blob: comparison text/blobdef.tex

equal deleted inserted replaced

-:4b64f9c6313f
+:76c301fdf0a2
 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$.
 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 $\bc_0(X)$.
+We want the vector space $\bc_1(X)$ to capture `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
+Thus we say  a $1$-blob diagram consists of
-combinations of 1-blob diagrams, where a 1-blob diagram consists of
 \begin{itemize}
 \item An embedded closed ball (``blob") $B \sub X$.
-\item A field $r \in \cC(X \setmin B; c)$
+\item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$.
-(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$).
+\item A field $r \in \cC(X \setmin B; c)$.
-\item A local relation field $u \in U(B; c)$
+\item A local relation field $u \in U(B; c)$.
-(same $c$ as previous bullet).
 \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
 \[
 	\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) .
 \]
 The first direct sum is indexed by all blobs $B\subset X$, and the second
 by all boundary conditions $c \in \cC(\bd B)$.
 (but keeping the blob label $u$).
 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
+Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations (redundancies, syzygies) among the
-local relations encoded in $\bc_1(X)$.
+local relations encoded in $\bc_1(X)$'.
-More specifically, $\bc_2(X)$ is the space of all finite linear combinations of
+More specifically, a $2$-blob diagram, comes in one of two types, disjoint and nested.
-2-blob diagrams, of which there are 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 pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors.
-\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$
+\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) =
+Define $\bd(B_1, B_2, u_1, u_2, r) =
-(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$.
+(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.
 It's easy to check that $\bd^2 = 0$.
 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 pair of nested balls (blobs) $B_1 \sub B_2 \sub X$.
-\item A field $r \in \cC(X \setmin B_0; c_0)$
+\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 $c_2 \in \cC(\bdy B_2)$).
-(for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$.
+\item A field $r \in \cC(X \setminus B_2; c_2)$.
-\item A local relation field $u_0 \in U(B_0; c_0)$.
+\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 $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$
+Define $\bd(B_1, B_2, u, r', r) = (B_2, u\bullet r', r) - (B_1, u, r' \bullet r)$.
-(for some $c_1 \in \cC(B_1)$) and
-$r' \in \cC(X \setmin B_1; c_1)$.
-Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$.
 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
+As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is
-(officially)
 \begin{eqnarray*}
 	\bc_2(X) & \deq &
 	\left(
-		\bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1}
+		\bigoplus_{B_1, B_2 \text{disjoint}} \bigoplus_{c_1, c_2}
-			U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1)
+			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}
+		\bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2}
-			U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0)
+			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.
 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.)
 If a blob has no other blobs strictly contained in it, we call it a twig blob.
 on $\bd B$.  this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$
 and blobs are allowed to meet $\bd X$.}
 \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 $c^t \in \cC(\bd B^t)$
 is determined by the $c_i$'s.
 $r$ is required to be splittable along the boundaries of all blobs, twigs or not.
 \[
 	\bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}}
 		\left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) .
 \]
 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above.
-$\overline{c}$ runs over all boundary conditions, again as described above.
+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 $\overline{B}$.
-$j$ 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
 \[
 	\bd : \bc_k(X) \to \bc_{k-1}(X)
 \]
 if removing $B_j$ creates new twig blobs.
 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$,
 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 $b$, $\supp(b) \sub X$,
 to be the union of the blobs of $b$.
 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram),
 We can associate an element $p(b)$ of $P$ to each blob diagram $b$
 (equivalently, to each rooted tree) according to the following rules:
 \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.
 (This correspondence works best if we thing of each twig label $u_i$ as having the form
 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map,

changeset 321	76c301fdf0a2
parent 313	ef8fac44a8aa
child 332	160ca7078ae9