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\newcommand{\Kh}{{\text{\it Kh}}}

\def\navigator{{Dror Bar-Natan: Pensieve: 2013-04: AltTan @ \today, \ampmtime}}

\def\talkurl{{\url{http://drorbn.net/AcademicPensieve/2013-04/AltTan/}}}

\def\Theorems{{\raisebox{0mm}{\parbox[t]{4in}{
{\bf Theorem 1.} If $T$ is non-split alternating, $\Kh(T)$ is coherently
diagonal.

{\bf Theorem 2.} If $\{\Omega_i\}$ are coherently diagonal and $D$ is
alternating planar, then $D(\Omega_1,\Omega_2,\dots)$ is coherently
diagonal.
}}}}

\def\RotationNumber{{\raisebox{0mm}{\parbox[t]{4in}{
{\bf Rotation Numbers.}
$R(\alpha):=\frac{1}{2k}\left[t_\alpha-h_\alpha\right]_{2k}-\frac12$, where
$\left[j\right]_{2k}:=\begin{cases}j &\text{if }j>0 \\ j+2k &\text{if }j<0
\end{cases}$, and $R(\circlearrowleft)=+1$ and $R(\circlearrowright)=-1$.
Also, $R(\alpha\{q\}):= R(\alpha)+q$.
Examples:
}}}}

\def\DiagonalComplex{{\raisebox{-1.5mm}{\parbox[t]{4in}{
{\bf Diagonal Complexes.} $\Omega\colon \cdots \to
\left[\sigma^r_j\right]_j \to \left[\sigma^{r+1}_j\right]_j \to \cdots$
such that $2r-R(\sigma^r_j)$ is a constant $C(\Omega)$.

{\bf Coherently Diagonal Complexes.} All partial closures can be {\em
reduced} to diagonal, with $C(U(\Omega))=C(\Omega)-C(D_U)$.
}}}}

\def\AltPlanAlg{{\raisebox{0mm}{\parbox[t]{4in}{
\parshape 6 0in 3in 0in 3in 0in 3in 0in 3in 0in 3in 0in 4in
{\bf Alternating Planar Algebra.} All input/output boundaries are
connected via the arcs, ``in'' and ``out'' strands alternate on all
boundaries. A ``rotation number'' $R_D$ can be defined.

{\bf Proposition 3.2.} $\displaystyle
  R(D(\sigma_1,\ldots,\sigma_d))=R_D+\sum_{i=1}^d R(\sigma_i)
$.
}}}}

\def\MainLemma{{\raisebox{0mm}{\parbox[t]{4in}{
{\bf ``Main'' Lemma 6.2.} The pairing $D(\Omega_1, \Omega_2)$ via an arc
diagram that has at least one boundary arc coming from its first input
of a coherently diagonal complex $\Omega_1$ and a diagonal complex
$\Omega_2$ is coherently diagonal.
}}}}

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