# Spectrification is incompatible with Szabó spectral sequence

Benjamin Cooper  ,  Pravakar Paul  and  Nicholas Seguin  University of Iowa,
Department of Mathematics, 14 MacLean Hall, Iowa City, IA 52242-1419 USA  ben-
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The Lipshitz-Sarkar Steenrod operations on Khovanov homology are incompatible
with Szabó’s geometric spectral sequence.

## Introduction

Variations on constructions of knot homology theories have determined a rich
mathematical structure, reflecting both their many origins and the surprising
capacity for dissimilar settings to carry the complexity of knot theory. In
the presence of many different, but ultimately equivalent definitions, it is
common to use the extra structure available in one setting to predict new
structure in another. Here we show that extra structure present in one setting
is incompatible with extra structure from another.

R. Lipshitz and S. Sarkar introduced a spectrum-level refinement of Khovanov
homology which determines Steenrod operations
$Sq^{n}:Kh^{t,q}(K;\mathbb{F}_{2})\to Kh^{t+n,q}(K;\mathbb{F}_{2})$ for $n\geq
1$ [LS14a, LS14b]. Z. Szabó defined a spectral sequence which interpolates
from Khovanov homology to a knot homology theory (conjecturally) isomorphic to
the Heegaard-Floer homology $\widehat{HF}(-\Sigma(K)\\#(S^{1}\times S^{2}))$
of the double branched cover $\Sigma(K)$ connected sum $S^{1}\times S^{2}$
[OS05, Sza15]. We prove that for each $n\geq 1$, there is a knot $K_{n}$ so
that $Sq^{n}$ does not commute with the differentials of Szabó’s spectral
sequence.

Compatibility between these constructions is a litmus test for the existence
of a spectrum-level refinement of Szabó’s spectral sequence or its terminus,
Heegaard-Floer homology. The incompatibility observed in this paper is
evidence of a richer story.

## Spectral sequences

Suppose that $(CKh(K),d)$ is the Khovanov chain complex associated to a link
$K$. A map $\delta:CKh(K)\to CKh(K)$ of chain complexes is a differential
when $(d+\delta)^{2}=0$. Associated to such a differential $\delta$ is a
spectral sequence $\\{E_{n},d_{n}\\}_{n=2}^{\infty}$, consisting of pages
$E_{n}=\oplus_{(i,j)\in\mathbb{Z}\times\mathbb{Z}}E^{i,j}_{n}$ and
differentials $d_{n}:E_{n}\to E_{n}$ such that $E_{n+1}:=H(E_{n},d_{n})$, from
the Khovanov homology $E_{2}=Kh(K)$, (where $Kh(K):=H(CKh(K),d)$), to the
homology of the total complex $E_{\infty}=H(CKh(K),d+\delta)$.

######  Definition.

An endomorphism $f_{m}:E_{m}\to E_{m}$ acting on the $E_{m}$-page gives rise
to an operation on $\\{E_{n}\\}_{n=m}^{\infty}$ when there is a sequence of
linear maps $f_{n}:E_{n}\to E_{n}$ for $n>m$ which satisfy the two properties
below.

  1. (1)

The map $f_{n}:E_{n}\to E_{n}$ commutes with the differential on the
$E_{n}$-page: $f_{n}d_{n}=d_{n}f_{n}$.

  2. (2)

The map $f_{n+1}:E_{n+1}\to E_{n+1}$ agrees with the homology of $f_{n}$ on
the $E_{n}$-page: $f_{n+1}=H(f_{n},d_{n}):H(E_{n},d_{n})\to H(E_{n},d_{n})$.

The Khovanov chain complex $(CKh(K),d)$ is defined as a direct sum of tensor
products of the Frobenius algebra $\mathbb{F}_{2}[x]/(x^{2})$. Choosing any
point on the knot $K$ allows us to adjoin a handle corresponding to
multiplication by $x$. This gives rise to a chain map $X:CKh(K)\to CKh(K)$ and
an induced map $X_{*}$ on homology [Kho03, §3]. A. Shumakovitch introduced a
decomposition of the form

$Kh(K;\mathbb{F}_{2})\cong\widetilde{Kh}(K;\mathbb{F}_{2})\oplus
X_{*}\widetilde{Kh}(K;\mathbb{F}_{2})$ (1.1)

where $\widetilde{Kh}(K;\mathbb{F}_{2})\subset Kh(K;\mathbb{F}_{2})$,
$\widetilde{Kh}(K;\mathbb{F}_{2})\cong
Kh(K;\mathbb{F}_{2})\otimes_{\mathbb{F}_{2}[x]/(x^{2})}\mathbb{F}_{2}$ is the
reduced Khovanov homology [Shu14, §3].

