\begin{scope}[yshift = - 7cm]
\draw (0,0) rectangle (1,2);
\draw (2,0) rectangle (3,2);

%\draw (1,2.75) to [out = 0, in = 180]
%(1.5,2.6) to [out = 0, in = 180]
\draw (1,.25) --(2,.25);
\begin{knot}[
%draft mode = crossings,
consider self intersections,
clip width = 5,
ignore endpoint intersections = true,
end tolerance = 2pt
% \flipcrossings{1,4}
% \strand (1,2.75) to [out = 0, in = 180] (2,1.75);
%\strand (2,2.75) to [out = 180, in =0] (1,1.75);

\strand (1,1.75) to [out = 0, in =180] (2,0.75);
\strand (1,0.75) to [out = 0, in = 180] (2,1.75);

\end{knot}

\draw (1.5, 1.5) circle (2.5cm);
\draw (7.5, 1.5) circle (2.5cm);
\draw (-4.5, 1.5) circle (2.5cm);
\draw (5.73,3.27) to [out = 135, in = 0]
(3,2.75) to [out = 180, in = 0]
(0,2.75) to [out = 180 , in = 45] (-2.73,3.27);
\draw (0,1.75) to [out = 180, in = -45] (-2.73, -.27);
\draw (3,2.75) to [out = 0, in = 135] (5.73,3.27);
\draw (3,1.75) to [out = 0, in = 225] (5.73,-.27);
\draw (-6.27,3.27) to [out = 135, in = 0] (-7.5,3.2);
\draw (-6.27,-.27) to [out = 225, in = 0] (-7.5,-.2);
\draw (9.27,3.27) to [out = 45, in = 180] (10.5,3.2);
\draw (9.27,-.27) to [out = -45, in = 180] (10.5,-.2);

\draw (-4.5,1.5) node{$R_{2i-1}$};
\draw (7.5,1.5) node{$R_{2i+1}$};
\draw (-3,3) node{\tiny{$+$}};
\draw (-3,0) node{\tiny{$-$}};
\draw (-6,3) node{\tiny{$-$}};
\draw (-6,0) node{\tiny{$+$}};
\begin{scope}[xshift = 12cm]
\draw (-3,3) node{\tiny{$+$}};
\draw (-3,0) node{\tiny{$-$}};
\draw (-6,3) node{\tiny{$-$}};
\draw (-6,0) node{\tiny{$+$}};
\end{scope}

\draw (.25,1.75) node{\tiny{$+$}};
%\draw (.25,2.75) node{\tiny{$+$}};
\draw (.75,.25) node{\tiny{$-$}};
\draw (.75,.75) node{\tiny{$+$}};
\draw (.75,1.75) node{\tiny{$-$}};
%\draw (.75,2.75) node{\tiny{$-$}};

\draw (2.25,.25) node{\tiny{$+$}};
\draw (2.25,.75) node{\tiny{$-$}};
\draw (2.25,1.75) node{\tiny{$+$}};
%\draw (2.25,2.75) node{\tiny{$+$}};
\draw (2.75,1.75) node{\tiny{$-$}};
%\draw (2.75,2.75) node{\tiny{$-$}};

\draw (2.5,.75)  node[yscale=-1]{$S$};
\draw (.5,.75) node[yscale=-1]{$T$};

\draw (1.5,-2) node{$\widetilde{D_A}$};

\end{scope}

\end{tikzpicture}\]

The diagrams $D_A$ and $\widetilde{D}_A$ in case 3 when both of the regions joined by the $A$-resolution of the top crossing are exterior faces. The diagram $\widetilde{D}_A$ has the same format as the standard Turaev genus one diagram in Figure <ref>.

The diagram $D_B$ is not $A$-adequate. Let $D_{BA}$ and $D_{BB}$ be the resolutions of $D_B$ at the bottom crossing, as in Figure <ref>. Since $D_{BA}$ is $A$-adequate, it follows that $\ukh^{1,-s_A(D_{BA})}(D_{BA})$ is trivial. The diagram $D_{BB}$ is a connected sum of alternating diagrams. Therefore $\ukh^{0,-s_A(D_{BB})}(D_{BB})\cong\mathbb{Z}$ and is generated by the all-$A$ Kauffman state $S_{BB}^A$ with $-$ labels on all components. The long exact sequence for $D_{BB}$, $D_B$, and $D_{BA}$ is given by

\[ \cdots\to \ukh^{0,-s_A(D_{BB})}(D_{BB})\xrightarrow{f^*} \ukh^{1,1-s_A(D_B)}(D_B)
\to  \ukh^{1,-s_A(D_{BA})}(D_{BA})\to\cdots.\]

\[\begin{tikzpicture}[scale=.6]

\draw (0,0) rectangle (1,3);
\draw (2,0) rectangle (3,3);

\draw (1,2.75) to [out = 0, in = 0, looseness = 1.3] (1,1.75);
\draw (2,2.75) to [out = 180, in = 180, looseness = 1.3] (2,1.75);
\draw (1.5,-1.5) node{$D_B$};

\begin{knot}[
%draft mode = crossings,
consider self intersections,
clip width = 5,
ignore endpoint intersections = true,
end tolerance = 2pt
% \flipcrossings{1,4}
% \strand (1,2.75) to [out = 0, in = 180] (2,1.75);
%\strand (2,2.75) to [out = 180, in =0] (1,1.75);

\strand (1,1.25) to [out = 0, in =180] (2,0.25);
\strand (1,0.25) to [out = 0, in = 180] (2,1.25);

\end{knot}

\draw (1.5, 1.5) circle (2.5cm);
\draw (7.5, 1.5) circle (2.5cm);
\draw (-4.5, 1.5) circle (2.5cm);
\draw (0,2.75) to [out = 180 , in = 45] (-2.73,3.27);
\draw (0,.25) to [out = 180, in = -45] (-2.73, -.27);
\draw (3,2.75) to [out = 0, in = 135] (5.73,3.27);
\draw (3,.25) to [out = 0, in = 225] (5.73,-.27);
\draw (-6.27,3.27) to [out = 135, in = 0] (-7.5,3.2);
\draw (-6.27,-.27) to [out = 225, in = 0] (-7.5,-.2);
\draw (9.27,3.27) to [out = 45, in = 180] (10.5,3.2);
\draw (9.27,-.27) to [out = -45, in = 180] (10.5,-.2);

\draw (-4.5,1.5) node{$R_{2i-1}$};
\draw (7.5,1.5) node{$R_{2i+1}$};
\draw (-3,3) node{\tiny{$+$}};
\draw (-3,0) node{\tiny{$-$}};
\draw (-6,3) node{\tiny{$-$}};
\draw (-6,0) node{\tiny{$+$}};
\begin{scope}[xshift = 12cm]
\draw (-3,3) node{\tiny{$+$}};
\draw (-3,0) node{\tiny{$-$}};
\draw (-6,3) node{\tiny{$-$}};
\draw (-6,0) node{\tiny{$+$}};
\end{scope}

\draw (.25,.25) node{\tiny{$-$}};
\draw (.25,2.75) node{\tiny{$+$}};
\draw (.75,.25) node{\tiny{$+$}};
\draw (.75,1.25) node{\tiny{$-$}};
\draw (.75,1.75) node{\tiny{$+$}};
\draw (.75,2.75) node{\tiny{$-$}};

\draw (2.25,.25) node{\tiny{$-$}};
\draw (2.25,1.25) node{\tiny{$+$}};
\draw (2.25,1.75) node{\tiny{$-$}};
\draw (2.25,2.75) node{\tiny{$+$}};
\draw (2.75,.25) node{\tiny{$+$}};
\draw (2.75,2.75) node{\tiny{$-$}};

%\draw (1.5,5) node{$N$};
%\draw (1.5,-2) node{$S$};

\begin{scope}[yshift=-7cm]
\draw (0,0) rectangle (1,3);
\draw (2,0) rectangle (3,3);

\draw (1,2.75) to [out = 0, in = 0, looseness = 1.3] (1,1.75);
\draw (2,2.75) to [out = 180, in = 180, looseness = 1.3] (2,1.75);

\draw (1,1.25) to [out = 0, in = 180] (1.5,1.1) to [out =0, in = 180] (2,1.25);
\draw (1,.25) to [out = 0, in = 180] (1.5,.4) to [out =0, in = 180] (2,.25);

\draw (1.5,-1.5) node{$D_{BA}$};

\begin{knot}[
%draft mode = crossings,
consider self intersections,
clip width = 5,
ignore endpoint intersections = true,
end tolerance = 2pt
% \flipcrossings{1,4}
% \strand (1,2.75) to [out = 0, in = 180] (2,1.75);
%\strand (2,2.75) to [out = 180, in =0] (1,1.75);

%\strand (1,1.25) to [out = 0, in =180] (2,0.25);
%\strand (1,0.25) to [out = 0, in = 180] (2,1.25);

\end{knot}

\draw (1.5, 1.5) circle (2.5cm);
\draw (7.5, 1.5) circle (2.5cm);
\draw (-4.5, 1.5) circle (2.5cm);
\draw (0,2.75) to [out = 180 , in = 45] (-2.73,3.27);
\draw (0,.25) to [out = 180, in = -45] (-2.73, -.27);
\draw (3,2.75) to [out = 0, in = 135] (5.73,3.27);
\draw (3,.25) to [out = 0, in = 225] (5.73,-.27);
\draw (-6.27,3.27) to [out = 135, in = 0] (-7.5,3.2);
\draw (-6.27,-.27) to [out = 225, in = 0] (-7.5,-.2);
\draw (9.27,3.27) to [out = 45, in = 180] (10.5,3.2);
\draw (9.27,-.27) to [out = -45, in = 180] (10.5,-.2);

\draw (-4.5,1.5) node{$R_{2i-1}$};
\draw (7.5,1.5) node{$R_{2i+1}$};
\draw (-3,3) node{\tiny{$+$}};
\draw (-3,0) node{\tiny{$-$}};
\draw (-6,3) node{\tiny{$-$}};
\draw (-6,0) node{\tiny{$+$}};
\begin{scope}[xshift = 12cm]
\draw (-3,3) node{\tiny{$+$}};
\draw (-3,0) node{\tiny{$-$}};
\draw (-6,3) node{\tiny{$-$}};
\draw (-6,0) node{\tiny{$+$}};
\end{scope}

\draw (.25,.25) node{\tiny{$-$}};
\draw (.25,2.75) node{\tiny{$+$}};
\draw (.75,.25) node{\tiny{$+$}};
\draw (.75,1.25) node{\tiny{$-$}};
\draw (.75,1.75) node{\tiny{$+$}};
\draw (.75,2.75) node{\tiny{$-$}};

