $\smash{\widetilde{\mathcal{C}}}$ (see
[Sak86]) to the group of $(\tau_{K},\iota_{K})$-complexes:

$\mathit{CFK}_{UV}:\smash{\widetilde{\mathcal{C}}}\rightarrow\mathfrak{K}_{\tau,\iota},\quad[K]\mapsto[(\mathit{CFK}(K),\iota_{K},\tau_{K})]$

In [BI21, Section 3], it is noted that there is a doubling homomorphism from
$\mathcal{C}$ to $\smash{\widetilde{\mathcal{C}}}$. Here, we produce the
algebraic analogue of this doubling map:

######  Lemma 5.3.

There exists a map
$\mathcal{D}\colon\mathfrak{K}_{\iota}\rightarrow\mathfrak{K}_{\tau,\iota}$
such that the following square commutes.

$\textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{K\mapsto
K\\#K^{r}}$$\scriptstyle{\mathit{CFK}_{UV}}$$\textstyle{\widetilde{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathit{CFK}_{UV}}$$\textstyle{\mathfrak{K}_{\iota}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{D}}$$\textstyle{\mathfrak{K}_{\tau,\iota}}$

###### Proof.

Given an $\iota_{K}$-complex $C=(C,\iota)$, define $C^{r}$ to be the complex

$C^{r}=C\otimes_{\mathbb{F}[\mathcal{U},\mathcal{V}]}\mathbb{F}[\mathcal{V},\mathcal{U}].$

Precomposing $\iota\colon C\rightarrow C$ with the Sarkar map gives a map
$\varsigma\circ\iota\colon C\rightarrow C$. This induces a map

$(\varsigma\circ\iota)^{r}\colon C^{r}\rightarrow C^{r}$

which makes $C^{r}$ into an $\iota_{K}$-complex. Let

$\mathcal{D}(C)=(C\otimes C^{r},\tau_{\otimes},\iota_{\otimes}),$

where

$\begin{split}\tau_{\otimes}&=(\mathrm{id}\otimes\mathrm{id}+\Psi_{C}\otimes\Phi_{C^{r}})\circ\tau_{\mathrm{exch}}\\\
\iota_{\otimes}&=\varsigma_{C\otimes
C^{r}}\circ(\mathrm{id}\otimes\mathrm{id}+\Psi_{C}\otimes\Phi_{C^{r}})\circ(\iota\otimes(\varsigma\circ\iota)^{r})\end{split}$

as in 3.1. Here, $\tau_{\mathrm{exch}}$ is defined by

$\tau_{\mathrm{exch}}(x\otimes y)=y\otimes x.$

Note that since the identity map $C\rightarrow C^{r}$ is a skew-isomorphism,
$\tau_{\mathrm{exch}}$ is a skew-isomorphism as well.

To show that $\mathcal{D}$ is well-defined, we must show that $\mathcal{D}$
maps $\iota_{K}$-locally equivalent complexes to
$(\tau_{K},\iota_{K})$-locally equivalent complexes. To prove this, let
$f:C_{1}\rightarrow C_{2}$ be an $\iota_{K}$-local map. Then $f$ induces an
$\iota_{K}$-local map $f^{r}:C^{r}_{1}\rightarrow C^{r}_{2}$. Consider the map

$f\otimes f^{r}:C_{1}\otimes C^{r}_{1}\rightarrow C_{2}\otimes C^{r}_{2}.$

In [Zem19a, Lemma 2.8], it is shown that $\Phi$ and $\Psi$ commute with all
chain maps up to homotopy. It follows that $f\otimes f^{r}$ homotopy commutes
with $\iota_{\otimes}$. Furthermore, by definition of $f^{r}$, we have

$\tau_{\mathrm{exch}}\circ(f\otimes f^{r})=(f\otimes
f^{r})\circ\tau_{\mathrm{exch}}.$

Again using the fact that $\Phi$ and $\Psi$ commute with all chain maps, this
shows $f\otimes f^{r}$ homotopy commutes with $\tau_{\otimes}$. Thus $f\otimes
f^{r}$ is a $(\tau_{K},\iota_{K})$-local map, as desired.∎

Lemma 5.3 should be compared with [Mal22, Theorem 1.8], in which the large-
surgery actions of $\tau_{K}$ and $\iota_{K}$ on $\mathit{CFK}(K\\#K^{r})$ are
used to define concordance invariants for $K$.

