55 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
55 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
56 just erasing the blob from the picture |
56 just erasing the blob from the picture |
57 (but keeping the blob label $u$). |
57 (but keeping the blob label $u$). |
58 |
58 |
59 Note that the skein space $A(X)$ |
59 Note that the skein space $A(X)$ |
60 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. This is Property \ref{property:skein-modules}, and also used in the second half of Property \ref{property:contractibility}. |
60 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
61 |
61 This is Property \ref{property:skein-modules}, and also used in the second |
62 Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations (redundancies, syzygies) among the |
62 half of Property \ref{property:contractibility}. |
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63 |
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64 Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations |
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65 (redundancies, syzygies) among the |
63 local relations encoded in $\bc_1(X)$'. |
66 local relations encoded in $\bc_1(X)$'. |
64 More specifically, a $2$-blob diagram, comes in one of two types, disjoint and nested. |
67 More specifically, a $2$-blob diagram, comes in one of two types, disjoint and nested. |
65 A disjoint 2-blob diagram consists of |
68 A disjoint 2-blob diagram consists of |
66 \begin{itemize} |
69 \begin{itemize} |
67 \item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors. |
70 \item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors. |
83 It's easy to check that $\bd^2 = 0$. |
86 It's easy to check that $\bd^2 = 0$. |
84 |
87 |
85 A nested 2-blob diagram consists of |
88 A nested 2-blob diagram consists of |
86 \begin{itemize} |
89 \begin{itemize} |
87 \item A pair of nested balls (blobs) $B_1 \sub B_2 \sub X$. |
90 \item A pair of nested balls (blobs) $B_1 \sub B_2 \sub X$. |
88 \item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$ (for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$). |
91 \item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$ |
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92 (for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$). |
89 \item A field $r \in \cC(X \setminus B_2; c_2)$. |
93 \item A field $r \in \cC(X \setminus B_2; c_2)$. |
90 \item A local relation field $u \in U(B_1; c_1)$. |
94 \item A local relation field $u \in U(B_1; c_1)$. |
91 \end{itemize} |
95 \end{itemize} |
92 (See Figure \ref{blob2ndiagram}.) |
96 (See Figure \ref{blob2ndiagram}.) |
93 \begin{figure}[t]\begin{equation*} |
97 \begin{figure}[t]\begin{equation*} |
112 \bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} |
116 \bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} |
113 U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2) |
117 U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2) |
114 \right) . |
118 \right) . |
115 \end{eqnarray*} |
119 \end{eqnarray*} |
116 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
120 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
117 (rather than a new, linearly independent 2-blob diagram). \nn{Hmm, I think we should be doing this for nested blobs too -- we shouldn't force the linear indexing of the blobs to have anything to do with the partial ordering by inclusion -- this is what happens below} |
121 (rather than a new, linearly independent 2-blob diagram). |
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122 \nn{Hmm, I think we should be doing this for nested blobs too -- |
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123 we shouldn't force the linear indexing of the blobs to have anything to do with |
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124 the partial ordering by inclusion -- this is what happens below} |
118 |
125 |
119 Now for the general case. |
126 Now for the general case. |
120 A $k$-blob diagram consists of |
127 A $k$-blob diagram consists of |
121 \begin{itemize} |
128 \begin{itemize} |
122 \item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. |
129 \item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. |
156 \[ |
163 \[ |
157 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
164 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
158 \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
165 \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
159 \] |
166 \] |
160 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
167 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
161 The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. |
168 The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs. |
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169 The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. |
162 |
170 |
163 The boundary map |
171 The boundary map |
164 \[ |
172 \[ |
165 \bd : \bc_k(X) \to \bc_{k-1}(X) |
173 \bd : \bc_k(X) \to \bc_{k-1}(X) |
166 \] |
174 \] |
178 \bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b). |
186 \bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b). |
179 } |
187 } |
180 The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. |
188 The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. |
181 Thus we have a chain complex. |
189 Thus we have a chain complex. |
182 |
190 |
183 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition. A homeomorphism acts in an obvious on blobs and on fields. |
191 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition. |
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192 A homeomorphism acts in an obvious on blobs and on fields. |
184 |
193 |
185 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
194 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
186 to be the union of the blobs of $b$. |
195 to be the union of the blobs of $b$. |
187 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
196 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
188 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
197 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
193 and cones, and which contains the point. |
202 and cones, and which contains the point. |
194 We can associate an element $p(b)$ of $P$ to each blob diagram $b$ |
203 We can associate an element $p(b)$ of $P$ to each blob diagram $b$ |
195 (equivalently, to each rooted tree) according to the following rules: |
204 (equivalently, to each rooted tree) according to the following rules: |
196 \begin{itemize} |
205 \begin{itemize} |
197 \item $p(\emptyset) = pt$, where $\emptyset$ denotes a 0-blob diagram or empty tree; |
206 \item $p(\emptyset) = pt$, where $\emptyset$ denotes a 0-blob diagram or empty tree; |
198 \item $p(a \du b) = p(a) \times p(b)$, where $a \du b$ denotes the distant (non-overlapping) union of two blob diagrams (equivalently, join two trees at the roots); and |
207 \item $p(a \du b) = p(a) \times p(b)$, where $a \du b$ denotes the distant (non-overlapping) union |
199 \item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). |
208 of two blob diagrams (equivalently, join two trees at the roots); and |
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209 \item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which |
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210 encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). |
200 \end{itemize} |
211 \end{itemize} |
201 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
212 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
202 a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
213 a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
203 (This correspondence works best if we thing of each twig label $u_i$ as having the form |
214 (This correspondence works best if we thing of each twig label $u_i$ as having the form |
204 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map, |
215 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map, |