\draw (2.25,.25) node{\tiny{$-$}};
\draw (2.25,1.25) node{\tiny{$+$}};
\draw (2.25,1.75) node{\tiny{$-$}};
\draw (2.25,2.75) node{\tiny{$+$}};
\draw (2.75,.25) node{\tiny{$+$}};
\draw (2.75,2.75) node{\tiny{$-$}};

\end{scope}

\begin{scope}[yshift=-14cm]
\draw (0,0) rectangle (1,3);
\draw (2,0) rectangle (3,3);

\draw (1,2.75) to [out = 0, in = 0, looseness = 1.3] (1,1.75);
\draw (2,2.75) to [out = 180, in = 180, looseness = 1.3] (2,1.75);

\draw (1,1.25) to [out = 0, in = 0, looseness = 1.3] (1,.25);
\draw (2,1.25) to [out = 180, in = 180, looseness = 1.3] (2,.25);

\draw (1.5,-1.5) node{$D_{BB}$};

\begin{knot}[
%draft mode = crossings,
consider self intersections,
clip width = 5,
ignore endpoint intersections = true,
end tolerance = 2pt
% \flipcrossings{1,4}
% \strand (1,2.75) to [out = 0, in = 180] (2,1.75);
%\strand (2,2.75) to [out = 180, in =0] (1,1.75);

%\strand (1,1.25) to [out = 0, in =180] (2,0.25);
%\strand (1,0.25) to [out = 0, in = 180] (2,1.25);

\end{knot}

\draw (1.5, 1.5) circle (2.5cm);
\draw (7.5, 1.5) circle (2.5cm);
\draw (-4.5, 1.5) circle (2.5cm);
\draw (0,2.75) to [out = 180 , in = 45] (-2.73,3.27);
\draw (0,.25) to [out = 180, in = -45] (-2.73, -.27);
\draw (3,2.75) to [out = 0, in = 135] (5.73,3.27);
\draw (3,.25) to [out = 0, in = 225] (5.73,-.27);
\draw (-6.27,3.27) to [out = 135, in = 0] (-7.5,3.2);
\draw (-6.27,-.27) to [out = 225, in = 0] (-7.5,-.2);
\draw (9.27,3.27) to [out = 45, in = 180] (10.5,3.2);
\draw (9.27,-.27) to [out = -45, in = 180] (10.5,-.2);

\draw (-4.5,1.5) node{$R_{2i-1}$};
\draw (7.5,1.5) node{$R_{2i+1}$};
\draw (-3,3) node{\tiny{$+$}};
\draw (-3,0) node{\tiny{$-$}};
\draw (-6,3) node{\tiny{$-$}};
\draw (-6,0) node{\tiny{$+$}};
\begin{scope}[xshift = 12cm]
\draw (-3,3) node{\tiny{$+$}};
\draw (-3,0) node{\tiny{$-$}};
\draw (-6,3) node{\tiny{$-$}};
\draw (-6,0) node{\tiny{$+$}};
\end{scope}

\draw (.25,.25) node{\tiny{$-$}};
\draw (.25,2.75) node{\tiny{$+$}};
\draw (.75,.25) node{\tiny{$+$}};
\draw (.75,1.25) node{\tiny{$-$}};
\draw (.75,1.75) node{\tiny{$+$}};
\draw (.75,2.75) node{\tiny{$-$}};

\draw (2.25,.25) node{\tiny{$-$}};
\draw (2.25,1.25) node{\tiny{$+$}};
\draw (2.25,1.75) node{\tiny{$-$}};
\draw (2.25,2.75) node{\tiny{$+$}};
\draw (2.75,.25) node{\tiny{$+$}};
\draw (2.75,2.75) node{\tiny{$-$}};

\end{scope}

\end{tikzpicture}\]

The diagrams $D_B$, $D_{BA}$, and $D_{BB}$ from case 3.

Because $\ukh^{1,-s_A(D_{BA})}(D_{BA})$ is trivial, if the image of $f^*$ is trivial, then $\ukh^{1,-s_A(D_B)}(D_B)$ is also trivial. Since $\ukh^{0,-s_A(D_{BB})}(D_{BB})\cong \mathbb{Z}$, it suffices to check that $f^*[S_{BB}^A]=[f(S_{BB}^A)]$ is trivial. The enhanced state $f(S_{BB}^A)$ is the state of $D_B$ where the bottom crossing is an $B$-resolution, all other crossings are $A$-resolutions, and every component is marked with a $-$. Let $S_{B}^A$ be the enhanced state of $D_B$ where every resolution is an $A$-resolution and every component is marked with a $-$. There is exactly one crossing (the bottom crossing) that when changed from $A$ to $B$ results in more components. Therefore $d(S_{B}^A)=\pm S_{B}^A$, and thus $f(S_{BB}^A)$ is in the image of the Khovanov differential. Therefore $f^*$ is the zero map, and thus $\ukh^{1,-s_A(D_B)}(D_B)$ is trivial.

Recall that the long exact sequence for $D_B$, $D_A$, and $D$ is

\[\cdots \ukh^{1,-s_A(D_B)}(D_B)\to\ukh^{2,2-s_A(D)}(D)\to\ukh^{2,2-s_A(D_A)}(D_A)\to\cdots,\]

and that both $\ukh^{1,-s_A(D_B)}(D_B)$ and $\ukh^{2,2-s_A(D_A)}(D_A)$ are trivial. Therefore $\ukh^{2,2-s_A(D)}(D)$ is also trivial.

Case 4: Suppose that $\girth(G_A(D))=2$, $D$ contains exactly four alternating tangles, and both $R_1$ and $R_3$ have a single crossing in the boundary of their respective eastern and western faces. Figure <ref> depicts the diagram $D$. If either of the regions joined by the $A$-resolution are completely contained inside $R_1$, then $D_A$ is  a reduced, $A$-adequate, Turaev genus one diagram, and $\ukh^{2,2-s_A(D_A)}(D_A)$ is trivial by induction. If both of the regions joined by the $A$-resolution intersect the boundary of $R_1$, then $D_A$ is the connected sum of alternating diagrams, and thus $\ukh^{2,2-s_A(D_A)}(D_A)$ is trivial.

\[\begin{tikzpicture}[scale = .6]

\draw (0,0) rectangle (1.5,1.5);
\draw (0,2.5) rectangle (1.5,4);

\draw (5,2) circle (2cm);

\draw (8.5,0) rectangle (10,1.5);
\draw (8.5,2.5) rectangle (10,4);

\draw (13.5,2) circle (2cm);

\draw (.75,.75) node{$R_1^2$};
\draw (.75,3.25) node{$R_1^1$};

\draw (.25,.25) node{\tiny{$+$}};
\draw (1.25,.25) node{\tiny{$-$}};
\draw (.25,1.25) node{\tiny{$-$}};
\draw (1.25,1.25) node{\tiny{$+$}};

\draw (.25,2.75) node{\tiny{$+$}};
\draw (1.25,2.75) node{\tiny{$-$}};
\draw (.25,3.75) node{\tiny{$-$}};
\draw (1.25,3.75) node{\tiny{$+$}};

\begin{scope}[xshift = 8.5cm]
\draw (.25,.25) node{\tiny{$+$}};
\draw (1.25,.25) node{\tiny{$-$}};
\draw (.25,1.25) node{\tiny{$-$}};
\draw (1.25,1.25) node{\tiny{$+$}};

\draw (.25,2.75) node{\tiny{$+$}};
\draw (1.25,2.75) node{\tiny{$-$}};
\draw (.25,3.75) node{\tiny{$-$}};
\draw (1.25,3.75) node{\tiny{$+$}};

\draw (3.8,3.15) node{$+$};
\draw (3.8, .85)node{$-$};
\draw (6.2,3.15) node{$-$};
\draw (6.2,.85) node{$+$};
\end{scope}

\draw (5,2) node{$R_2$};

\draw (3.8,3.15) node{$+$};
\draw (3.8, .85)node{$-$};
\draw (6.2,3.15) node{$-$};
\draw (6.2,.85) node{$+$};

\draw (3.586,3.414) to [out = 150, in = 0] (1.5,3.75);
\draw (3.586,.586) to [out = 210, in = 0] (1.5,.25);
\draw (6.414,3.414) to [out = 30, in = 180] (8.5,3.75);
\draw (6.414,.586) to [out = -30, in = 180] (8.5,.25);

\begin{scope}[xshift = 8.5cm]
\draw (3.586,3.414) to [out = 150, in = 0] (1.5,3.75);
\draw (3.586,.586) to [out = 210, in = 0] (1.5,.25);
\end{scope}

\draw (9.25,.75) node{$R_3^2$};
\draw (9.25,3.25) node{$R_3^1$};
\draw (13.5,2) node{$R_4$};

\draw (0,.25) arc (90:270:.5cm);
\draw (0,-.75) -- (14.7,-.75);

\draw (0,3.75) arc (270:90:.5cm);
\draw (0,4.75) -- (14.7,4.75);

\draw (14.914, .586) to [out = -45, in = 0, looseness = 1.5] (14.7,-.75);
\draw (14.914,3.414) to [out = 45, in = 0, looseness=1.5] (14.7,4.75);

\begin{knot}[
%draft mode = crossings,
consider self intersections,
clip width = 5,
ignore endpoint intersections = true,
end tolerance = 2pt
% \flipcrossings{1,4}
% \strand (.25,1.5) to [out = 90, in = 270] (1.25,2.5);
% \strand (.25,2.5) to [out = 270, in = 90] (1.25,1.5);

\strand (8.75,1.5) to [out = 90, in = 270] (9.75,2.5);
\strand (8.75,2.5) to [out = 270, in = 90] (9.75,1.5);

\end{knot}

\draw (.25,2.5) to [out = 270, in = 90] (.4,2) to [out = 270, in = 90] (.25,1.5);
\draw (1.25,2.5) to [out = 270, in = 90] (1.1,2) to [out = 270, in = 90] (1.25,1.5);
\draw (7.5, -1.25) node{$D_A$};

\begin{scope}[yshift=-7cm]
\draw (0,0) rectangle (1.5,1.5);
\draw (0,2.5) rectangle (1.5,4);

\draw (5,2) circle (2cm);

\draw (8.5,0) rectangle (10,1.5);
\draw (8.5,2.5) rectangle (10,4);

\draw (13.5,2) circle (2cm);

\draw (.75,.75) node{$R_1^2$};
\draw (.75,3.25) node{$R_1^1$};