###### Proof of 1.2.

If $\tau(K)\neq 0$, then it follows from the cabling formula of [Hom14,
Theorem 1] that $\tau(K_{2,1})\neq 0$ and thus $K_{2,1}$ is not slice. (Here,
$\tau$ is the concordance invariant of [OS03].) Hence we may assume
$\tau(K)=0$. Under the assumptions of the theorem, it then follows from [HM17,
Section 8] that $\mathit{CFK}(K)$ is $\iota_{K}$-locally equivalent to
$\mathit{CFK}(4_{1})$. By Lemma 5.3, this shows that $\mathit{CFK}(K\\#K^{r})$
is $(\tau_{K},\iota_{K})$-locally equivalent to
$\mathit{CFK}(4_{1}\\#4_{1}^{r})$. Now, if $\mathit{CFK}(K_{1})$ and
$\mathit{CFK}(K_{2})$ are $(\tau_{K},\iota_{K})$-locally equivalent, then
$A_{0}(K_{1})$ and $A_{0}(K_{2})$ are easily seen to be $(\tau,\iota)$-locally
equivalent. Hence the same argument as in Theorem 1.1 applies equally well to
show that there is no local map from the trivial complex to
$CF^{-}(\Sigma_{2}(K_{2,1}))$. Likewise, Lemma 5.2 applies to show that there
exists a local map from $CF^{-}(\Sigma_{2}(K_{2,1}))$ into the trivial
complex. The same argument as in the latter part of Theorem 1.1 completes the
proof. ∎

######  Remark 5.4.

Our proof of Theorem 1.2 shows that $K_{2,1}$ is not slice in any
$\mathbb{Z}/2\mathbb{Z}$-homology ball. Indeed, if a knot admits a slice disk
$D$ in such a ball, then it follows from the proof of [CG78, Lemma 4.2] that
we can form the branched double cover over $D$ and that this double cover is
again a $\mathbb{Z}/2\mathbb{Z}$-homology ball. Hence the same obstruction as
discussed in Section 2 applies. Likewise, the argument of Theorem 1.2 shows
that $K_{2,1}$ is not torsion in the $\mathbb{Z}/2\mathbb{Z}$-homology
concordance group.

######  Remark 5.5.

For the reader familiar with [DHM22], we emphasize that considering the
$\tau$\- and $\iota$-actions on $A_{0}(4_{1}\\#4_{1}^{r})$ simultaneously is
crucial. Indeed, it is straightforward to check that the actions of $\tau$,
$\iota$, and $\tau\circ\iota$ are all individually locally trivial on
$A_{0}(4_{1}\\#4_{1}^{r})$. The formalism used in this article is thus a
slight enhancement of the setup of [DHM22] and [Mal22, Theorem 1.8], in which
the action of $\iota$ is incorporated only through the consideration of
$\tau\circ\iota$. Following [DMS22, Definition 2.10], it is possible to define
a $(\tau,\iota)$-local equivalence group; further applications of this to
problems involving sliceness and concordance will appear in forthcoming work
by the authors.

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Master integrals contributing to two-loop leading colour QCD helicity
amplitudes for top-quark pair production in the gluon fusion channel

Ekta Chaubey

Dipartimento di Fisica and Arnold-Regge Center, Università di Torino, and
INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy

<EMAIL_ADDRESS>
15th International Symposium on Radiative Corrections:
Applications of Quantum Field Theory to Phenomenology,
FSU, Tallahasse, FL, USA, 17-21 May 2021 10.21468/SciPostPhysProc.?

## Abstract

We present the master integrals relevant for computing the complete set of
leading-colour analytic helicity amplitudes for top-quark pair production via
gluon fusion at two-loops in QCD. We include corrections due to massive
fermion loops which give rise to integrals involving elliptic curves. We also
elaborate on the structure of singularities that play an important role in the
numerical evaluation of the iterated integrals and their analytic
continuation.

