documentclass{article}
usepackage{tikz}
usepackage{amsmath}
begin{document}
[
begin{aligned}
begin{tikzpicture}
draw (0,0)
    node [above] {$A$}
    to [out=down, in=down] (1,0)
    node [above] {$A^*$};
end{tikzpicture}
end{aligned}
=
begin{aligned}
begin{tikzpicture}
draw (0,0)
    node [above] {$A$}
    to [out=down, in=up] (1,-1)
    to [out=down, in=down] (0,-1)
    to [out=up, in=down] (1,0)
    node [above] {$A^*$};
end{tikzpicture}
end{aligned}
]
end{document}

catcode`@=11
input lambda.sty
input tikz

defUnravel#1{#1Unravel@{}}
defUnravel@#1{#1FoldrSpaceize{}}
defSpaceize#1#2{ #1#2}

newdimentallest
newcountindex

defTypesetThem#1{%
  advanceindex by 1
  TeXif{ifoddindex }{vcenter totallest{hbox{$#1$}vss}}{#1}}

defFindTallest#1{%
  setbox0vbox{hbox{$#1$}}%
  ifdimdimexprht0+dp0>tallesttallest=dimexprht0+dp0fi}

defSameHeight#1{%
  index=0
  tallest=0pt
  Unravel{MapFindTallest{#1}}
  Unravel{MapTypesetThem{#1}}}

defMyTikZList{Listize[
  {tikzdraw (0,0)
    node[above] {$A$}
    to[out=down,in=down] (1,0)
    node[above] {$A^*$};},
  =,
  {tikzdraw (0,0)
    node[above] {$A$}
    to[out=down,in=up] (1,-1)
    to[out=down,in=down] (0,-1)
    to[out=up,in=down] (1,0)
    node[above] {$A^*$};},
  +,
  {tikzdraw (0,0)
    node[above] {$A$}
    to[out=down,in=up] (1,-1)
    to[out=down,in=down] (0,-1)
    to[out=up,in=down] (1,0)
    node[above] {$A^*$};},
  -,
  {tikzdraw (0,0)
    node[above] {$A$}
    to[out=down,in=down] (1,0)
    node[above] {$A^*$};},
  ]}

$$
  SameHeightMyTikZList
$$

bye

documentclass{article}
usepackage{tikz}
usepackage{amsmath}
usetikzlibrary{calc}
begin{document}

An entangled equation and required meaningless text just to justify this sentence length.
begin{equation}
begin{tikzpicture}[baseline=(current bounding box.center)]
node (A) at (0,0) {tikz{draw (0,0)
    node [above] {$A$}
    to [out=down, in=down] (1,0)
    node [above] {$A^*$};}};
node[anchor=north west] (B) at ($(A.north east)+(1em,0)$) {
tikz{draw (0,0)
    node [above] {$A$}
    to [out=down, in=up] (1,-1)
    to [out=down, in=down] (0,-1)
    to [out=up, in=down] (1,0)
    node [above] {$A^*$};
}
};
node[left] (equa) at (B.west) {=};
end{tikzpicture}
end{equation}
end{document}

begin{equation}
    smash{raisebox{-height/2}{begin{tikzpicture}
    draw (0,0)
        node [above] {$A$}
        to [out=down, in=down] (1,0)
        node [above] {$A^*$};
    end{tikzpicture}}}
    = smash{raisebox{-height/2}{begin{tikzpicture}
    draw (0,0)
        node [above] {$A$}
        to [out=down, in=up] (1,-1)
        to [out=down, in=down] (0,-1)
        to [out=up, in=down] (1,0)
        node [above] {$A^*$};
    end{tikzpicture}}}
end{equation}