\draw (.25,.25) node{\tiny{$+$}};
\draw (1.25,.25) node{\tiny{$-$}};
\draw (.25,1.25) node{\tiny{$-$}};
\draw (1.25,1.25) node{\tiny{$+$}};

\draw (.25,2.75) node{\tiny{$+$}};
\draw (1.25,2.75) node{\tiny{$-$}};
\draw (.25,3.75) node{\tiny{$-$}};
\draw (1.25,3.75) node{\tiny{$+$}};

\begin{scope}[xshift = 8.5cm]
\draw (.25,.25) node{\tiny{$+$}};
\draw (1.25,.25) node{\tiny{$-$}};
\draw (.25,1.25) node{\tiny{$-$}};
\draw (1.25,1.25) node{\tiny{$+$}};

\draw (.25,2.75) node{\tiny{$+$}};
\draw (1.25,2.75) node{\tiny{$-$}};
\draw (.25,3.75) node{\tiny{$-$}};
\draw (1.25,3.75) node{\tiny{$+$}};

\draw (3.8,3.15) node{$+$};
\draw (3.8, .85)node{$-$};
\draw (6.2,3.15) node{$-$};
\draw (6.2,.85) node{$+$};
\end{scope}

\draw (5,2) node{$R_2$};

\draw (3.8,3.15) node{$+$};
\draw (3.8, .85)node{$-$};
\draw (6.2,3.15) node{$-$};
\draw (6.2,.85) node{$+$};

\draw (3.586,3.414) to [out = 150, in = 0] (1.5,3.75);
\draw (3.586,.586) to [out = 210, in = 0] (1.5,.25);
\draw (6.414,3.414) to [out = 30, in = 180] (8.5,3.75);
\draw (6.414,.586) to [out = -30, in = 180] (8.5,.25);

\begin{scope}[xshift = 8.5cm]
\draw (3.586,3.414) to [out = 150, in = 0] (1.5,3.75);
\draw (3.586,.586) to [out = 210, in = 0] (1.5,.25);
\end{scope}

\draw (9.25,.75) node{$R_3^2$};
\draw (9.25,3.25) node{$R_3^1$};
\draw (13.5,2) node{$R_4$};

\draw (0,.25) arc (90:270:.5cm);
\draw (0,-.75) -- (14.7,-.75);

\draw (0,3.75) arc (270:90:.5cm);
\draw (0,4.75) -- (14.7,4.75);

\draw (14.914, .586) to [out = -45, in = 0, looseness = 1.5] (14.7,-.75);
\draw (14.914,3.414) to [out = 45, in = 0, looseness=1.5] (14.7,4.75);

\begin{knot}[
%draft mode = crossings,
consider self intersections,
clip width = 5,
ignore endpoint intersections = true,
end tolerance = 2pt
% \flipcrossings{1,4}
% \strand (.25,1.5) to [out = 90, in = 270] (1.25,2.5);
% \strand (.25,2.5) to [out = 270, in = 90] (1.25,1.5);

\strand (8.75,1.5) to [out = 90, in = 270] (9.75,2.5);
\strand (8.75,2.5) to [out = 270, in = 90] (9.75,1.5);

\end{knot}

\draw (.25,2.5) to [out = 270, in = 270, looseness=1.3] (1.25,2.5);
\draw (.25,1.5) to [out = 90, in = 90, looseness=1.3] (1.25,1.5);
\draw (7.5, -1.25) node{$D_B$};
\end{scope}

\end{tikzpicture}\]

Diagrams of $D_A$ and $D_B$ in case $4$.

Let $D_{BA}$ and $D_{BB}$ be the diagrams obtained by resolving the pictured crossing of $D$ in $R_3$, as in Figure <ref>. One can repeat the argument in case 3 verbatim to conclude that $\ukh^{1,-s_A(D_B)}(D_B)$ is trivial. The long exact sequence for $D_B$, $D_A$, and $D$ is

\[\cdots \ukh^{1,-s_A(D_B)}(D_B)\to\ukh^{2,2-s_A(D)}(D)\to\ukh^{2,2-s_A(D_A)}(D_A)\to\cdots.\]

Since both $\ukh^{1,-s_A(D_B)}(D_B)$ and $\ukh^{2,2-s_A(D_A)}(D_A)$ are trivial, it follows that $\ukh^{2,2-s_A(D)}(D)$ is also trivial.

\[\begin{tikzpicture}[scale=.6]
\draw (0,0) rectangle (1.5,1.5);
\draw (0,2.5) rectangle (1.5,4);

\draw (5,2) circle (2cm);

\draw (8.5,0) rectangle (10,1.5);
\draw (8.5,2.5) rectangle (10,4);

\draw (13.5,2) circle (2cm);

\draw (.75,.75) node{$R_1^2$};
\draw (.75,3.25) node{$R_1^1$};

\draw (.25,.25) node{\tiny{$+$}};
\draw (1.25,.25) node{\tiny{$-$}};
\draw (.25,1.25) node{\tiny{$-$}};
\draw (1.25,1.25) node{\tiny{$+$}};

\draw (.25,2.75) node{\tiny{$+$}};
\draw (1.25,2.75) node{\tiny{$-$}};
\draw (.25,3.75) node{\tiny{$-$}};
\draw (1.25,3.75) node{\tiny{$+$}};

\begin{scope}[xshift = 8.5cm]
\draw (.25,.25) node{\tiny{$+$}};
\draw (1.25,.25) node{\tiny{$-$}};
\draw (.25,1.25) node{\tiny{$-$}};
\draw (1.25,1.25) node{\tiny{$+$}};

\draw (.25,2.75) node{\tiny{$+$}};
\draw (1.25,2.75) node{\tiny{$-$}};
\draw (.25,3.75) node{\tiny{$-$}};
\draw (1.25,3.75) node{\tiny{$+$}};

\draw (3.8,3.15) node{$+$};
\draw (3.8, .85)node{$-$};
\draw (6.2,3.15) node{$-$};
\draw (6.2,.85) node{$+$};
\end{scope}

\draw (5,2) node{$R_2$};

\draw (3.8,3.15) node{$+$};
\draw (3.8, .85)node{$-$};
\draw (6.2,3.15) node{$-$};
\draw (6.2,.85) node{$+$};

\draw (3.586,3.414) to [out = 150, in = 0] (1.5,3.75);
\draw (3.586,.586) to [out = 210, in = 0] (1.5,.25);
\draw (6.414,3.414) to [out = 30, in = 180] (8.5,3.75);
\draw (6.414,.586) to [out = -30, in = 180] (8.5,.25);

\begin{scope}[xshift = 8.5cm]
\draw (3.586,3.414) to [out = 150, in = 0] (1.5,3.75);
\draw (3.586,.586) to [out = 210, in = 0] (1.5,.25);
\end{scope}

\draw (9.25,.75) node{$R_3^2$};
\draw (9.25,3.25) node{$R_3^1$};
\draw (13.5,2) node{$R_4$};

\draw (0,.25) arc (90:270:.5cm);
\draw (0,-.75) -- (14.7,-.75);

\draw (0,3.75) arc (270:90:.5cm);
\draw (0,4.75) -- (14.7,4.75);

\draw (14.914, .586) to [out = -45, in = 0, looseness = 1.5] (14.7,-.75);
\draw (14.914,3.414) to [out = 45, in = 0, looseness=1.5] (14.7,4.75);

\draw (8.75,2.5) to [out = 270, in =90] (8.9,2) to [out=270, in=90] (8.75,1.5);
\draw (9.75,2.5) to [out = 270, in = 90] (9.6,2) to [out = 270, in=90] (9.75,1.5);

\draw (.25,2.5) to [out = 270, in = 270, looseness=1.3] (1.25,2.5);
\draw (.25,1.5) to [out = 90, in = 90, looseness=1.3] (1.25,1.5);
\draw (7.5, -1.25) node{$D_{BA}$};

\begin{scope}[yshift = -7cm]
\draw (0,0) rectangle (1.5,1.5);
\draw (0,2.5) rectangle (1.5,4);

\draw (5,2) circle (2cm);

\draw (8.5,0) rectangle (10,1.5);
\draw (8.5,2.5) rectangle (10,4);

\draw (13.5,2) circle (2cm);

\draw (.75,.75) node{$R_1^2$};
\draw (.75,3.25) node{$R_1^1$};

\draw (.25,.25) node{\tiny{$+$}};
\draw (1.25,.25) node{\tiny{$-$}};
\draw (.25,1.25) node{\tiny{$-$}};
\draw (1.25,1.25) node{\tiny{$+$}};

\draw (.25,2.75) node{\tiny{$+$}};
\draw (1.25,2.75) node{\tiny{$-$}};
\draw (.25,3.75) node{\tiny{$-$}};
\draw (1.25,3.75) node{\tiny{$+$}};

\begin{scope}[xshift = 8.5cm]
\draw (.25,.25) node{\tiny{$+$}};
\draw (1.25,.25) node{\tiny{$-$}};
\draw (.25,1.25) node{\tiny{$-$}};
\draw (1.25,1.25) node{\tiny{$+$}};

\draw (.25,2.75) node{\tiny{$+$}};
\draw (1.25,2.75) node{\tiny{$-$}};
\draw (.25,3.75) node{\tiny{$-$}};
\draw (1.25,3.75) node{\tiny{$+$}};

\draw (3.8,3.15) node{$+$};
\draw (3.8, .85)node{$-$};
\draw (6.2,3.15) node{$-$};
\draw (6.2,.85) node{$+$};
\end{scope}

\draw (5,2) node{$R_2$};

\draw (3.8,3.15) node{$+$};
\draw (3.8, .85)node{$-$};
\draw (6.2,3.15) node{$-$};
\draw (6.2,.85) node{$+$};

\draw (3.586,3.414) to [out = 150, in = 0] (1.5,3.75);
\draw (3.586,.586) to [out = 210, in = 0] (1.5,.25);
\draw (6.414,3.414) to [out = 30, in = 180] (8.5,3.75);
\draw (6.414,.586) to [out = -30, in = 180] (8.5,.25);

\begin{scope}[xshift = 8.5cm]
\draw (3.586,3.414) to [out = 150, in = 0] (1.5,3.75);
\draw (3.586,.586) to [out = 210, in = 0] (1.5,.25);
\end{scope}

\draw (9.25,.75) node{$R_3^2$};
\draw (9.25,3.25) node{$R_3^1$};
\draw (13.5,2) node{$R_4$};

\draw (0,.25) arc (90:270:.5cm);
\draw (0,-.75) -- (14.7,-.75);

\draw (0,3.75) arc (270:90:.5cm);
\draw (0,4.75) -- (14.7,4.75);

\draw (14.914, .586) to [out = -45, in = 0, looseness = 1.5] (14.7,-.75);
\draw (14.914,3.414) to [out = 45, in = 0, looseness=1.5] (14.7,4.75);

\draw (8.75,2.5) to [out = 270, in = 270, looseness=1.3] (9.75,2.5);
\draw (8.75,1.5) to [out = 90, in = 90, looseness=1.3] (9.75,1.5);

\draw (.25,2.5) to [out = 270, in = 270, looseness=1.3] (1.25,2.5);
\draw (.25,1.5) to [out = 90, in = 90, looseness=1.3] (1.25,1.5);
\draw (7.5, -1.25) node{$D_{BB}$};
\end{scope}

\end{tikzpicture}\]

Diagrams of $D_{BA}$ and $D_{BB}$ in case $4$.