###### Contents

  1. 1 Introduction
  2. 2 Definitions
    1. 2.1 Iterated integrals
    2. 2.2 Elliptic curves
  3. 3 Master integrals
    1. 3.1 Elliptic curves in the master integrals
    2. 3.2 New master integrals
  4. 4 Numerical evaluations of elliptic iterated integrals
    1. 4.1 Choice of path
      1. 4.1.1 Technical challenges
      2. 4.1.2 Singularity structure of the differential system
  5. 5 Conclusion

##  1 Introduction

The top quark is an important ingredient in understanding the fundamental
forces and in particular of the electroweak symmetry breaking mechanism, due
to its large mass. The Higgs potential depends on its value and hence a
precise understanding of the top quark is imperative.

The only complete two-loop amplitudes for QCD corrections to top-quark pair
production were available numerically [1, 2, 3] until this year. The main
reason for this is the appearance of a more complicated class of special
functions in amplitudes containing internal masses. These functions have been
identified to involve integrals over elliptic curves [4, 5, 6, 7, 8, 9, 10,
11, 12], and lie on the boundary of our current mathematical understanding. We
recently published a set of helicity amplitudes for top-quark pair production
in the leading colour approximation, in [13]. For the first time, we also
included the contributions from heavy fermion loops, which give rise to
iterated integrals involving multiple elliptic curves. Most of the master
integrals for these heavy loop corrections were published in [6, 5]. Apart
from obtaining compact analytic helicity amplitudes by sampling Feynman
diagrams with finite field arithmetic [14, 15, 16, 17, 18, 19], we also
studied the numerical evaluation of the final amplitudes. The helicity
amplitudes presented contain complete information about top quark decays in
the narrow width approximation.

Perturbative computations at two-loop order with massive internal lines
contain a large number of steps as usual, each with some technical bottlenecks
to overcome. More details about the amplitude construction and finite field
reconstruction are available in ref. [13]. The master integrals contributing
to these amplitudes also included master integrals from one final two-loop
integral topology that were not available previous to the publication [13],
which also contained two elliptic master integrals. We obtained the (canonical
form) differential equation [20, 21, 22, 23, 24, 9] and the solution was
expressed in terms of iterated integrals [25]. In these proceedings, we
elaborate on the analytic computations of the planar master integrals and
discuss the technical bottlenecks in the numerical evaluation with elliptic
kernels containing multiple elliptic curves.

##  2 Definitions

In this section, we review the definition of Chen’s iterated integrals [26]
and an elliptic curve.

###  2.1 Iterated integrals

Consider a path $\gamma$ on an $n$-dimensional manifold $M$ with starting
point $x_{i}=\gamma(0)$ and end point $x_{f}=\gamma(1)$,
$\gamma\;:\;[0,1]\rightarrow M$. Let us also consider a set of differential
1-forms $\\{\omega_{i}\\}$ and their pullbacks to the interval $[0,1]$, which
we denote by $f_{j}(\lambda)\;d\lambda=\gamma^{*}\omega_{j}$. The $k$-fold
iterated integral over $\omega_{1},...,\omega_{k}$ along $\gamma$ is defined
as

$\displaystyle I_{\gamma}(\omega_{1},...,\omega_{k};\lambda)\;$
$\displaystyle=\;\int_{0}^{\lambda}d\lambda_{1}f_{1}(\lambda_{1})\int_{0}^{\lambda_{1}}d\lambda_{2}f_{2}(\lambda_{2})\;...\int_{0}^{\lambda_{k-1}}d\lambda_{k}f_{k}(\lambda_{k})$
(1)
$\displaystyle=\;\int_{0}^{\lambda}d\lambda_{1}f_{1}(\lambda_{1})\,I_{\gamma}(\omega_{2},...,\omega_{k};\lambda_{1})\,,$
(2)

where we define the $0$-fold integrals as $I_{\gamma}(;\lambda)\;=\;1$. The
iterated integrals defined in Eq. (1) have a lot of useful properties [27].
The most important property for our work is concerning the decomposition of
the path $\gamma$. Let $\alpha,\beta$: $[0,1]\rightarrow M$ be two paths such
that $\beta(0)=\alpha(1)$ and let $\gamma=\alpha\beta$ be the path obtained by
concatenating $\alpha$ and $\beta$. Then we can decompose