The lemma below shows that the first part of this story extends to Szabó’s
spectral sequence.

######  Lemma.

The map $X_{*}:Kh(K)\to Kh(K)$ gives rise to an operation on the Szabó’s
spectral sequence.

###### Proof.

We construct a map $X_{Sz}:CKh(K)\to CKh(K)$ which commutes with Szabó’s
differential $\delta_{Sz}$ and agrees with $X$ on the associated graded of the
filtration defining the spectral sequence. This is accomplished by refactoring
Szabó’s proof of invariance under the Reidemeister 1 move [Sza15, Thm. 7.2].

Pick a point $p$ on $K$. Adding a kink in the knot at $p$ gives a knot diagram
$K^{\sharp}$. By resolving the crossing at the kink, the chain complex
$\langle K^{\sharp}\rangle$

$\left[\begin{minipage}{25.29494pt} \includegraphics[scale={.45}]{PKink}
\end{minipage}\,\,\,\right]_{(\langle
K^{\sharp}\rangle,\delta_{Sz})}$$=$$Cone\Bigg{(}\left[\begin{minipage}{25.29494pt}
\includegraphics[scale={.45}]{PZero}
\end{minipage}\,\,\right]_{(C_{0},\delta_{0})}$$\left[\begin{minipage}{25.29494pt}
\includegraphics[scale={.45}]{POne}
\end{minipage}\,\,\right]_{(C_{1},\delta_{1})}\Bigg{)}$$S$

associated to the diagram $K^{\sharp}$, can be written as a cone on a map $S$,
so $\delta_{Sz}^{2}=0$ implies $S\delta_{0}=\delta_{1}S$. Now the disjoint
circle in the diagram for $C_{0}$ produces a decomposition $C_{0}\cong
C_{0}^{+}\oplus C_{0}^{-}$ where $C_{0}^{-}$ consists of elements divisible by
$x$ (in the Frobenius algebra associated to the disjoint circle) and
$C_{0}^{+}$ those elements which are not divisible by $x$. Under this
isomorphism $S$ is a sum of two maps $S=X_{Sz}+1$ where $X_{Sz}:C_{0}^{-}\to
C_{1}$ and $1:C_{0}^{+}\to C_{1}$. Szabó observes that the map $1$ is the
identity map by construction. So $S$ and $1$ are chain maps, which implies
that $X_{Sz}$ must also commute with Szabó differentials.

Since the map $X_{Sz}$ has positive homological degree or $t$-degree, it
preserves the filtration $F_{k}C:=\\{y\in C:t(y)\geq k\\}$ defining the Szabó
spectral sequence and induces an operation on the spectral sequence. Lastly,
since the first order term of the Szabó differential is the Khovanov
differential, the first order term of the map $X_{Sz}$ is $X$, so $X$ (and
$X_{*}=H(X,d)$) extend to operations on the spectral sequence. ∎

The map $X_{Sz}$ can be written as $X_{Sz}=\sum_{n=0}^{\infty}X_{n}$ where
$X_{0}=X$ and $X_{n}:=\sum_{p+q=n}E_{p,q}$ for $n>0$ where $E_{p,q}$ is the
assignment from [Sza15, Def. 4.5]. This formula depends on an orientation of
the saddle $S$, but any two choices are homotopic [Sza15, §5.1].

## The counterexample

We now combine the materials above with the output of the computer programs
KnotKit by C. Seed and JavaKh by D. Bar-Natan and J. Green [Cotb, Cota, BN07]
to produce an incompatibility between the Bockstein $Sq^{1}:Kh(K)\to Kh(K)$
and the Szabó spectral sequence. This occurs on the $E_{3}$-page of the
spectral sequence associated to the torus knot $T(4,5)$.

The Poincaré polynomial of the unreduced $\mathbb{F}_{2}$-Khovanov homology of
the torus knot $T(4,5)$ is given by

$\displaystyle P_{2}$
$\displaystyle=(q^{11}+q^{13})t^{0}+(q^{15}+q^{17})t^{2}+(q^{17}+q^{19})t^{3}+(q^{17}+q^{19})t^{4}+(q^{21}+q^{23})t^{5}$
$\displaystyle\quad\quad+(q^{19}+2q^{21}+q^{23})t^{6}+(q^{21}+2q^{23}+q^{25})t^{7}+(q^{23}+q^{25})t^{8}$
$\displaystyle\quad\quad+(q^{25}+2q^{27}+q^{29})t^{9}+(q^{27}+q^{29})t^{10}$

This is the Poincaré polynomial of the $E_{2}$-page of the Szabó spectral
sequence. The polynomials associated to the $E_{3}$ and $E_{4}=E_{\infty}$
pages are given below.