Theorem <ref> and <ref> combine to prove Theorem <ref>.

Let $D$ be an $A$-Turaev genus one diagram of $L$. Then $D$ is either $A$-almost alternating or $D$ is $A$-adequate and has Turaev genus one. In the former case, Theorem <ref> implies the result, and in the latter case Theorem <ref> implies the result.

If $D$ is a $B$-Turaev genus one diagram of $L$, then the mirror $\overline{D}$ is $A$-Turaev genus one, and Equation <ref> implies the result.

Let $K$ be the $14$-crossing knot depicted in Figure <ref>. The Khovanov homology of $K$ is computed in Table <ref>. The Khovanov homology of $K$ in its maximum polynomial grading is $Kh^{*,j_{\max}(K)}(K) \cong Kh^{8,19}(K)\cong \mathbb{Z}^2$. The Khovanov homology of $K$ in its minimum polynomial grading is $Kh^{*,j_{\min}(K)}(K) \cong Kh^{-2,-1}(K) \cong \mathbb{Z}$; also $Kh^{i_{\min}(K)+2,j_{\min}(K)+2}(K) \cong Kh^{0,1}(K) \cong \mathbb{Z}$, which is nontrivial. Therefore Theorem <ref> implies that $K$ has Turaev genus at least two. No previous result would obstruct this knot from having Turaev genus one.

\[\begin{tikzpicture}[scale = .3]
\begin{knot}[
%draft mode = crossings,
consider self intersections,
clip width = 10,
ignore endpoint intersections = true,
end tolerance = 2pt
\flipcrossings{1,4,5,9,10,12}
\strand[thick] (16,1) to [out = 0, in = 240]
(21,4) to [out = 60, in = -30]
(20,10) to [out = 150, in = 0]
(14,11) to [out = 180, in = -30]
(8,13) to [out = 150, in = 240]
(2.5,20.5) to [out = 60, in = 180]
(7,23) to [out = 0, in =80]
%(14,21) to [out = -60, in = 80]
(17,16) to [out = 260, in = 120]
(16,9) to [out = 300, in = 180]
(19,7) to [out = 0, in = 270]
(26,13) to [out = 90, in = 30]
(17,19) to [out = 210, in = 90]
(14,15) to [out = 270, in = 225]
(19,15) to [out = 45, in = -60]
(20,23) to [out = 120, in = 0]
(14,26) to [out = 180, in = 90]
%(7,19) to [out = 225, in = 90]
(4,9) to [out = 270, in = 180]
(9,4) to [out = 0, in = 225]
(15,7) to [out = 45, in = 270]
(19,13) to [out = 90, in = 45, looseness=1.5]
(10,14) to [out = 225, in=180]
\end{knot}

%\fill (14,19) circle (.5cm);
\end{tikzpicture}\]

Theorem <ref> implies this knot has Turaev genus at least two.

$j \backslash i$    -2     -1     0     1     2     3     4     5     6     7     8

19                                             2

17                                          1     $2_2$

15                                      2     $2, 1_2$

13                              1     3     $1,2_2$

11                              $1,1_2$     $2,3_2$

9                      1     4     $3,1_2$

7                  1     $1,1_2$     $1,3_2$

5                  $1,1_2$     $3,1_2$

3          1     2     1

1         $1_2$    red 1

-1      1

The Khovanov homology of the 14-crossing knot in Figure <ref>. An entry of $k$ indicates a $\mathbb{Z}^k$ summand, and an entry of $k_2$ indicates a $\mathbb{Z}_2^k$ summand.

§ RASMUSSEN'S INVARIANT AND $4$-GENUS

In this section, we prove Theorems <ref> and <ref>. Theorem <ref> gives formulas the Rasmussen $s$ invariant of some Turaev genus one knots, and Theorem <ref> gives bounds on the smooth $4$-genus of the same Turaev genus one knots.

Let $D$ be an $A$-Turaev genus one diagram of a knot $K$. Theorem <ref> implies that one summand of Khovanov homology in the third nontrivial homological grading is trivial. If the third nontrivial homological grading in $Kh(D)$ is $i=0$, then Theorem <ref> and its proof below show how to compute the Rasmussen $s$ invariant, which gives a lower bound on the smooth $4$-genus $g_4(K)$ of the knot $K$. If $D$ is $A$-almost alternating, then the third nontrivial homological grading is $i=0$ when $D$ has three negative crossings, and if $D$ is an $A$-adequate diagram with Turaev genus one, then the third nontrivial homological grading is $i=0$ when $D$ has two negative crossings.

The Seifert or oriented resolution of an oriented crossing $~ \tikz[baseline=.6ex, scale = .4]{
\draw[->] (0,0) -- (1,1);
\draw[->] (.3,.7) -- (0,1);
\draw (.7,.3) -- (1,0);
}~$ or $~ \tikz[baseline=.6ex, scale = .4]{
\draw[->] (.7,.7) -- (1,1);
\draw[->] (1,0) -- (0,1);
\draw (0,0) -- (.3,.3);
}~$ is given by $~ \tikz[baseline=.6ex, scale = .4]{
\draw[->] (0,0) to [out = 60, in = -60] (0,1);
\draw[->] (1,0) to [out = 120, in = 240] (1,1);
}.$ The state obtained by taking the Seifert resolution at every crossing is the Seifert state. Let $f(D)$ denote the number of components in the Seifert state of the link diagram $D$. A link is positive if it has a diagram where every crossing is positive. Nakamura [29] proved the following theorem about the $4$-genus of positive links.

Let $D$ be a diagram of $L$ such that every crossing in $D$ is positive. The four-genus of $L$ is given by

\[g_4(L) = \frac{2-\mu(L) - f(D) +c(D)}{2}\]

where $\mu(L)$ is the number of components of $L$, $f(D)$ is the number of Seifert circles of $D$, and $c(D)$ is the number of crossings of $D$.

Rasmussen [30] proved that if $K$ is positive, then $s(K) = 2g_4(K)$. Tagami [32] proved that if $K$ is almost-positive, that is, if $K$ has a diagram with one negative crossing, then $s(K) = 2g_4(K)$. Theorem <ref> can be seen as an extension of this work in the case that the knot has Turaev genus one.

The following result is a well-known consequence of Rasmussen's construction of $s(K)$; see, for instance, Proposition 2.2 in Tagami [32].

For any knot $K$, the Khovanov homology groups $Kh^{0,s(K)\pm 1}(K;\mathbb{Q})$ are nontrivial.

Theorems <ref> and <ref> combine to give us the proof of Theorem <ref>.

Let $D$ be an $A$-adequate diagram such that $g_T(D)=1$ and $c_-(D)=2$. Theorem <ref> implies that $\ukh^{2,2-s_A(D)}(D)\cong Kh^{0,c(D)-s_A(D)-4}(D)$ is trivial. Then Inequality <ref> implies that $Kh^{0,j}(D)$ is nontrivial only when $j=c(D)-s_A(D)-2$ or $c(D)-s_A(D)$. Hence Theorem <ref> implies that the Rasmussen $s$-invariant of $K$ is given by $s(K)=c(D)-s_A(D)-1$.

Now suppose that $D$ is an $A$-almost alternating diagram such that $c_-(D)=3$. Theorem <ref> implies that $\ukh^{3,4-s_A(D)}(D) \cong Kh^{0,c(D)-s_A(D)-5}(D)$ is trivial. Then Inequality <ref> implies that $Kh^{0,j}(D)$ is nontrivial only when $j=c(D)-s_A(D)-3$ or $c(D)-s_A(D)-1$. Finally Theorem <ref> implies that $s(K) = c(D)-s_A(D)-2$.

The proofs in the cases where $D$ is $B$-almost alternating or $B$-adequate are similar.

Theorem <ref> gives the lower bound $\frac{|s(K)|}{2}$ for $g_4(K)$, but it remains to show that with the assumptions of Theorem <ref>, $\frac{|s(K)|}{2}+1$ is an upper bound for $g_4(K)$. We begin by analyzing properties of diagrams of Turaev genus one with a small number of negative crossings.

Let $D$ be an oriented link diagram, and let $\Gamma=\Gamma(D)$ be the $4$-valent graph obtained by considering the crossings of $D$ as vertices in $\Gamma$ and the arcs of $D$ going between the crossings as the edges of $\Gamma$. The orientation of $D$ induces a direction on each edge of $\Gamma$ so that $\Gamma$ becomes a directed graph. If $D$ is $A$-adequate, the all-$A$ Kauffman state induces a cycle decomposition of $\Gamma$. A vertex of $\Gamma$ can be labeled as positive or negative according to whether the corresponding crossing in $D$ is positive or negative.
See Figure <ref> for an example of this construction.

\[\begin{tikzpicture}[scale = .5, thick]

\begin{knot}[
%draft mode = crossings,
consider self intersections,
clip width = 4,
ignore endpoint intersections = true,
end tolerance = 1pt
\flipcrossings{9,1,4,6,7}
\strand (8,0) to [out = 0, in = -60]
(9,5) to [out = 120, in =-30]
(5,12) to [out = 150, in = 60]
(1,11) to [out = 240, in = 210]
(3,7) to [out = 30, in = 225]
(5,14) to [out = 45, in = 100]
(13,12) to [out = -80, in=0]
(11.5,8) to [out = 180, in = 0]
(5,10) to [out = 180, in = 120, looseness=1.5]
(1,4) to [out = -60, in = 225]
(12,5) to [out = 45, in =270]
(14,9) to [out = 90, in = 80]
(6,4) to [out = 260, in= 180]

\end{knot}
\draw (7.5,.5) -- (8,0) -- (7.5,-.5);