$\displaystyle
I_{\gamma}(\omega_{1},...,\omega_{k};\lambda)\;=\sum_{i=0}^{n}I_{\beta}(\omega_{1},...,\omega_{i};\lambda)\,I_{\alpha}(\omega_{i+1},...,\omega_{n};\lambda)\,.$
(3)

Well-known examples of iterated integrals are the Multiple polylogarithms
(MPLs) [28], which are a special class of functions, where the $1$-forms
$\omega_{i}$ are such that $\gamma^{*}\omega_{j}=\frac{d\lambda}{\lambda-c}$,
for some $c\in\mathbb{C}$. The class of MPLs is quite well understood. There
exist many tools for their algebraic manipulation and numerical evaluation
[29, 30, 31]. However, it is not always possible to express all master
integrals in terms of these functions. This is especially true in computations
involving massive internal particles. Here we often encounter elliptic curves
in the function space of the master integrals.

###  2.2 Elliptic curves

A generic quartic case of an elliptic curve $E$ can be described by the
equation

$\displaystyle
E\;:\;w^{2}-\left(z-z_{1}\right)\left(z-z_{2}\right)\left(z-z_{3}\right)\left(z-z_{4}\right)=0,$
(4)

where the $z_{i}\in\mathbb{C}$. The $z_{i}$ define the properties of the
elliptic curve and are generally functions of the kinematic variables
$x=(x_{1},...,x_{n})$: $z_{j}\;=\;z_{j}\left(x\right),\;j\in\\{1,2,3,4\\}\,.$
An elliptic curve is associated with two elliptic periods $\psi_{i}$. In order
to define these elliptic periods, let us introduce the auxiliary variables
$Z_{i}$ as

$\displaystyle
Z_{1}\;=\;\left(z_{2}-z_{1}\right)\left(z_{4}-z_{3}\right),\;\;\;\;\;\;Z_{2}\;=\;\left(z_{3}-z_{2}\right)\left(z_{4}-z_{1}\right),\;\;\;\;\;\;Z_{3}\;=\;\left(z_{3}-z_{1}\right)\left(z_{4}-z_{2}\right)\,.$
(5)

The $Z_{i}$ satisfy the relation $Z_{1}+Z_{2}\;=\;Z_{3}$, and can be used to
define the modulus $k^{2}$ and the complementary modulus $\bar{k}^{2}$ of the
elliptic curve $E$ as $k^{2}\;=\;\frac{Z_{1}}{Z_{3}}$, and
$\bar{k}^{2}\;=\;1-k^{2}\;=\;\frac{Z_{2}}{Z_{3}}\,.$ Then we can choose the
two elliptic periods $\psi_{i}$ associated to the elliptic curve $E$ as
$\psi_{1}\;=\;\frac{4K\left(k\right)}{Z_{3}^{\frac{1}{2}}}$, and
$\psi_{2}\;=\;\frac{4iK\left(\bar{k}\right)}{Z_{3}^{\frac{1}{2}}}\,,$ where
$K$ is the complete elliptic integral of the first kind
$K(\lambda)=\int_{0}^{1}\frac{dt}{\sqrt{(1-t^{2})(1-\lambda t^{2})}}\,.$

##  3 Master integrals

To compute the finite remainder for the helicity amplitude, we need the
results for all the master integrals appearing in the one- and two- loop
amplitudes. All the one- and two-loop master integrals appearing in the
amplitudes not involving a closed top-loop, for example, the first two
topologies from Fig. 1, can be expressed in terms of MPLs [32, 33]. For the
two-loop amplitude involving a single top-quark closed-loop, for example for
the last three topologies from Fig. 1, the master integrals are associated
with elliptic curves [5, 6].

Figure 1: Master integral topologies. Black-solid lines represent massive
particles, red-dashed lines represent massless particles.

###  3.1 Elliptic curves in the master integrals

In our work, the dependence of iterated integrals on elliptic curves enters
through the appearance of elliptic periods in the integration kernels. The
topbox diagram (Fig. 2(b)) has three elliptic sub-sectors corresponding to
three different elliptic curves. These curves can be obtained from the maximal
cuts in the Baikov representation [34, 35, 36]. We identify the three
different elliptic curves by the labels $a,b,$ and $c$ according to the
diagrams depicted in Fig. 2.