$\displaystyle P_{3}$
$\displaystyle=(q^{11}+q^{13})t^{0}+(q^{17}+q^{19})t^{3}+(q^{19}+2q^{21}+q^{23})t^{6}+(q^{21}+q^{23})t^{7}$
$\displaystyle\quad\quad+(q^{23}+q^{25})t^{8}+(q^{25}+2q^{27}+q^{29})t^{9}+(q^{27}+q^{29})t^{10}$
$\displaystyle P_{4}$
$\displaystyle=(q^{11}+q^{13})t^{0}+(q^{19}+q^{21})t^{6}+(q^{21}+q^{23})t^{7}$
$\displaystyle\quad\quad+(q^{23}+q^{25})t^{8}+(q^{25}+2q^{27}+q^{29})t^{9}+(q^{27}+q^{29})t^{10}$

The diagram below also contains this information. In the diagram, the non-zero
$d_{2}$ differentials are denoted by solid arrows and non-zero $d_{3}$
differentials are denoted by dashed arrows. The $(t,q)$-bidegree of the
differential $d_{n}:E_{n}\to E_{n}$ is $(n,2n-2)$.

$t$$q$$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$13$$15$$17$$19$$21$$23$$25$$27$$29$$2$$2$$2$

The non-zero maps $Sq^{1}$ and $Sq^{2}$ of $(t,q)$-degrees $(1,0)$ and $(2,0)$
are represented by double arrows. A black dot is a generator which is in the
image of the operation $X_{*}$ and a white dot is a generator which is not in
the image of $X_{*}$, as in Eqn. (1.1). The vector spaces of rank 2 are
labelled by the number $2$. In the picture above, each such vector space
consists of one black dot and one white dot.

######  Theorem.

The Bockstein $Sq^{1}:Kh(T(4,5))\to Kh(T(4,5))$ does not give rise to an
operation on the Szabó spectral sequence.

###### Proof.

Assume that the Bockstein gives rise to an operation on the Szabó spectral
sequence. Then there are maps $Sq^{1}_{n}:E^{i,j}_{n}\to E^{i+1,j}_{n}$ for
$n\geq 2$ which satisfy the two properties in the definition above. But this
cannot be true!

First observe that, of the vector spaces: $E_{2}^{4,17}$, $E_{2}^{3,17}$,
$E_{2}^{6,21}$ and $E_{2}^{7,21}$, only $E_{2}^{4,17}$ interacts with a non-
zero $d_{2}$ differential. In this way, $E_{3}^{i,j}=H(E_{2},d_{2})$ implies
the following isomorphisms

$E_{3}^{4,17}\cong 0\quad\textnormal{ and }\quad E_{3}^{3,17}\cong
E_{2}^{3,17},\quad E_{3}^{6,21}\cong E_{2}^{6,21},\quad E_{3}^{7,21}\cong
E_{2}^{7,21}.$

By assumption, the value of $Sq^{1}_{3}$ agrees with $Sq^{1}_{2}$ under the
correspondences established by the last two of these isomorphisms. So the
diagram below appears on the $E_{3}$-page.