\draw (8,-2) node{$D$};

\begin{scope}[xshift= 16cm]

\begin{knot}[
%draft mode = crossings,
consider self intersections,
clip width = 4,
ignore endpoint intersections = true,
end tolerance = 1pt
\flipcrossings{9,1,4,6,7}
\strand (8,0) to [out = 0, in = -60]
(9,5) to [out = 120, in =-30]
(5,12) to [out = 150, in = 60]
(1,11) to [out = 240, in = 210]
(3,7) to [out = 30, in = 225]
(5,14) to [out = 45, in = 100]
(13,12) to [out = -80, in=0]
(11.5,8) to [out = 180, in = 0]
(5,10) to [out = 180, in = 120, looseness=1.5]
(1,4) to [out = -60, in = 225]
(12,5) to [out = 45, in =270]
(14,9) to [out = 90, in = 80]
(6,4) to [out = 260, in= 180]

\end{knot}
\draw (7.5,.5) -- (8,0) -- (7.5,-.5);

\fill[red] (5.7, 1.3) circle (.2cm);
\fill[red] (10,3) circle (.2cm);
\fill[red] (13.5,9) circle (.2cm);
\fill[red] (9.3,8) circle (.2cm);
\fill[red](8.1,8) circle (.2cm);
\fill[red] (7,9) circle (.2cm);
\fill[red] (4.8,9.2) circle (.2cm);
\fill[red] (1,7.2) circle (.2cm);

\fill[red] (1.2,9.5) circle (.2cm);
\fill[red] (3.7,10.5) circle (.2cm);
\fill[red] (3.7, 12) circle (.2cm);

\fill[red] (6.5,2.6) circle (.2cm);
\fill[red] (9.1,3.4) circle (.2cm);
\fill[red] (7.9,6.6) circle (.2cm);

\fill[red] (4.7,12.5) circle (.2cm);
\fill[red] (7.5,10) circle (.2cm);
\fill[red] (9.3,9.2) circle (.2cm);
\fill[red] (12.7,10.6) circle (.2cm);

\begin{scope}[very thick,red, decoration={
mark=at position 0.5 with {\arrow{>}}}
\draw[postaction={decorate}] (5.7,1.3) to [out=-80, in =-80, looseness=2] (10,3);
\draw[postaction={decorate}] (10,3) to [out=20, in =-90, looseness=1] (13.5,9);
\draw[postaction={decorate}] (13.5,9) to [out=225, in =-30, looseness=1.3] (9.3,8);
\draw[postaction={decorate}] (9.3,8) to (8.1,8);
\draw[postaction={decorate}] (8.1,8) to (7,9);
\draw[postaction={decorate}] (7,9) to [out=170, in =10, looseness=1] (4.8,9.2);
\draw[postaction={decorate}] (1,7.2) to [out=-60, in =260, looseness=1.5] (4.8,9.2);
\draw[postaction={decorate}] (1,7.2) to [out=-100, in =190, looseness=1.5] (5.7,1.3);

\draw[postaction={decorate}] (3.7,10.5) to [out=190, in =20, looseness=1] (1.2,9.5);
\draw[postaction={decorate}] (3.7,12) to [out=190, in =90, looseness=1] (1.2,9.5);
\draw[postaction={decorate}] (3.7,10.5) -- (3.7,12);

\draw[postaction={decorate}] (6.5,2.6) to [out = 30, in = 210] (9.1,3.4);
\draw[postaction={decorate}] (9.1,3.4) to [out = 120, in = 300] (7.9,6.6);
\draw[postaction={decorate}] (7.9,6.6) to [out = 240, in = 90] (6.5,2.6);

\draw[postaction={decorate}] (4.7,12.5) to [out = 60, in = 100, looseness = 1.3] (12.7,10.6);
\draw[postaction={decorate}] (12.7,10.6) to [out = 180, in = 30, looseness = 1.2] (9.3,9.2);
\draw[postaction={decorate}] (9.3,9.2) to (7.5,10);
\draw[postaction={decorate}] (7.5,10) to [out = 120, in = -30, looseness = 1.2] (4.7,12.5);

\end{scope}

\draw (8,-2) node{$\mathcal{C}$};

\end{scope}

\end{tikzpicture}\]

A diagram $D$ of the knot $9_{43}$ and the cycle decomposition $\mathcal{C}$ of $\Gamma(D)$ induced by the all-$A$ state of $D$. The diagram $D$ is $A$-adequate and has two negative crossings; the negative crossings correspond to sinks or sources in $\mathcal{C}$.

Let $D$ be an oriented $A$-adequate link diagram. Each cycle in the cycle decomposition of $\Gamma(D)$ induced by the all-$A$ state of $D$ contains an even number of negative vertices.

Let $\mathcal{C}$ be the cycle decomposition of $\Gamma(D)$ induced by the all-$A$ state. Figure <ref> shows that a negative crossing becomes two vertices in distinct cycles of $\mathcal{C}$ where one is a source and the other is a sink. A positive crossing becomes two vertices in distinct cycles of $\mathcal{C}$, neither of which is a sink or a source. Because every cycle whose edges are directed has an equal number of sinks and sources, the result follows.

\[\begin{tikzpicture}
\draw (0,0) -- (.8,.8);
\draw[->] (1.2,1.2) -- (2,2);
\draw[->] (2,0) -- (0,2);

\fill[red] (1,1.4) circle (0.05cm);
\draw[red] (1,1.4) -- (1.6,2);
\draw[red,->] (1,1.4) -- (1.3,1.7);
\draw[red] (1,1.4) -- (.4,2);
\draw[red,->] (1,1.4) -- (.7,1.7);
\fill[red] (1,.6) circle (0.05cm);
\draw[red] (1.6,0) -- (1,.6);
\draw[red,->] (1.6,0) -- (1.3,.3);
\draw[red] (.4,0) -- (1,.6);
\draw[red,->] (.4,0) -- (.7,.3);

\draw (1,-.5) node{negative crossing};

\begin{scope}[xshift = 4cm]
\draw[->] (0,0) -- (2,2);
\draw (2,0) -- (1.2,.8);
\draw[->] (.8,1.2) -- (0,2);
\draw (1,-.5) node{positive crossing};

\fill[red] (1.4,1) circle (0.05cm);
\fill[red] (.6,1) circle (0.05cm);
\draw[red] (2,.4) -- (1.4,1) -- (2,1.6);
\draw[red] (0,.4) -- (.6,1) -- (0,1.6);
\draw[red,->] (0,.4) -- (.3,.7);
\draw[red,->] (.6,1) -- (.3,1.3);
\draw[red,->] (2,.4) -- (1.7,.7);
\draw[red,->] (1.4,1) -- (1.7,1.3);

\end{scope}

\end{tikzpicture}\]

A negative crossing results in a sink or source in the cycle decomposition induced by the all-$A$ state while a positive crossing does not.

In order to prove Theorem <ref>, we use the following lemma describing the structure of one type of knot that satisfies the assumptions of the theorem.

\[\begin{tikzpicture}[scale = 1]

\draw (0,0) rectangle (1.5,1.5);
\draw (0,2.5) rectangle (1.5,4);
\draw (2.5,0) rectangle (4,1.5);
\draw (2.5,2.5) rectangle (4,4);

\draw (.75,3.25) node{$R_1$};
\draw (3.25,3.25) node{$R_2$};
\draw (3.25,.75) node{$R_3$};
\draw (.75,.75) node{$R_4$};

\draw (.4,1.25) node{\tiny{$-$}};
\draw (1.1,1.25) node{\tiny{$+$}};
\draw (1.25,1.1) node{\tiny{$-$}};
\draw (1.25,.4) node{\tiny{$+$}};

\draw (.4,2.75) node{\tiny{$+$}};
\draw (1.1,2.75) node{\tiny{$-$}};
\draw (1.25,2.9) node{\tiny{$+$}};
\draw (1.25,3.6) node{\tiny{$-$}};

\draw (2.9,1.25) node{\tiny{$-$}};
\draw (3.6,1.25) node{\tiny{$+$}};
\draw (2.75,1.1) node{\tiny{$+$}};
\draw (2.75,.4) node{\tiny{$-$}};

\draw (2.9,2.75) node{\tiny{$+$}};
\draw (3.6,2.75) node{\tiny{$-$}};
\draw (2.75,2.9) node{\tiny{$-$}};
\draw (2.75,3.6) node{\tiny{$+$}};

\begin{knot}[    %draft mode = crossings,
consider self intersections,
clip width = 5,
ignore endpoint intersections = true,
end tolerance = 1pt

\strand[->]
(1.1,2.5) to [out = 270, in =90 ] (.4,1.5);
\strand[->]
(1.1,1.5) to [out = 90, in =270 ] (.4,2.5);

\strand[->] (2.5,3.6) to [out = 180, in =0] (1.5,2.9);
\strand[->] (1.5,3.6) to [out = 0, in= 180] (2.5,2.9);

\strand[->] (2.9,1.5) to [out = 90, in =270] (3.6,2.5);
\strand [->] (2.9,2.5) to [out = 270, in =90] (3.6,1.5);

\strand[->] (2.5,.4) to [out = 180, in=0] (1.5,1.1);
\strand[->] (1.5,.4) to [out = 0, in = 180] (2.5,1.1);

\end{knot}

\draw[red, dashed] (2,3.25) circle (.3cm);

\begin{scope}[xshift = 5cm]

\draw (0,0) rectangle (6,1.5);
\draw (0,2.5) rectangle (6,4);

\draw (3,.75) node{$R_2$};
\draw (3,3.25) node{$R_1$};

\draw (.25,1.25) node{\tiny{$-$}};
\draw (1.25,1.25) node{\tiny{$+$}};
\draw (1.75,1.25) node{\tiny{$-$}};
\draw (2.75,1.25) node{\tiny{$+$}};
\draw (3.25,1.25) node{\tiny{$-$}};
\draw (4.25,1.25) node{\tiny{$+$}};
\draw (4.75,1.25) node{\tiny{$-$}};
\draw (5.75,1.25) node{\tiny{$+$}};

\draw (.25,2.75) node{\tiny{$+$}};
\draw (1.25,2.75) node{\tiny{$-$}};
\draw (1.75,2.75) node{\tiny{$+$}};
\draw (2.75,2.75) node{\tiny{$-$}};
\draw (3.25,2.75) node{\tiny{$+$}};
\draw (4.25,2.75) node{\tiny{$-$}};
\draw (4.75,2.75) node{\tiny{$+$}};
\draw (5.75,2.75) node{\tiny{$-$}};

\begin{knot}[    %draft mode = crossings,
consider self intersections,
clip width = 5,
ignore endpoint intersections = true,
end tolerance = 1pt
\strand[->] (.25,2.5) to [out = 270,in=90] (1.25,1.5);
\strand[->] (.25,1.5) to [out = 90, in=270] (1.25,2.5);

\strand[->] (1.75,1.5) to [out = 90, in = 270] (2.75,2.5);
\strand[->] (1.75, 2.5) to [out = 270, in =90] (2.75,1.5);

\strand[->] (3.25,1.5) to [out = 90, in = 270] (4.25,2.5);
\strand[->] (3.25, 2.5) to [out = 270, in =90] (4.25,1.5);

\strand[->] (4.75,1.5) to [out = 90, in = 270] (5.75,2.5);
\strand[->] (4.75, 2.5) to [out = 270, in =90] (5.75,1.5);

\end{knot}

\draw[red, dashed] (.75,2) circle (.3cm);

\end{scope}

\end{tikzpicture}\]

The two possible formats for an $A$-almost alternating diagram with positive dealternator and three negative crossings. Each tangle $R_i$ is a positive alternating tangle. In each case, the dealternator is circled.