(a) The sunrise topology

(b) The topbox topology

(c) The bubblebox topology

Figure 2: Topologies in the topbox family that are associated with the
elliptic curves $E^{(a)}$, $E^{(b)}$ and $E^{(c)}$. Black-solid lines
represent massive particles, red-dashed lines represent massless particles.

The sunrise graph (Fig. 2(a)) is associated with the elliptic curve $E^{(a)}$.
The iterated integrals involving this topology can be cast in terms of
iterated integrals of modular forms, and are well-suited for numerical
evaluation [7, 11, 12, 37]. The second elliptic curve $E^{(b)}$ is associated
with the topbox sector itself (Fig. 2(b)). The third elliptic curve $E^{(c)}$
is associated with another elliptic sub-sector, the bubblebox sector (Fig.
2(c)). All the corresponding quartics ($z_{i}$’s) are given by

$\displaystyle z^{(a)}_{1}\;$
$\displaystyle=\;\frac{t-4m^{2}}{\mu^{2}},\;\;\;z^{(a)}_{2}\;=\;\frac{-m^{2}-2m\sqrt{t}}{\mu^{2}},\;\;\;z^{(a)}_{3}\;=\;\frac{-m^{2}+2m\sqrt{t}}{\mu^{2}},\;\;\;z^{(a)}_{4}\;=\;\frac{t}{\mu^{2}},$
$\displaystyle z^{(b)}_{1}\;$
$\displaystyle=\;\frac{t-4m^{2}}{\mu^{2}},\;\;\;z^{(b)}_{2}\;=\;\frac{-m^{2}-2m\sqrt{t+\frac{\left(m^{2}-t\right)^{2}}{s}}}{\mu^{2}},\;\;\;z^{(b)}_{3}\;=\;\frac{-m^{2}+2m\sqrt{t+\frac{\left(m^{2}-t\right)^{2}}{s}}}{\mu^{2}},$
$\displaystyle z^{(b)}_{4}\;$ $\displaystyle=\;\frac{t}{\mu^{2}},$
$\displaystyle z^{(c)}_{1}\;$ $\displaystyle=\;\frac{t-4m^{2}}{\mu^{2}},\qquad
z^{(c)}_{2}\;=\;\frac{1}{\mu^{2}}\left(-m^{2}\frac{\left(s+4t\right)}{\left(s-4m^{2}\right)}-\frac{2}{4m^{2}-s}\sqrt{sm^{2}\left(st+\left(m^{2}-t\right)^{2}\right)}\right),$
$\displaystyle z^{(c)}_{3}\;$
$\displaystyle=\;\frac{1}{\mu^{2}}\left(-m^{2}\frac{\left(s+4t\right)}{\left(s-4m^{2}\right)}+\frac{2}{4m^{2}-s}\sqrt{sm^{2}\left(st+\left(m^{2}-t\right)^{2}\right)}\right),\qquad
z^{(c)}_{4}\;=\;\frac{t}{\mu^{2}}.$ (6)

###  3.2 New master integrals

A contribution to the amplitudes from the closed top loop sector comes from
the so-called penta-triangle topology [13]. The kinematic variables that we
used are $x$ and $y$, defined by

$-\frac{s}{m_{t}^{2}}=\frac{(1-x)^{2}}{x},\qquad\frac{t}{m_{t}^{2}}=y,$ (7)

to rationalize the square root,$\sqrt{s\;(s-4m_{t}^{2})}$. We bring all the
master integrals of this topology in a canonical form [24, 9],

$d\vec{\mathcal{J}}\;=\;\epsilon A\vec{\mathcal{J}},\quad
A=A_{x}\;dx+A_{y}\;dy,$ (8)

where $A$ does not depend on $\epsilon$ and is rational in the kinematic
variables $(x,y)$. The new master integrals (the last two topologies in Fig.
1) can be computed order by order in $\epsilon$ from the differential equation
given in Eq. (8). We integrate the differential equation from a base point
$(x_{0},y_{0})=(0,1)$ (corresponding to $s=\infty$ and $t=m^{2}$) and express
the results of the master integrals iteratively in the expansion parameter
$\epsilon$. Both of these integrals receive contributions from the elliptic
topbox sub-sectors and hence contain elliptic iterated integrals themselves.
The boundary values of the integrals at different orders in $\epsilon$ are
obtained from the regularity of the differential system at the point
$x,y=(1,1)$, where these two master integrals vanish. The fourth integral from
Fig. 1 has a sunrise in its sub-sectors and hence is associated with the
elliptic curve $a$. The fifth integral from Fig. 1, on the other hand,
contains an elliptic subsector from the topbox which is associated with the
elliptic curve $b$ [6] and hence is itself elliptic. The full results for
these integrals are provided with the ancillary files of [13].