$(6,21)\,\,\leavevmode\hbox to6.09pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 3.04544pt\lower-3.04544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.84544pt}{0.0pt}\pgfsys@curveto{2.84544pt}{1.5715pt}{1.5715pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-1.5715pt}{2.84544pt}{-2.84544pt}{1.5715pt}{-2.84544pt}{0.0pt}\pgfsys@curveto{-2.84544pt}{-1.5715pt}{-1.5715pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{1.5715pt}{-2.84544pt}{2.84544pt}{-1.5715pt}{2.84544pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox
to5.69pt{\vbox to5.69pt{\pgfpicture\makeatletter\hbox{\hskip
2.84544pt\lower-2.84544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.84544pt}{0.0pt}\pgfsys@curveto{2.84544pt}{1.5715pt}{1.5715pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-1.5715pt}{2.84544pt}{-2.84544pt}{1.5715pt}{-2.84544pt}{0.0pt}\pgfsys@curveto{-2.84544pt}{-1.5715pt}{-1.5715pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{1.5715pt}{-2.84544pt}{2.84544pt}{-1.5715pt}{2.84544pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$$(7,21)\,\,\leavevmode\hbox
to5.69pt{\vbox to5.69pt{\pgfpicture\makeatletter\hbox{\hskip
2.84544pt\lower-2.84544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.84544pt}{0.0pt}\pgfsys@curveto{2.84544pt}{1.5715pt}{1.5715pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-1.5715pt}{2.84544pt}{-2.84544pt}{1.5715pt}{-2.84544pt}{0.0pt}\pgfsys@curveto{-2.84544pt}{-1.5715pt}{-1.5715pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{1.5715pt}{-2.84544pt}{2.84544pt}{-1.5715pt}{2.84544pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$$(3,17)\,\,\leavevmode\hbox
to5.69pt{\vbox to5.69pt{\pgfpicture\makeatletter\hbox{\hskip
2.84544pt\lower-2.84544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.84544pt}{0.0pt}\pgfsys@curveto{2.84544pt}{1.5715pt}{1.5715pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-1.5715pt}{2.84544pt}{-2.84544pt}{1.5715pt}{-2.84544pt}{0.0pt}\pgfsys@curveto{-2.84544pt}{-1.5715pt}{-1.5715pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{1.5715pt}{-2.84544pt}{2.84544pt}{-1.5715pt}{2.84544pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$$(4,17)\,\,0$$Sq^{1}_{3}\neq
0$$Sq^{1}_{3}\neq 0$$Sq^{1}_{3}=0$$d_{3}\neq 0$$d_{3}=0$

Again, $E_{3}^{4,17}=0$ implies that $d_{3}Sq^{1}_{3}=0$. On the other hand,
we shall see that $Sq^{1}_{3}d_{3}\neq 0$. To understand this composition,
first observe that the lemma above implies that $d_{3}(\leavevmode\hbox
to5.69pt{\vbox to5.69pt{\pgfpicture\makeatletter\hbox{\hskip
2.84544pt\lower-2.84544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.84544pt}{0.0pt}\pgfsys@curveto{2.84544pt}{1.5715pt}{1.5715pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-1.5715pt}{2.84544pt}{-2.84544pt}{1.5715pt}{-2.84544pt}{0.0pt}\pgfsys@curveto{-2.84544pt}{-1.5715pt}{-1.5715pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{1.5715pt}{-2.84544pt}{2.84544pt}{-1.5715pt}{2.84544pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=d_{3}X_{3}(\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04544pt\lower-3.04544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.84544pt}{0.0pt}\pgfsys@curveto{2.84544pt}{1.5715pt}{1.5715pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-1.5715pt}{2.84544pt}{-2.84544pt}{1.5715pt}{-2.84544pt}{0.0pt}\pgfsys@curveto{-2.84544pt}{-1.5715pt}{-1.5715pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{1.5715pt}{-2.84544pt}{2.84544pt}{-1.5715pt}{2.84544pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=X_{3}d_{3}(\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04544pt\lower-3.04544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.84544pt}{0.0pt}\pgfsys@curveto{2.84544pt}{1.5715pt}{1.5715pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-1.5715pt}{2.84544pt}{-2.84544pt}{1.5715pt}{-2.84544pt}{0.0pt}\pgfsys@curveto{-2.84544pt}{-1.5715pt}{-1.5715pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{1.5715pt}{-2.84544pt}{2.84544pt}{-1.5715pt}{2.84544pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\leavevmode\hbox
to5.69pt{\vbox to5.69pt{\pgfpicture\makeatletter\hbox{\hskip
2.84544pt\lower-2.84544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.84544pt}{0.0pt}\pgfsys@curveto{2.84544pt}{1.5715pt}{1.5715pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-1.5715pt}{2.84544pt}{-2.84544pt}{1.5715pt}{-2.84544pt}{0.0pt}\pgfsys@curveto{-2.84544pt}{-1.5715pt}{-1.5715pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{1.5715pt}{-2.84544pt}{2.84544pt}{-1.5715pt}{2.84544pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$, where
$X_{3}=H(X_{2},d_{2})$ acts non-trivially by Eqn. (1.1). Second,
$Sq^{1}_{3}=[Sq^{1}_{2}]=[Sq^{1}]$ is non-zero on this generator by virtue of
the computer computation. So $d_{3}Sq^{1}_{3}\neq Sq^{1}_{3}d_{3}$, which
contradicts the assumption that $Sq^{1}$ gives rise to an operation. ∎

######  Corollary.

There is no integral lift of the Szabó spectral sequence for which the
$E_{2}$-page agrees with (even) Khovanov homology.