Let $D$ be an $A$-almost alternating diagram such that $c_-(D)=3$ and the dealternator is a positive crossing. Then $D$ is one of the two diagrams depicted in Figure <ref> where each tangle $R_i$ is an alternating tangle with all positive crossings.

Let $D_{\alt}$ be the alternating diagram obtained from $D$ by changing the dealternator. Since the dealternator is positive, $D_{\alt}$ has four negative crossings, corresponding to four negative vertices $v_1$, $v_2$, $v_3$, and $v_4$ in $\Gamma(D_{alt})$. Since each vertex in $\Gamma(D_{\alt})$ appears twice in the cycle decomposition $\mathcal{C}$ induced by the all-$A$ state of $D_{\alt}$, it follows that the cycle decomposition $\mathcal{C}$ contains eight negative vertices, with at most four being contained in a single cycle.  Lemma <ref> states that each cycle in $\mathcal{C}$ contains an even number of negative vertices, and thus there are three possibilities for the cycles containing negative crossings.

* Two cycles contain four negative vertices each.

* One cycle contains four negative vertices and two cycles contain two negative vertices each.

* Four cycles contain two negative vertices each.

If there are two cycles with four negative vertices each in $\mathcal{C}$, then $D_{\alt}$ is in the format of $D_1$ in Figure <ref>, and changing one of the negative crossings in $D_1$ results in the first diagram in Figure <ref>. If there is one cycle with four negative vertices and two cycles with two negative vertices each in $\mathcal{C}$, then $D_{\alt}$ is in the format $D_3$ in Figure <ref>. However, changing any of the four negative crossings in such a diagram results in the diagram not being $A$-almost alternating because there will be a region that shares a crossing in its border with both the regions $u_1$ and $u_2$.

Suppose that $\mathcal{C}$ has four cycles each with two negative vertices. Either there are two vertices $v_i$ and $v_j$ that are both contained in two different cycles of $\mathcal{C}$ or no such pair of vertices exist. In the former case, the $D_{\alt}$ is in the format of $D_4$ in Figure <ref>. Changing any of the four negative crossings in such a diagram results in the regions $u_1$ and $u_2$ being adjacent, and hence the resulting diagram is not $A$-almost alternating. In the latter case, $D_{\alt}$ is in the format of $D_2$ in Figure <ref>, and changing one of the negative crossings in $D_2$ results in the second diagram of Figure <ref>.

\[\begin{tikzpicture}[scale = 1]

\draw (2,-.5) node{$D_1$};

\draw (0,0) rectangle (1.5,1.5);
\draw (0,2.5) rectangle (1.5,4);
\draw (2.5,0) rectangle (4,1.5);
\draw (2.5,2.5) rectangle (4,4);

\draw (.75,3.25) node{$R_1$};
\draw (3.25,3.25) node{$R_2$};
\draw (3.25,.75) node{$R_3$};
\draw (.75,.75) node{$R_4$};

\draw (.4,1.25) node{\tiny{$-$}};
\draw (1.1,1.25) node{\tiny{$+$}};
\draw (1.25,1.1) node{\tiny{$-$}};
\draw (1.25,.4) node{\tiny{$+$}};

\draw (.4,2.75) node{\tiny{$+$}};
\draw (1.1,2.75) node{\tiny{$-$}};
\draw (1.25,2.9) node{\tiny{$+$}};
\draw (1.25,3.6) node{\tiny{$-$}};

\draw (2.9,1.25) node{\tiny{$-$}};
\draw (3.6,1.25) node{\tiny{$+$}};
\draw (2.75,1.1) node{\tiny{$+$}};
\draw (2.75,.4) node{\tiny{$-$}};

\draw (2.9,2.75) node{\tiny{$+$}};
\draw (3.6,2.75) node{\tiny{$-$}};
\draw (2.75,2.9) node{\tiny{$-$}};
\draw (2.75,3.6) node{\tiny{$+$}};

\begin{knot}[    %draft mode = crossings,
consider self intersections,
clip width = 5,
ignore endpoint intersections = true,
end tolerance = 1pt

\strand[->]
(1.1,1.5) to [out = 90, in =270 ] (.4,2.5);
\strand[->]
(1.1,2.5) to [out = 270, in =90 ] (.4,1.5);

\strand[->] (2.5,3.6) to [out = 180, in =0] (1.5,2.9);
\strand[->] (1.5,3.6) to [out = 0, in= 180] (2.5,2.9);

\strand[->] (2.9,1.5) to [out = 90, in =270] (3.6,2.5);
\strand [->] (2.9,2.5) to [out = 270, in =90] (3.6,1.5);

\strand[->] (2.5,.4) to [out = 180, in=0] (1.5,1.1);
\strand[->] (1.5,.4) to [out = 0, in = 180] (2.5,1.1);

\end{knot}

%\draw[red, dashed] (2,3.25) circle (.3cm);

\begin{scope}[xshift = 5cm]

\draw (3,-.5) node{$D_2$};

\draw (0,0) rectangle (6,1.5);
\draw (0,2.5) rectangle (6,4);

\draw (3,.75) node{$R_2$};
\draw (3,3.25) node{$R_1$};

\draw (.25,1.25) node{\tiny{$-$}};
\draw (1.25,1.25) node{\tiny{$+$}};
\draw (1.75,1.25) node{\tiny{$-$}};
\draw (2.75,1.25) node{\tiny{$+$}};
\draw (3.25,1.25) node{\tiny{$-$}};
\draw (4.25,1.25) node{\tiny{$+$}};
\draw (4.75,1.25) node{\tiny{$-$}};
\draw (5.75,1.25) node{\tiny{$+$}};

\draw (.25,2.75) node{\tiny{$+$}};
\draw (1.25,2.75) node{\tiny{$-$}};
\draw (1.75,2.75) node{\tiny{$+$}};
\draw (2.75,2.75) node{\tiny{$-$}};
\draw (3.25,2.75) node{\tiny{$+$}};
\draw (4.25,2.75) node{\tiny{$-$}};
\draw (4.75,2.75) node{\tiny{$+$}};
\draw (5.75,2.75) node{\tiny{$-$}};

\begin{knot}[    %draft mode = crossings,
consider self intersections,
clip width = 5,
ignore endpoint intersections = true,
end tolerance = 1pt

\strand[->] (.25,1.5) to [out = 90, in=270] (1.25,2.5);
\strand[->] (.25,2.5) to [out = 270,in=90] (1.25,1.5);

\strand[->] (1.75,1.5) to [out = 90, in = 270] (2.75,2.5);
\strand[->] (1.75, 2.5) to [out = 270, in =90] (2.75,1.5);

\strand[->] (3.25,1.5) to [out = 90, in = 270] (4.25,2.5);
\strand[->] (3.25, 2.5) to [out = 270, in =90] (4.25,1.5);

\strand[->] (4.75,1.5) to [out = 90, in = 270] (5.75,2.5);
\strand[->] (4.75, 2.5) to [out = 270, in =90] (5.75,1.5);

\end{knot}

%    \draw[red, dashed] (.75,2) circle (.3cm);

\end{scope}

\begin{scope}[yshift = -5cm, xshift = -1.5cm]

\draw (2.75,-.5) node{$D_3$};

\draw (0,0) rectangle (1,3);
\draw (2,0) rectangle (3.5,3);
\draw (4.5,0) rectangle (5.5,3);
\draw (.4,1.5) node{$R_1$};
\draw (2.75,1.5) node{$R_2$};
\draw (5.1,1.5) node{$R_3$};

\draw (.75,.25) node{\tiny{$+$}};
\draw (.75,1.25) node{\tiny{$-$}};
\draw (.75,1.75) node{\tiny{$+$}};
\draw (.75,2.75) node{\tiny{$-$}};

\draw (2.25,.25) node{\tiny{$-$}};
\draw (2.25,1.25) node{\tiny{$+$}};
\draw (2.25,1.75) node{\tiny{$-$}};
\draw (2.25,2.75) node{\tiny{$+$}};

\draw (3.25,.25) node{\tiny{$+$}};
\draw (3.25,1.25) node{\tiny{$-$}};
\draw (3.25,1.75) node{\tiny{$+$}};
\draw (3.25,2.75) node{\tiny{$-$}};

\draw (4.75,.25) node{\tiny{$-$}};
\draw (4.75,1.25) node{\tiny{$+$}};
\draw (4.75,1.75) node{\tiny{$-$}};
\draw (4.75,2.75) node{\tiny{$+$}};

\begin{knot}[    %draft mode = crossings,
consider self intersections,
clip width = 5,
ignore endpoint intersections = true,
end tolerance = 1pt

\strand[->] (2,.25) to [out = 180, in=0] (1,1.25);
\strand[->] (1,.25) to [out = 0,in=180] (2,1.25);

\strand[->] (2,1.75) to [out = 180, in=0] (1,2.75);
\strand[->] (1,1.75) to [out = 0,in=180] (2,2.75);

\strand[->] (3.5,2.75) to [out = 0, in = 180] (4.5,1.75);
\strand[->] (4.5,2.75) to [out =180, in = 0] (3.5,1.75);

\strand[->] (3.5,1.25) to [out = 0, in = 180] (4.5,.25);
\strand[->] (4.5,1.25) to [out =180, in = 0] (3.5,.25);

\end{knot}

\end{scope}

\begin{scope}[xshift = 5cm, yshift = -5cm]

\draw (3,-.5) node{$D_4$};

\draw (0,0) rectangle (6,4);
\draw (.5,.5) rectangle (2.5,3.5);
\draw (3.5,.5) rectangle (5.5,3.5);
\draw (1,1.5) rectangle (2,2.5);
\draw (4,1.5) rectangle (5,2.5);

