--- 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.