###### Proof.

Suppose there is such a chain complex
$(CKh(T(4,5);\mathbb{Z}),\tilde{\delta}_{Sz}))$, so

$(CKh(T(4,5);\mathbb{F}_{2}),\delta_{Sz})\cong(CKh(T(4,5);\mathbb{Z}),\tilde{\delta}_{Sz}))\otimes_{\mathbb{Z}}\mathbb{F}_{2}$

and $\tilde{\delta}_{Sz}=d+d_{1}+d_{2}+\cdots$ where $d$ is the usual (even)
Khovanov differential. Then the Bockstein $Sq^{1}$ associated to
$\tilde{\delta}_{Sz}$ must give an operation on the Szabó spectral sequence
which agrees with the Bockstein $Sq^{1}$ associated to the (even) Khovanov $d$
differential on the $E_{2}$-page. The theorem above shows that this is not
possible. ∎

######  Remark.

In contrast to the corollary above, there is a preprint suggesting that
Szabó’s spectral sequence lifts to odd integral Khovanov homology [Bei]. The
second author has shown that the odd Bockstein map extends to an operation on
the Szabó spectral sequence [Pau22].

The Cartan formula allows us to construct examples for which $Sq^{n}$ is not
an operation on the Szabó spectral sequence.

######  Corollary.

For each $n\geq 1$, there is link $K_{n}$ for which the Steenrod operation
$Sq^{n}$ does not gives rise to an operation on Szabó’s geometric spectral
sequence.

###### Proof.

Set $K_{n}:=\sqcup_{i=1}^{n}T(4,5)$ so that $CKh(K_{n})\cong
CKh(T(4,5))^{\otimes n}$. The Szabó differential $\delta_{Sz}$ respects this
isomorphism and the Künneth formula shows $E_{m}(K_{n})\cong
E_{m}(T(4,5))^{\otimes n}$ and $E_{m+1}(T(4,5))^{\otimes n}\cong
H(E_{m}(T(4,5))^{\otimes n},d_{m})$.

Now let $a\in E^{3,17}_{3}$ be the element which satisfies $d_{3}a=b\in
E^{6,21}_{3}$ as in the proof of the previous theorem. In what follows, set
$Sq^{n}:=Sq^{n}_{3}$. We have $Sq^{n}a=0$ for all $n\geq 1$ and $Sq^{n}b=0$
for all $n>1$. Consider that $d_{3}(a\otimes b\otimes\cdots\otimes
b)=b\otimes\cdots\otimes b$ and the Steenrod operation is

$Sq^{n}(b\otimes\cdots\otimes
b)=\sum_{i_{1}+\cdots+i_{n}=n}Sq^{i_{1}}b\otimes\cdots\otimes
Sq^{i_{n}}b=Sq^{1}b\otimes\cdots\otimes Sq^{1}b$

which ensures that the composition $Sq^{n}d_{3}(a\otimes b\otimes\cdots\otimes
b)$ is non-zero. On the other hand, $Sq^{n}(a\otimes b\otimes\cdots\otimes
b)=Sq^{1}a\otimes Sq^{1}b\otimes\cdots\otimes Sq^{1}b=0$ because $Sq^{1}a=0$.
Therefore, $d_{3}Sq^{n}\neq Sq^{n}d_{3}$. ∎

## A pattern

In the first few dozen knot table examples one can observe a relationship
between $Sq^{2}$ and $d_{2}$. It appears that every non-zero $d_{2}$ leads to
a non-zero $Sq^{2}$, they occur together in the butterfly configuration
depicted below.

$t$$q$$t$$t+1$$t+2$$q$$q+2$$q+4$

See for example, $(t,q)=(2,15)$ in $T(4,5)$. More formally, this pattern
suggests that

$d_{2}(xm)=xm^{\prime}\quad\textnormal{ implies }\quad
Sq^{2}(m)=xm^{\prime}+\cdots.$

The converse is false as $T(4,5)$ contains a $Sq^{2}$-operation at
$(t,q)=(5,21)$ of a different sort.

## Acknowledgments

The authors thank R. Lipshitz for mentioning this question at the Banff
workshop in June 2021. The first author thanks the organizers M. Aganagić, S.
Krushkal and B. Webster for inviting his participation. This paper was
partially funded by Simons Award #638089.

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