\draw (1.5,2) node{$R_1$};
\draw (4.5,2) node{$R_2$};
\draw (3,2) node{$R_3$};

\draw (1.2,.3) node{\tiny{$-$}};
\draw (1.8,.3) node{\tiny{$+$}};

\draw (1.2,1.7) node{\tiny{$+$}};
\draw (1.8,1.7) node{\tiny{$-$}};

\draw (1.2,2.3) node{\tiny{$-$}};
\draw (1.8,2.3) node{\tiny{$+$}};

\draw (1.2,3.7) node{\tiny{$+$}};
\draw (1.8,3.7) node{\tiny{$-$}};

\draw (4.2,.3) node{\tiny{$-$}};
\draw (4.8,.3) node{\tiny{$+$}};

\draw (4.2,1.7) node{\tiny{$+$}};
\draw (4.8,1.7) node{\tiny{$-$}};

\draw (4.2,2.3) node{\tiny{$-$}};
\draw (4.8,2.3) node{\tiny{$+$}};

\draw (4.2,3.7) node{\tiny{$+$}};
\draw (4.8,3.7) node{\tiny{$-$}};

\begin{knot}[    %draft mode = crossings,
consider self intersections,
clip width = 5,
ignore endpoint intersections = true,
end tolerance = 1pt

\strand[->] (1.8,1.5) to [out = 270, in=90] (1.2,.5);
\strand[->] (1.8,.5) to [out = 90,in=270] (1.2,1.5);

\strand[->] (1.2,2.5) to [out = 90, in=270] (1.8,3.5);
\strand[->] (1.2,3.5) to [out = 270,in=90] (1.8,2.5);

\strand[->] (4.8,1.5) to [out = 270, in=90] (4.2,.5);
\strand[->] (4.8,.5) to [out = 90,in=270] (4.2,1.5);

\strand[->] (4.2,2.5) to [out = 90, in=270] (4.8,3.5);
\strand[->] (4.2,3.5) to [out = 270,in=90] (4.8,2.5);

\end{knot}

\end{scope}

\end{tikzpicture}\]

Reduced alternating diagrams with four negative crossings. The tangle $R_i$ is positive and alternating.

In order to prove Theorem <ref>, we need two strategies that yield upper bounds on $g_4(K)$. The first is that $g_4(K)\leq g_3(K)$ where $g_3(K)$ is the Seifert genus of $K$, i.e. the minimum genus of any Seifert surface of $K$. If $D$ is a diagram of $K$, then define $g_3(D)$ to be the genus of the surface obtained by applying Seifert's algorithm to $D$. We have

\[g_3(D) = \frac{1}{2}(1 + c(D) - f(D))\]

where $c(D)$ is the number of crossings of $D$ and $f(D)$ is the number of components in the oriented or Seifert state of $D$. It follows that if $D$ is a diagram of $K$, then

\begin{equation}
\label{eq:g4bound1}
g_4(K) \leq \frac{1}{2}(1 + c(D)-f(D)).
\end{equation}

The second strategy we use is modifying a diagram via a saddle move, that is replacing two adjacent and oppositely oriented arcs in a diagram as in Figure <ref>. Let $K_1$ be a knot with diagram $D_1$. Let $D_2$ be the diagram obtained after performing two saddle moves to $D_1$, and let $K_2$ be the link with diagram $D_2$. A saddle move changes the number of components of a link by one, and thus the link $K_2$ will either have one or three components. Figure <ref> indicates the following inequality

\begin{equation}
\label{eq:g4bound2}
g_4(K_1) \leq \begin{cases} g_4(K_2) + 1&\text{if $K_2$ is a knot,}\\
g_4(K_2) + 2 & \text{if $K_2$ is a link with three components.}
\end{cases}
\end{equation}

\[\begin{tikzpicture}

\begin{scope}[decoration={
mark=at position 0.55 with {\arrow{>}}}
\draw[postaction={decorate}] (0,0) to [out=45, in =135] (1,0);
\draw[postaction={decorate}] (1,1) to [out=225, in =-45] (0,1);

\draw[postaction={decorate}] (0,2) to [out=45, in =-45] (0,3);
\draw[postaction={decorate}] (1,3) to [out=225, in =135] (1,2);

\end{scope}

\draw[<->] (.5,1.2) -- (.5,1.8);

\draw (3.5,3) ellipse (.5cm and .1cm);
\draw (2,1.5) arc (180:360:.5cm and .1cm);
\draw[dashed] (2,1.5) arc (180:0:.5cm and .1cm);

\draw (4,1.5) arc (180:360:.5cm and .1cm);
\draw[dashed] (4,1.5) arc (180:0:.5cm and .1cm);

\draw (3,0) arc (180:360:.5cm and .1cm);
\draw[dashed] (3,0) arc (180:0:.5cm and .1cm);

\draw (3,3) to [out = 270, in = 90] (2,1.5) to [out = 270, in = 90] (3,0);
\draw (4,3) to [out = 270, in = 90] (5,1.5) to [out = 270, in = 90] (4,0);
\draw (3,1.5) to [out = 90, in = 90, looseness = 1.2] (4,1.5) to [out = 270, in = 270, looseness=1.2] (3,1.5);

\draw (3.5,3.5) node{$K_1$};
\draw (3.5,-.5) node{$K_2$};

\begin{scope}[xshift = 5cm]
\draw (3.5,3) ellipse (.5cm and .1cm);
\draw (2,1.5) arc (180:360:.5cm and .1cm);
\draw[dashed] (2,1.5) arc (180:0:.5cm and .1cm);

\draw (4,1.5) arc (180:360:.5cm and .1cm);
\draw[dashed] (4,1.5) arc (180:0:.5cm and .1cm);

\draw (3,0) arc (180:360:.5cm and .1cm);
\draw[dashed] (3,0) arc (180:0:.5cm and .1cm);

\draw (1,0) arc (180:360:.5cm and .1cm);
\draw[dashed] (1,0) arc (180:0:.5cm and .1cm);

\draw (5,0) arc (180:360:.5cm and .1cm);
\draw[dashed] (5,0) arc (180:0:.5cm and .1cm);

\draw(3,3) to [out = 270, in = 90] (2,1.5) to [out = 270, in = 90] (1,0);
\draw (4,3) to [out = 270, in = 90] (5,1.5) to [out = 270, in = 90] (6,0);
\draw (5,0) to [out = 90, in = 270] (4,1.5) to [out = 90, in = 90,looseness = 1.2] (3,1.5) to [out = 270, in = 90] (4,0);
\draw (3,0) to [out = 90, in = 90, looseness = 1.2] (2,0);

\draw (3.5,3.5) node{$K_1$};
\draw (3.5,-.5) node{$K_2$};

\end{scope}

\end{tikzpicture}\]

A saddle move and the induced resulting cobordism between $K_1$ and $K_2$ in the cases where $K_2$ is a knot or a three component link.

Suppose that $D$ is $A$-adequate, has Turaev genus one, and has two negative crossings. Since $D$ is $A$-adequate, the cycle decomposition of $\Gamma(D)$ induced by the all-$A$ states has two cycles each with two negative vertices. For positive crossings, the $A$-resolution and the oriented resolution agree. The Seifert state for $D$ and the all-$A$ state for $D$ are related by changing the two oriented or $B$-resolutions at the negative crossings. Because the vertices corresponding to those circles are in two cycles, performing both switches preserves the number of components in the state. Thus the number $f(D)$ of components in the Seifert state of $D$ is the same as the number $s_A(D)$ of components in the all-$A$ state of $D$. Equation <ref> implies

\[g_4(K) \leq g_3(D) = \frac{1}{2}(1+c(D) - f(D)) = \frac{1}{2}(1+c(D) - s_A(D)) =\frac{s(K)}{2}+1,\]

where the last equality follows from Theorem <ref>.

Suppose that $D$ is $A$-almost alternating and has three negative crossings. Furthermore, suppose the dealternator of $D$ is negative, and let $D_{\alt}$ be the diagram obtained from $D$ be changing the dealternator. Then $D_{\alt}$ is a reduced alternating, and thus adequate, diagram with two negative crossings. Hence $s_A(D_{\alt})=f(D_{\alt})$ as in the previous paragraph. Because $D$ and $D_{\alt}$ are related by a crossing change, the genera of the surfaces obtained by Seifert's algorithm are the same, that is $g_3(D) = g_3(D_{\alt})$. Also, $s_A(D_{\alt}) = s_A(D) + 1$. Thus

\begin{align*}
g_3(D) = & \; g_3(D_{\alt})\\
= & \; \frac{1}{2}(1+c(D_{\alt}) - f(D_{\alt}))\\
= & \; \frac{1}{2}(1+c(D_{\alt}) - s_A(D_{\alt}) )\\
= &\;  \frac{1}{2}(c(D) -s_A(D)).
\end{align*}

Hence Equation <ref> implies

\[g_4(K) \leq g_3(D) = \frac{1}{2}(c(D) -s_A(D)) = \frac{s(K)}{2}+1,\]

where again the last equality follows from Theorem <ref>.

Suppose that $D$ is $A$-almost alternating and has three negative crossings, but now also suppose the dealternator of $D$ is positive. Lemma <ref> states that $D$ is of the format of one of the two diagrams in Figure <ref>. Suppose that $D$ has the format of the diagram on the left of Figure <ref>. Performing a flype and a Reidemeister 2 move around the tangle $R_1$ results in a positive knot, and thus $g_4(K)=s(K)$ in this case.

Finally suppose that $D$ has the format of the diagram on the right in Figure <ref>. Unfortunately, the approach using Equation <ref> yields $g_4(K) \leq \frac{s(K)}{2} + 2$, and so in order to prove the result, we use a strategy involving Inequality <ref>. As shown in Figure <ref>, one can perform two saddle moves followed by a flype and transform $D$ into a positive diagram $D_{\pos}$. Let $L_{\pos}$ be the link whose diagram is $D_{\pos}$. By Theorem <ref>,

\[g_4(L_{\pos}) = \frac{1}{2}(2-\mu(L_{\pos}) - f(D_{\pos}) + c(D_{pos}))\]

where the number of components $\mu(L_{\pos})$ of $L_{\pos}$ is either one or three. We have $c(D_{\pos}) = c(D) -4$ and $f(D_{\pos})=s_A(D_{\pos}) = s_A(D)-1$. If $L_{\pos}$ is a knot, then Inequality <ref> implies that

\[g_4(K) \leq g_4(L_{\pos})+1 = \frac{1}{2}(3 + c(D_{\pos}) - f(D_{\pos})) = \frac{1}{2}(c(D) - s_A(D)) = \frac{s(K)}{2}+1,\]

where the last equality follows from Theorem <ref>. If $L_{\pos}$ is a link with three components, then Inequality <ref> implies that

\[g_4(K) \leq g_4(L_{\pos})+2 = \frac{1}{2}(3+c(D_{\pos}) - f(D_{\pos}))=\frac{s(K)}{2}+1,\]

as above.