##  4 Numerical evaluations of elliptic iterated integrals

In this section, we discuss how we numerically compute the master integrals
involving multiple elliptic curves. The integrals containing only MPLs can be
evaluated to high precision, for example, using Ginac [38]. The numerical
evaluation of the iterated integrals associated with elliptic curves is a
complicated task, specifically due to the presence of transcendental functions
in the form of elliptic periods and their derivatives. The iterated integrals
are integrated from the base point $(x=0,y=1)$ to any $x$ and $y$. For this,
we use a path involving multiple line segments and use the path decomposition
formula to patch together the contribution over all the segments. We need to
take into consideration both spurious as well as physical singularities of the
system while choosing the path segments. We explain how we choose the paths to
evaluate the iterated integrals in the Euclidean and the physical region.

In the Euclidean region the iterated integrals have only real contributions.
To compute the iterated integrals in this region, we start from a small
neighborhood of the boundary point. This makes our task relatively
straightforward. We split the integration path from the boundary point $(0,1)$
to any point $(x,y)$ and integrate over each of these segments, where we
series expand the kernels around some points along the paths. Using the path
decomposition formula Eq. (3), we patch together the contribution to obtain
the final result. The following path segments were chosen: $\alpha$, from
$(0,0)$ to $(x,0)$ and $\beta$, from $(x,0)$ to $(x,y)$. Over the first
segment, all the master integrals in our case compute to only MPLs, which we
can evaluate to very high precision, as mentioned before. To integrate along
the second segment, we expand all integration kernels around $y=1$. To
integrate to a point $y$ not close to $1$, we may use more segments along $y$
and use the path decomposition formula recursively.

Now we discuss the computations in the physical region. In our case, the
physical region is governed by the following equations ($m_{t}=1$):

$\displaystyle s\geq 4,\qquad G(p_{1},p_{2},p_{3})\geq
0\;\implies\;s(-t^{2}-st+2t-1)\geq 0,$

where $G$ is the Gram-determinant, see point $P$ in Fig. 3. For this, we need
to analytically continue the iterated integrals around the physical branch
points. We series expand all the integrands and use multiple (one-dimensional)
path segments. Multiple path segments are needed to get a better convergence
of the results. The choice of path is controlled by the radius of convergence
of the series. This in turn depends on the singularities present in the
kernels. It is important to properly choose the number of path segments and
their sizes. This is to make sure that the series solution converges properly
on each of these segments and also that the computation does not become too
slow due to the presence of too many segments.

Figure 3: Integrating to a point $P$ in the physical region

For evaluating these integrals numerically in the physical region we need to
analytically continue the associated kernels, also the ones containing
elliptic curves, across all the branch cuts appropriately. The analytic
continuation of integrals having one parameter dependence has already been
discussed in the literature [10, 12, 37]. The analytic continuation of all the
integrals in our case, which includes the topbox integral associated to three
different elliptic curves, is however much more involved. We choose the
correct i$\epsilon$ prescription near the singular points to analytically
continue the kernels involving periods of the elliptic curves and their
derivatives. We now present some more details of the singularity structure of
our differential system, which plays a crucial role in the numerical
evaluation of the master integrals.

###  4.1 Choice of path

The choice of path is even more important to evaluate the master integrals
written in terms of iterated integrals, in the physical region. Below we
explain the observations that motivate our choice for the path.

