diff -r 4b64f9c6313f -r 76c301fdf0a2 text/blobdef.tex --- a/text/blobdef.tex Wed Jun 02 08:43:12 2010 -0700 +++ b/text/blobdef.tex Wed Jun 02 11:45:19 2010 -0700 @@ -25,23 +25,21 @@ We of course define $\bc_0(X) = \lf(X)$. (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. We'll omit this sort of detail in the rest of this section.) -In other words, $\bc_0(X)$ is just the vector space of all (linearized) fields on $X$. +In other words, $\bc_0(X)$ is just the vector space of fields on $X$. -$\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. -Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear -combinations of 1-blob diagrams, where a 1-blob diagram consists of +We want the vector space $\bc_1(X)$ to capture `the space of all local relations that can be imposed on $\bc_0(X)$'. +Thus we say a $1$-blob diagram consists of \begin{itemize} \item An embedded closed ball (``blob") $B \sub X$. -\item A field $r \in \cC(X \setmin B; c)$ -(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). -\item A local relation field $u \in U(B; c)$ -(same $c$ as previous bullet). +\item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$. +\item A field $r \in \cC(X \setmin B; c)$. +\item A local relation field $u \in U(B; c)$. \end{itemize} (See Figure \ref{blob1diagram}.) \begin{figure}[t]\begin{equation*} \mathfig{.6}{definition/single-blob} \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} -In order to get the linear structure correct, we (officially) define +In order to get the linear structure correct, the actual definition is \[ \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . \] @@ -61,26 +59,24 @@ Note that the skein space $A(X)$ 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}. -$\bc_2(X)$ is, roughly, the space of all relations (redundancies, syzygies) among the -local relations encoded in $\bc_1(X)$. -More specifically, $\bc_2(X)$ is the space of all finite linear combinations of -2-blob diagrams, of which there are two types, disjoint and nested. - +Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations (redundancies, syzygies) among the +local relations encoded in $\bc_1(X)$'. +More specifically, a $2$-blob diagram, comes in one of two types, disjoint and nested. A disjoint 2-blob diagram consists of \begin{itemize} -\item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. -\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ +\item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors. +\item A field $r \in \cC(X \setmin (B_1 \cup B_2); c_1, c_2)$ (where $c_i \in \cC(\bd B_i)$). -\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.} +\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. \end{itemize} (See Figure \ref{blob2ddiagram}.) \begin{figure}[t]\begin{equation*} \mathfig{.6}{definition/disjoint-blobs} \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} -We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; +We also identify $(B_1, B_2, u_1, u_2, r)$ with $-(B_2, B_1, u_2, u_1, r)$; reversing the order of the blobs changes the sign. -Define $\bd(B_0, B_1, u_0, u_1, r) = -(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. +Define $\bd(B_1, B_2, u_1, u_2, r) = +(B_2, u_2, u_1\bullet r) - (B_1, u_1, u_2\bullet r) \in \bc_1(X)$. In other words, the boundary of a disjoint 2-blob diagram is the sum (with alternating signs) of the two ways of erasing one of the blobs. @@ -88,48 +84,42 @@ A nested 2-blob diagram consists of \begin{itemize} -\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. -\item A field $r \in \cC(X \setmin B_0; c_0)$ -(for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$. -\item A local relation field $u_0 \in U(B_0; c_0)$. +\item A pair of nested balls (blobs) $B_1 \sub B_2 \sub X$. +\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 $c_2 \in \cC(\bdy B_2)$). +\item A field $r \in \cC(X \setminus B_2; c_2)$. +\item A local relation field $u \in U(B_1; c_1)$. \end{itemize} (See Figure \ref{blob2ndiagram}.) \begin{figure}[t]\begin{equation*} \mathfig{.6}{definition/nested-blobs} \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} -Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ -(for some $c_1 \in \cC(B_1)$) and -$r' \in \cC(X \setmin B_1; c_1)$. -Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. +Define $\bd(B_1, B_2, u, r', r) = (B_2, u\bullet r', r) - (B_1, u, r' \bullet r)$. Note that the requirement that -local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. +local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$. As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating sum of the two ways of erasing one of the blobs. -If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. +When we erase the inner blob, the outer blob inherits the label $u\bullet r'$. It is again easy to check that $\bd^2 = 0$. -As with the 1-blob diagrams, in order to get the linear structure correct it is better to define -(officially) +As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is \begin{eqnarray*} \bc_2(X) & \deq & \left( - \bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1} - U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1) + \bigoplus_{B_1, B_2 \text{disjoint}} \bigoplus_{c_1, c_2} + U(B_1; c_1) \otimes U(B_2; c_2) \otimes \lf(X\setmin (B_1\cup B_2); c_1, c_2) \right) \\ && \bigoplus \left( - \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} - U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) + \bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} + U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2) \right) . \end{eqnarray*} -The final $\lf(X\setmin B_0; c_0)$ above really means fields splittable along $\bd B_1$, -but we didn't feel like introducing a notation for that. For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign -(rather than a new, linearly independent 2-blob diagram). +(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} Now for the general case. A $k$-blob diagram consists of \begin{itemize} -\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. +\item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or $B_i \sub B_j$ or $B_j \sub B_i$. (The case $B_i = B_j$ is allowed. @@ -141,7 +131,7 @@ \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. (These are implied by the data in the next bullets, so we usually suppress them from the notation.) -$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ +The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ if the latter space is not empty. \item A field $r \in \cC(X \setmin B^t; c^t)$, where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ @@ -168,8 +158,7 @@ \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . \] Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. -$\overline{c}$ runs over all boundary conditions, again as described above. -$j$ 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 $\overline{B}$. +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 $\overline{B}$. The boundary map \[ @@ -186,9 +175,9 @@ where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. Finally, define \eq{ - \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). + \bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b). } -The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. +The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. Thus we have a chain complex. We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, @@ -205,7 +194,7 @@ \begin{itemize} \item $p(\emptyset) = pt$, where $\emptyset$ denotes a 0-blob diagram or empty tree; \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 -\item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which encloses all the others. +\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). \end{itemize} For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while a diagram of $k$ disjoint blobs corresponds to a $k$-cube.