117 \nn{need to also require that $KLM$ antirefines to $KM$, etc.} |
117 \nn{need to also require that $KLM$ antirefines to $KM$, etc.} |
118 Then we have a filtration degree 2 chain, as shown in Figure \ref{zzz5}, which does the trick. |
118 Then we have a filtration degree 2 chain, as shown in Figure \ref{zzz5}, which does the trick. |
119 (Each small triangle in Figure \ref{zzz5} can be filled with a filtration degree 2 chain.) |
119 (Each small triangle in Figure \ref{zzz5} can be filled with a filtration degree 2 chain.) |
120 |
120 |
121 \begin{figure}[!ht] |
121 \begin{figure}[!ht] |
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122 %\begin{equation*} |
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123 %\mathfig{1.0}{tempkw/zz5} |
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124 %\end{equation*} |
122 \begin{equation*} |
125 \begin{equation*} |
123 \mathfig{1.0}{tempkw/zz5} |
126 \begin{tikzpicture} |
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127 \node(M) at (0,0) {$M$}; |
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128 \foreach \angle/\label in {0/K', 45/K'L, 90/L, 135/KL, 180/K, 225/KL', 270/L', 315/K'L'} { |
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129 \node(\label) at (\angle:4) {$\label$}; |
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130 } |
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131 \foreach \label in {K', L, K, L'} { |
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132 \node(\label M) at ($(M)!0.6!(\label)$) {$\label M$}; |
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133 \draw[->] (\label M)--(M); |
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134 \draw[->] (\label M)--(\label); |
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135 } |
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136 \foreach \k in {K, K'} { |
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137 \foreach \l in {L, L'} { |
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138 \node(\k \l M) at (intersection cs: first line={(\k M)--(\l)}, second line={(\l M)--(\k)}) {$\k \l M$}; |
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139 \draw[->] (\k \l M)--(M); |
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140 \draw[->] (\k \l M)--(\k \l ); |
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141 \draw[->] (\k \l M)--(\k M); |
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142 \draw[->] (\k \l M)--(\l); |
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143 \draw[->] (\k \l M)--(\l M); |
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144 \draw[->] (\k \l M)--(\k); |
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145 } |
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146 } |
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147 \draw[->] (K'L') to[bend right=10] (K'); |
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148 \draw[->] (K'L') to[bend left=10] (L'); |
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149 \draw[->] (KL') to[bend left=10] (K); |
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150 \draw[->] (KL') to[bend right=10] (L'); |
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151 \draw[->] (K'L) to[bend left=10] (K'); |
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152 \draw[->] (K'L) to[bend right=10] (L); |
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153 \draw[->] (KL) to[bend right=10] (K); |
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154 \draw[->] (KL) to[bend left=10] (L); |
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155 \end{tikzpicture} |
124 \end{equation*} |
156 \end{equation*} |
125 \caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} |
157 \caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} |
126 \label{zzz5} |
158 \label{zzz5} |
127 \end{figure} |
159 \end{figure} |
128 |
160 |