####  4.1.1 Technical challenges

Let us start by integrating the iterated integrals in our system first along
$x$ ($y=constant$) and then along $y$ ($x=constant$), also for the physical
region, as we do for the Euclidean region. We immediately hit a problem. On
the line x=y, we encounter some functions which are not elementary. The
general form of the integrals arising from these functions has the form

$\int\frac{t^{m}\;\text{log}(-\text{log}(t))}{\text{log}(t)^{n}}dt,\int\frac{t^{m}\;Ei(m\text{log}(t))}{\text{log}(t)}dt,\int\frac{t^{4}\;\text{LogIntegral}(t)}{\text{log}(t)^{n}}dt,$
(9)

where $Ei$ is the Exponential Integral. There is no power series expansion
(specifically around the point of expansions), but only an asymptotic series
expansion [39], of LogIntegral($x$) or $Ei(\text{log}(x)n$), for any integer
$n$. The occurrence of such functions can be understood as follows. Many of
our kernels contain periods in the denominator, which give rise to logarithms
when series expanded. These logarithms give rise to the functions above upon
integration. It is worth noting that these kernels cannot be handled in this
form by any program which performs a generalized series expansion of the
kernels, for example [40], as we will face the same problems. If we change our
order of integration such that we first integrate along $y$ and then along
$x$, we circumvent these problems. All the kernels giving rise to the type of
functions mentioned above drop out, or become simpler, because of the boundary
conditions. In addition, while crossing the line $y=0$, we get a singularity
for all the kernels having periods belonging to the sunrise topology. Whereas
for the elliptic kernels having both $x$ and $y$ dependence, there is a
singularity on the line $x=y$, which is the line crossing over to the physical
region. Both these types of singularities coincide, for the path shown in Fig.
3 at the point (0,0). Therefore it is easier to integrate all the iterated
integrals first along $y$ and then along $x$, motivating our choice of paths.

####  4.1.2 Singularity structure of the differential system

On the path shown in Fig. 3, after integrating along $x=0$, we need to
integrate along $y=constant$ line. On this second line, it is again important
to choose the point of expansion and the number of path segments carefully.
For the second line, the singularities of the kernels in $x$ change depending
on the value of $y$. A good rule of thumb is that the segments should not be
larger than half the radius of convergence of the series expansion [8]. Below
we show all the singularities, computed naively by considering zeroes of all
the denominators, of the iterated integrals at particular values of $y$. For
comparisons, we show the singularity structure for two values of $y$, $y=0.2$
and $-1.75$, before and after crossing over to the physical region. It is
clear from the pictures that for the point y=-1.75 (also true for other points
in this region), the radius of convergence is relatively big. This lets us
choose fewer segments on this path, eventually making the integration time
faster.

(a) Singularities in the $x$ plane for y=0.2

(b) Singularities in the $x$ plane for y=-1.75

Figure 4: Singularities of the differential system of topbox.

The downside of choosing a path like in Fig. 3 is that the kernels no longer
simplify to MPLs on either of the parts, $x=constant$ or $y=constant$, as was
the case for the path chosen for the Euclidean region. Using the path
decomposition formula also in this case, we can patch together the
contributions from the different segments and compute the iterated integrals
over the whole path. For the integrals belonging to the topbox family, we
analytically continue all the iterated integrals associated with the curves
$a$ and $b$ to the physical region, along with all the integrals depending
only on MPLs.

Analogously we can integrate into other phase-space regions. Another approach
to perform the integration in different phase space regions is to start
integrating from a point already in that region. The upside of this is that we
do not need to analytically continue across the branch cut. The downside is
that for this we need to compute the boundary conditions at new base points.

##  5 Conclusion

In these proceedings, we discussed the analytic form of master integrals
needed to compute a complete set of analytic helicity amplitudes for the
planar corrections to top quark pair production via gluon fusion at two loops
in QCD. These integrals give rise to a set of special functions which are
associated with three elliptic curves. The numerical evaluation of iterated
integrals with these kernels is complicated in the current form, and we
elaborated on some of the challenges faced. Nevertheless, it was the first
time that amplitude level expressions using the massive spinor-helicity
formalism were obtained for the planar corrections to top quark pair
production, including the top quark loops using analytic expressions for the
master integrals involving elliptic curves. Despite the growth in analytic
complexity, we presented the study of the numerical evaluation as well as
analytic simplification leading to complete cancellation of the universal IR
and UV poles.

## Acknowledgements

This project has received funding from the European Union’s Horizon 2020
research and innovation programmes New level of theoretical precision for LHC
Run 2 and beyond (grant agreement No 683211) and High precision multi-jet
dynamics at the LHC (grant agreement No 772009).

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