Classification of Links with Minimal Rank Khovanov Homology and a Conjecture of Shumakovitch

\newcommand\fsnmod{\fS_n\mathrm{-mod}} \newcommand\fS{\mathfrak{S}} \newcommand{\thin}{\mathrm{thin}} \newcommand{\KR}{\mathrm{KR}} \newcommand{\Exp}{\mathrm{Exp}} \newcommand{\sheaf}{\mathrm{sheaf}} \newcommand{\Aut}{\mathrm{Aut}} \newcommand{\Ind}{\mathrm{Ind}} \newcommand{\bad}{\mathrm{bad}} \newcommand{\cyc}{\mathrm{cyc}} \newcommand{\red}{\mathrm{red}} \newcommand{\im}{\mathrm{im}\,} \newcommand{\MBC}{\mathrm{M}_{\mathrm{BC}}} \newcommand{\ldp}{\lessdot^\prime} \newcommand{\simp}{\mathrm{simp}} \newcommand{\Ch}{\mathrm{Ch}} \newcommand{\ch}{\mathrm{ch}} \newcommand{\gr}{\mathrm{gr}} \newcommand{\FCH}{F^{\mathrm{Ch}}} \newcommand{\HCH}{H^{\mathrm{Ch}}} \newcommand{\FCS}{F^{\mathrm{CS}}} \newcommand{\HCS}{H^{\mathrm{CS}}} \newcommand{\FKH}{F^{\mathrm{Kh}}} \newcommand{\FVAND}{F^{\mathrm{Vand}}} \newcommand{\HVAND}{H_F^{\mathrm{Vand}}} \newcommand{\NBC}{\mathrm{NBC}} \newcommand{\BCC}{\mathrm{BC}} \newcommand{\cell}{\mathrm{cell}} \newcommand{\Kh}{\mathrm{Kh}} \newcommand{\hb}{\mathrm{hb}} \newcommand{\RGF}{\mathrm{RA}} \newcommand{\rank}{\mathrm{rank}} \newcommand{\Ob}{\mathrm{Ob}} \newcommand{\Decat}{\mathrm{Decat}} \newcommand{\qdim}{q\mathrm{dim} } \newcommand{\op}{\mathrm{op}} \newcommand{\Br}{\mathrm{Br}} \newcommand{\id}{\mathrm{id}} \newcommand{\Hom}{\mathrm{Hom}} \newcommand{\Int}{\mathrm{Int}} \newcommand{\Funct}{\mathrm{Funct}} \newcommand{\OP}{\mathcal{OP}} \newcommand{\PP}{\mathcal{P}} \newcommand{\rk}{\mathrm{rk}} \newcommand{\qrk}{q\mathrm{rk}} \newcommand{\N}{\mathbb{N}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\C}{\mathbb{C}} \newcommand{\F}{\mathbb{F}} \newcommand{\zp}{\mathbb{Z}_p} \newcommand{\pinch}{\hspace{.5mm}\usebox\opinch\hspace{.5mm}} \newcommand{\vcplx}{\mathcal{C}^*(D,\vec{x})} \newcommand{\vkcplx}{\mathcal{C}^k(D,\vec{x})} \newcommand{\vhcplx}{\mathcal{H}^*(D,\vec{x})} \newcommand{\vkhcplx}{\mathcal{H}^k(D,\vec{x})} \newcommand{\tvcplx}{\mathcal{C}^*(T_{2,n},\vec{x})} \newcommand{\tvkcplx}{\mathcal{C}^k(T_{2,n},\vec{x})} \newcommand{\tvhcplx}{\mathcal{H}^*(T_{2,n},\vec{x})} \newcommand{\tvkhcplx}{\mathcal{H}^k(T_{2,n},\vec{x})} \newcommand{\vsdiag}{[\![D]\!]} \newcommand{\inv}{\mathrm{inv}} \newcommand{\fix}{\mathrm{fix}} \newcommand{\sgn}{\mathrm{sgn}} \newcommand{\bott}{\mathrm{bot}} \newcommand{\topp}{\mathrm{top}} \newcommand{\ins}{\mathrm{ins}} \newcommand{\Zndiag}{\Z^n_+\mathrm{-}\mathbf{diag}} \newcommand{\Mor}{\mathrm{Mor}} \newcommand{\Cone}{\mathrm{Cone}} \newcommand{\wtilde}[1]{{\stackrel{\sim}{\smash{#1}\rule{0pt}{1.1ex}}}} \newcommand{\mc}[1]{\mathcal{#1}} \definecolor{forest}{rgb}{0.03, 0.47, 0.19} \newcommand{\ACcom}[1]{{\textcolor{forest}{{\sf AC:} #1}}} \newcommand{\Zmod}{\Z\textbf{-mod}} There is a spectral sequence given by Turner [(see here)](https://arxiv.org/abs/math/0411225) on $\Z_2$-Khovanov homology whose limiting page has rank $2^n$ where $n$ is the number of link components. As a consequence, we find that the rank of $\Z_2$-Khovanov homology of any link is at least $2^n$. In [this paper](https://arxiv.org/abs/1909.10032), Xie and Zhang give a classification of links with rank $2^n$. Their main result is that a link has rank $2^n$ Khovanov homology if and only if it is an iterated connect sum of unlinks and Hopf links. They refer to this family of links as "forests of unlinks". This family also appears in a conjecture of Shumakovitch. In [this paper](https://arxiv.org/abs/1806.05168), Shumakovitch conjectures that all links besides forests of unlinks contain torsion of order $2^r$ for some $r$ (even better, he claims $r=1$, but let us consider this weaker conjecture since it still gives unlink detection as a corollary). Shumakovitch also conjectures-in the same paper-that there is an algebraic relationship between the differentials $d_T^r$ on the $r$th page of Turner's spectral sequence, and the differential $d_B^r$ on the $r$th page of the $\Z_2$-Bockstein spectral sequence. He doesn't make this explicit, but I interpret this in the following way. **Conjecture 1: ** There exists maps $v_r$ of bidegree $(0,2r)$ on $\Z_2$-Khovanov homology such that $$d_T^r=d_B^r\circ v_r+v_r\circ d_T^r.$$ The $r=1$ case is shown [here](https://arxiv.org/abs/1806.05168). Now assume that the above conjecture holds, and let $L$ be a link which is not a forest of unlinks. Then the first page of the Turner spectral sequence has rank strictly greater than $2^n$ where $n$ is the number of components of $L$. It follows that there exists at least one non-zero Turner differential $d_T^r$. It follows from Conjecture 1 that $d_B^r$ is nonzero. Consequently, there is torsion of order $2^r$ in the integral Khovanov homology of $L$ (this is how the Bockstein spectral sequence works, nonzero maps on the $r$th page kill pairs of $\Z_2$ summands in $\Z_2$-Khovanov homology which come from a $\Z_{2^r}$ summand in integral Khovanov homology). However, let us consider the torus link $T(3,4)$. One can find the Khovanov homology [here](http://katlas.org/wiki/T(4,3)). It is clearly the case that $d_T^2$ is nonzero (this follows from the fact that the degree of $d_T^r$ is $(1,2r)$ and the infinity page of the Turner spectral sequence for $T(3,4)$ is concentrated in homological degree 0). Since there is no $\Z_4$-summand in the integral Khovanov homology of $T(3,4)$, we have found a counterexample to Conjecture 1.