Therefore $\frac{s(K)}{2} \leq g_4(K) \leq \frac{s(K)}{2}+1$ when $K$ is in case $1$ or $2$ of the theorem. The result is proved analogously when $K$ is in case $3$ or $4$.

\[\begin{tikzpicture}

\draw (0,0) rectangle (4,1);
\draw (0,2) rectangle (4,3);

\draw (.2,.8) node{\tiny{$-$}};
\draw (.8,.8) node{\tiny{$+$}};
\draw (1.2,.8) node{\tiny{$-$}};
\draw (1.8,.8) node{\tiny{$+$}};
\draw (2.2,.8) node{\tiny{$-$}};
\draw (2.8,.8) node{\tiny{$+$}};
\draw (3.2,.8) node{\tiny{$-$}};
\draw (3.8,.8) node{\tiny{$+$}};

\draw (2,.4) node{$R_2$};

\draw (.2,2.2) node{\tiny{$+$}};
\draw (.8,2.2) node{\tiny{$-$}};
\draw (1.2,2.2) node{\tiny{$+$}};
\draw (1.8,2.2) node{\tiny{$-$}};
\draw (2.2,2.2) node{\tiny{$+$}};
\draw (2.8,2.2) node{\tiny{$-$}};
\draw (3.2,2.2) node{\tiny{$+$}};
\draw (3.8,2.2) node{\tiny{$-$}};

\draw (2,2.6) node{$R_1$};

\begin{knot}[    %draft mode = crossings,
consider self intersections,
clip width = 5,
ignore endpoint intersections = true,
end tolerance = 1pt

\strand[->] (.2,2) to [out = 270, in = 90] (.8,1);
\strand[->] (.2,1) to [out = 90, in = 270] (.8,2);

\strand[->] (1.2,1) to [out = 90, in = 270] (1.8,2);
\strand[->] (1.2,2) to [out = 270, in = 90] (1.8,1);

\strand[->] (2.2,1) to [out = 90, in = 270] (2.8,2);
\strand[->] (2.2,2) to [out = 270, in = 90] (2.8,1);

\strand[->] (3.2,1) to [out = 90, in = 270] (3.8,2);
\strand[->] (3.2,2) to [out = 270, in = 90] (3.8,1);

\end{knot}

\draw (2,-.5) node{$D$};

\begin{scope}[xshift = 6cm]

\draw (.6,0) rectangle (3.4,1);
\draw (.6,2) rectangle (3.4,3);

\draw (.8,.2) node{\tiny{$-$}};
\draw (.8,.8) node{\tiny{$+$}};
\draw (1.2,.8) node{\tiny{$-$}};
\draw (1.8,.8) node{\tiny{$+$}};
\draw (2.2,.8) node{\tiny{$-$}};
\draw (2.8,.8) node{\tiny{$+$}};
\draw (3.2,.8) node{\tiny{$-$}};
\draw (3.2,.2) node{\tiny{$+$}};

\draw (.8,2.2) node{\tiny{$-$}};
\draw (.8,2.8) node{\tiny{$+$}};
\draw (1.2,2.2) node{\tiny{$+$}};
\draw (1.8,2.2) node{\tiny{$-$}};
\draw (2.2,2.2) node{\tiny{$+$}};
\draw (2.8,2.2) node{\tiny{$-$}};
\draw (3.2,2.8) node{\tiny{$-$}};
\draw (3.2,2.2) node{\tiny{$+$}};

\draw (2,2.6) node{$R_1$};
\draw (2,.4) node{$R_2$};

\begin{knot}[    %draft mode = crossings,
%consider self intersections,
clip width = 4,
ignore endpoint intersections = false,
end tolerance = 1pt

\strand[->] (.6,2.8) to [out = 180, in= 180,looseness=1.4] (.6,.8);
\strand[->] (.6,.2) to [out = 180, in = 180,looseness=1.4] (.6,2.2);

\strand[->] (1.2,1) to [out = 90, in = 270] (1.8,2);
\strand[->] (1.2,2) to [out = 270, in = 90] (1.8,1);

\strand[->] (2.2,1) to [out = 90, in = 270] (2.8,2);
\strand[->] (2.2,2) to [out = 270, in = 90] (2.8,1);

\strand[->] (3.4,.8) to [out=0, in=0, looseness = 1.4 ] (3.4,2.8);
\strand[->] (3.4,2.2) to [out=0, in=0, looseness = 1.4] (3.4,.2);

\end{knot}

\draw (2,-.5) node{$D$};

\end{scope}

\begin{scope}[yshift = -4cm]

\draw (.6,0) rectangle (3.4,1);
\draw (.6,2) rectangle (3.4,3);

\draw (.8,.2) node{\tiny{$-$}};
\draw (.8,.8) node{\tiny{$+$}};
\draw (1.2,.8) node{\tiny{$-$}};
\draw (1.8,.8) node{\tiny{$+$}};
\draw (2.2,.8) node{\tiny{$-$}};
\draw (2.8,.8) node{\tiny{$+$}};
\draw (3.2,.8) node{\tiny{$-$}};
\draw (3.2,.2) node{\tiny{$+$}};

\draw (.8,2.2) node{\tiny{$-$}};
\draw (.8,2.8) node{\tiny{$+$}};
\draw (1.2,2.2) node{\tiny{$+$}};
\draw (1.8,2.2) node{\tiny{$-$}};
\draw (2.2,2.2) node{\tiny{$+$}};
\draw (2.8,2.2) node{\tiny{$-$}};
\draw (3.2,2.8) node{\tiny{$-$}};
\draw (3.2,2.2) node{\tiny{$+$}};

\draw (2,2.6) node{$R_1$};
\draw (2,.4) node{$R_2$};

\begin{knot}[    %draft mode = crossings,
%consider self intersections,
clip width = 4,
ignore endpoint intersections = false,
end tolerance = 1pt

\strand[->] (.6,2.8) to [out = 180, in= 180,looseness=1.4] (.6,.8);
\strand[->] (.6,.2) to [out = 180, in = 180,looseness=1.4] (.6,2.2);

\strand[->] (3.4,.8) to [out=0, in=0, looseness = 1.4 ] (3.4,2.8);
\strand[->] (3.4,2.2) to [out=0, in=0, looseness = 1.4] (3.4,.2);

\end{knot}

\draw[->] (1.2,1) to [out = 90, in = 90] (1.8,1);
\draw[->] (1.2,2) to [out = 270, in = 270] (1.8,2);
\draw[->] (2.2,1) to [out = 90, in = 90] (2.8,1);
\draw[->] (2.2,2) to [out = 270, in = 270] (2.8,2);

\draw (2,-.5) node{$\widetilde{D}$};

\end{scope}

\begin{scope}[yshift = -4cm, xshift = 6cm]

\draw (.6,0) rectangle (3.4,1);
\draw (.6,2) rectangle (3.4,3);

\draw (.8,.2) node{\tiny{$-$}};
\draw (.8,.8) node{\tiny{$+$}};
\draw (1.2,.8) node{\tiny{$-$}};
\draw (1.8,.8) node{\tiny{$+$}};
\draw (2.2,.8) node{\tiny{$-$}};
\draw (2.8,.8) node{\tiny{$+$}};
\draw (3.2,.8) node{\tiny{$-$}};
\draw (3.2,.2) node{\tiny{$+$}};

\draw (.8,2.2) node{\tiny{$-$}};
\draw (.8,2.8) node{\tiny{$+$}};
\draw (1.2,2.8) node{\tiny{$-$}};
\draw (1.8,2.8) node{\tiny{$+$}};
\draw (2.2,2.8) node{\tiny{$-$}};
\draw (2.8,2.8) node{\tiny{$+$}};
\draw (3.2,2.8) node{\tiny{$-$}};
\draw (3.2,2.2) node{\tiny{$+$}};

\draw (2,2.4) node[yscale=-1]{$R_1$};
\draw (2,.4) node{$R_2$};

\draw[->] (1.2,1) to [out = 90, in = 90] (1.8,1);
\draw[->] (1.2,3) to [out = 90, in = 90] (1.8,3);
\draw[->] (2.2,1) to [out = 90, in = 90] (2.8,1);
\draw[->] (2.2,3) to [out = 90, in = 90] (2.8,3);

\draw[->] (.6,2.2) to [out =180, in = 180] (.6,.8);
\draw[->] (.6,.2) to [out = 180, in = 180] (.6,2.8);
\draw[->] (3.4,.8) to [out= 0, in=0] (3.4,2.2);
\draw[->] (3.4,2.8) to [out = 0, in = 0](3.4,.2);

\draw (2,-.5) node{$D_{\text{pos}}$};

\end{scope}

\end{tikzpicture}\]

Performing two saddle moves transforms $D$ into a positive diagram $D_{\pos}$.

Theorem <ref> gives information about the first three nontrivial homological gradings of Khovanov homology. One might expect to repeat the arguments of this section when $D$ is $A$-Turaev genus one and the $i=0$ homological grading occurs in the second nontrivial homological grading. In the case where $D$ is $A$-adequate and Turaev genus one, this happens when the knot has one negative crossing. Tagami [32] proved that $2g_4(K)=s(K)$ for such knots. In the case where $D$ is $A$-almost alternating, the $i=0$ homological grading occurs in the second homological grading when $D$ has two negative crossings. Alas, this does not yield new interesting examples because of the following theorem.

There are no $A$-almost alternating knots with exactly two negative crossings.

Let $D$ be an $A$-almost alternating diagram with two negative crossings, and let $D_{\alt}$ be the reduced alternating diagram obtained by changing the dealternator in $D$. Then $D_{\alt}$ either has one or three negative crossings. Since $D_{\alt}$ is a reduced alternating diagram, it is $A$-adequate. Lemma <ref> states that each cycle in the cycle decomposition of $\Gamma(D)$ has an even number of negative vertices, which is impossible if $D$ only has one negative crossing. If $D$ has three negative crossings, then there are three cycles in the cycle decomposition of $\Gamma(D)$, each of which contains two negative vertices. In this case, there will be a region in the alternating tangle of $D$ adjacent to both $u_1$ and $u_2$, and thus $D$ is not $A$-almost alternating. Thus there is no diagram $A$-almost alternating diagram $D$ with two negative crossings.

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