equal
deleted
inserted
replaced
65 where $u\bullet r$ denotes the field on $X$ obtained by gluing $u$ to $r$. |
65 where $u\bullet r$ denotes the field on $X$ obtained by gluing $u$ to $r$. |
66 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
66 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
67 just erasing the blob from the picture |
67 just erasing the blob from the picture |
68 (but keeping the blob label $u$). |
68 (but keeping the blob label $u$). |
69 |
69 |
70 \nn{it seems rather strange to make this a theorem} |
70 \nn{it seems rather strange to make this a theorem} \nn{it's a theorem because it's stated in the introduction, and I wanted everything there to have numbers that pointed into the paper --S} |
71 Note that directly from the definition we have |
71 Note that directly from the definition we have |
72 \begin{thm} |
72 \begin{thm} |
73 \label{thm:skein-modules} |
73 \label{thm:skein-modules} |
74 The skein module $A(X)$ is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
74 The skein module $A(X)$ is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
75 \end{thm} |
75 \end{thm} |
149 \item If $X'$ is obtained from $X$ by gluing, then any permissible configuration of blobs |
149 \item If $X'$ is obtained from $X$ by gluing, then any permissible configuration of blobs |
150 on $X$ gives rise to a permissible configuration on $X'$. |
150 on $X$ gives rise to a permissible configuration on $X'$. |
151 (This is necessary for Proposition \ref{blob-gluing}.) |
151 (This is necessary for Proposition \ref{blob-gluing}.) |
152 \end{itemize} |
152 \end{itemize} |
153 Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not |
153 Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not |
154 a manifold. \todo{example} |
154 a manifold. |
155 Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs. |
155 Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs. |
156 |
156 |
157 \begin{example} |
157 \begin{example} |
158 Consider the four subsets of $\Real^3$, |
158 Consider the four subsets of $\Real^3$, |
159 \begin{align*} |
159 \begin{align*} |
238 \item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which |
238 \item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which |
239 encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). |
239 encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). |
240 \end{itemize} |
240 \end{itemize} |
241 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
241 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
242 a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
242 a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
243 (This correspondence works best if we think of each twig label $u_i$ as having the form |
243 (When the fields come from an $n$-category, this correspondence works best if we think of each twig label $u_i$ as having the form |
244 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cF(B_i) \to C$ is the evaluation map, |
244 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cF(B_i) \to C$ is the evaluation map, |
245 and $s:C \to \cF(B_i)$ is some fixed section of $e$. \todo{This parenthetical remark mysteriously specialises to the category case}) |
245 and $s:C \to \cF(B_i)$ is some fixed section of $e$.) |
246 |
246 |
247 |
247 |