37 \item A local relation field $u \in U(B; c)$ |
37 \item A local relation field $u \in U(B; c)$ |
38 (same $c$ as previous bullet). |
38 (same $c$ as previous bullet). |
39 \end{itemize} |
39 \end{itemize} |
40 (See Figure \ref{blob1diagram}.) |
40 (See Figure \ref{blob1diagram}.) |
41 \begin{figure}[t]\begin{equation*} |
41 \begin{figure}[t]\begin{equation*} |
42 \mathfig{.9}{definition/single-blob} |
42 \mathfig{.6}{definition/single-blob} |
43 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
43 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
44 In order to get the linear structure correct, we (officially) define |
44 In order to get the linear structure correct, we (officially) define |
45 \[ |
45 \[ |
46 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
46 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
47 \] |
47 \] |
73 (where $c_i \in \cC(\bd B_i)$). |
73 (where $c_i \in \cC(\bd B_i)$). |
74 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. \nn{We're inconsistent with the indexes -- are they 0,1 or 1,2? I'd prefer 1,2.} |
74 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. \nn{We're inconsistent with the indexes -- are they 0,1 or 1,2? I'd prefer 1,2.} |
75 \end{itemize} |
75 \end{itemize} |
76 (See Figure \ref{blob2ddiagram}.) |
76 (See Figure \ref{blob2ddiagram}.) |
77 \begin{figure}[t]\begin{equation*} |
77 \begin{figure}[t]\begin{equation*} |
78 \mathfig{.9}{definition/disjoint-blobs} |
78 \mathfig{.6}{definition/disjoint-blobs} |
79 \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} |
79 \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} |
80 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
80 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
81 reversing the order of the blobs changes the sign. |
81 reversing the order of the blobs changes the sign. |
82 Define $\bd(B_0, B_1, u_0, u_1, r) = |
82 Define $\bd(B_0, B_1, u_0, u_1, r) = |
83 (B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
83 (B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
93 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$. |
93 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$. |
94 \item A local relation field $u_0 \in U(B_0; c_0)$. |
94 \item A local relation field $u_0 \in U(B_0; c_0)$. |
95 \end{itemize} |
95 \end{itemize} |
96 (See Figure \ref{blob2ndiagram}.) |
96 (See Figure \ref{blob2ndiagram}.) |
97 \begin{figure}[t]\begin{equation*} |
97 \begin{figure}[t]\begin{equation*} |
98 \mathfig{.9}{definition/nested-blobs} |
98 \mathfig{.6}{definition/nested-blobs} |
99 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
99 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
100 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
100 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
101 (for some $c_1 \in \cC(B_1)$) and |
101 (for some $c_1 \in \cC(B_1)$) and |
102 $r' \in \cC(X \setmin B_1; c_1)$. |
102 $r' \in \cC(X \setmin B_1; c_1)$. |
103 Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. |
103 Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. |
151 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
151 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
152 If $B_i = B_j$ then $u_i = u_j$. |
152 If $B_i = B_j$ then $u_i = u_j$. |
153 \end{itemize} |
153 \end{itemize} |
154 (See Figure \ref{blobkdiagram}.) |
154 (See Figure \ref{blobkdiagram}.) |
155 \begin{figure}[t]\begin{equation*} |
155 \begin{figure}[t]\begin{equation*} |
156 \mathfig{.9}{definition/k-blobs} |
156 \mathfig{.7}{definition/k-blobs} |
157 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
157 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
158 |
158 |
159 If two blob diagrams $D_1$ and $D_2$ |
159 If two blob diagrams $D_1$ and $D_2$ |
160 differ only by a reordering of the blobs, then we identify |
160 differ only by a reordering of the blobs, then we identify |
161 $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
161 $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |