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\def\arXiv#1{{\href{http://arxiv.org/abs/#1}{arXiv:#1}}}

\definecolor{Magenta}{HTML}{FF00FF}
\def\Magenta{\color{Magenta}}

\def\web#1{{\Magenta\href{http://drorbn.net/23-FC/#1}{$\omega\epsilon\beta$/#1}}}

\def\S#1#2#3{\fbox{
  \!\!\parbox[c]{10mm}{\includegraphics[width=10mm]{figs/#1.eps}}
  \hspace{-4mm}$\begin{array}{r}
    \scriptstyle #2 \\[-2pt]
    \scriptstyle #3 \\[-2pt]
    \scriptstyle #1
  \end{array}$\hspace{-3mm}
}}
\def\C#1{\hspace{-2mm}\includegraphics[width=6mm]{figs/C#1.eps}}
\def\neg{\hspace{-2mm}}

\def\la{\langle}
\def\ra{\rangle}

\def\bbF{{\mathbb F}}
\def\bbN{{\mathbb N}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\calC{{\mathcal C}}
\def\calF{{\mathcal F}}
\def\calH{{\mathcal H}}
\def\calO{{\mathcal O}}
\def\calX{{\mathcal X}}
\def\frakg{{\mathfrak g}}

\newcommand{\qdim}{\operatorname{{\it q}dim}}
\newcommand{\qed}{~\hfill$\square$}

\def\KB#1#2{{\left\la\eepic{figs/#1}{#2}\right\ra}}

\newcommand{\Kh}{\operatorname{Kh}}
\newcommand{\Span}{\operatorname{span}}

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\def\red{\color{red}}
\def\green{\color{green}}
\def\blue{\color{blue}}

\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes}{Classes}:
  \href{\myurl/classes/\#2223}{2022-23}:
  \href{\myurl/classes/23-FastComputations}{Fast Computations in Knot Theory}:
  {\bf Khovanov Homology For Knots}
}}

\def\SR{{$\bigcirc^k\mapsto (q+q^{-1})^k$}}

\def\Jones{{\scalebox{1}{$
  J:\overcrossing\mapsto q\smoothing-q^2\hsmoothing,
  \quad J:\undercrossing\mapsto-q^{-2}\hsmoothing+q^{-1}\smoothing,
$}}}

\newcommand{\leps}[2]{{\hspace{-3pt}\begin{array}{c}%
  \raisebox{-2.5pt}{\includegraphics[width=#1]{figs/#2.eps}}%
\end{array}\hspace{-3pt}}}
\def\s#1{\leps{3mm}{TM#1}}
\def\R2{{\raisebox{3mm}{\scalebox{0.79}{\parbox[t]{2.5in}{
\begin{eqnarray*}
  J:\s{}
  & \mapsto & -q^{-1}\s{00} + \s{01} + \s{10} - q\s{11} \\
  & = & -q^{-1}\hsmoothing + \smoothing +(q+q^{-1})\hsmoothing - q\hsmoothing
    = \smoothing.
\end{eqnarray*}
}}}}}

\def\RedLem{{\raisebox{0mm}{\scalebox{0.8}{\parbox[t]{5in}{
\noindent {\bf The Reduction Lemma.} If $\phi$ is an isomorphism then the
complex
\[
  \xymatrix@C=2cm{
    \left[C\right]
    \ar[r]^{\left(\begin{array}{c}\alpha \\ \beta\end{array}\right)} &
    {\left[\begin{array}{c}b_1 \\ D\end{array}\right]}
    \ar[r]^{\left(\begin{array}{cc}
      \phi & \delta \\ \gamma & \epsilon
    \end{array}\right)} &
    {\left[\begin{array}{c}b_2 \\ E\end{array}\right]}
    \ar[r]^{\left(\begin{array}{cc}
      \mu & \nu
    \end{array}\right)} &
    \left[F\right]
  }
\]
is isomorphic to the (direct sum) complex
\[
  \xymatrix@C=3cm{
    \left[C\right]
    \ar[r]^{\left(\begin{array}{c}0 \\ \beta\end{array}\right)} &
    {\left[\begin{array}{c}b_1 \\ D\end{array}\right]}
    \ar[r]^{\left(\begin{array}{cc}
      \phi & 0 \\ 0 & \epsilon-\gamma\phi^{-1}\delta
    \end{array}\right)} &
    {\left[\begin{array}{c}b_2 \\ E\end{array}\right]}
    \ar[r]^{\left(\begin{array}{cc}
      0 & \nu
    \end{array}\right)} &
    \left[F\right]
  }
\]
}}}}}

\def\MF{{\raisebox{3mm}{\parbox{2.75in}{
\[ \xymatrix@C=2cm{
  M^0 \ar[r]^A
    \ar@<-2pt>[d]_{U^0} \ar@<2pt>[d]^{V^0} &
  M^1 \ar[r]^B \ar[ld]_{h^1}
    \ar@<-2pt>[d]_{U^1} \ar@<2pt>[d]^{V^1} &
  M^0 \ar[ld]_{h^0}
    \ar@<-2pt>[d]_{U^0} \ar@<2pt>[d]^{V^0} \\
  N^0 \ar[r]^{A'} &
  N^1 \ar[r]^{B'} &
  N^0 \\
} \]
}}}}

\def\Rasmussen{{\red J.~Rasmussen:} Leads to a no-analysis proof of
a conjecture by Milnor.}

\def\webdef{{
{\Magenta $\omega\epsilon\beta:=$
\url{http://www.math.toronto.edu/drorbn/Talks/Montreal-1306}}
}}

\def\LieInvariants{{\raisebox{0mm}{\parbox[t]{1.75in}{
{\red Story/Theorem.} To every finite-dimensional metrized Lie algebra
$\frakg$ and a list $R_1,\ldots,R_n$ of representations thereof, there
is an associated invariant of $n$-component links, valued in Laurent
polynomials in a variable $q$ (really, in finite linear combinations of
characters on the space of metrics on $\frakg$).

{\red Queries.} Do you know how to tell/prove it? Really? With all
the details? Could you teach the story/proof leaving no black boxes in
a one-semester course on knot theory to students who are not already
experts on Lie algebras? Do I really need to know about Cartan subalgebras
and root and weight spaces? Is this theorem at all true?
}}}}

\def\whatis{{\raisebox{0mm}{\parbox[t]{3.0in}{
What is Categorification$=$Concretization$=$de-abstraction?
``3'' is $\{\text{\red cow}, \text{\green cow}, \text{\blue cow}\}$ and
$\{\text{\red pig}, \text{\green pig}, \text{\blue pig}\}$ and many other
things\ldots
\newline\null\hfill{\red \ldots categorification is choosing which 3 it is!}
}}}}

\def\catN{{\raisebox{2.5mm}{\parbox[t]{3.0in}{
{\red $\bbN$.} Natural numbers $\mapsto$ finite sets, equalities $\mapsto$
bijections, inequalities $\mapsto$ injections and surjections:
\par$\displaystyle
  \binom{2n}{n}=\sum\binom{n}{k}^2\mapsto$
\newline\null\hfill$\displaystyle
  \binom{X\times\{1,2\}}{|X|}
  \leftrightarrow\bigcup\binom{X}{k}\times\binom{X}{k}.
$
}}}}

\def\catZ{{\raisebox{1.5mm}{\parbox[t]{3.5in}{
{\red $\bbZ$.} Negative numbers:
}}}}

\def\weakercat{{\raisebox{1.5mm}{\parbox[t]{3.65in}{
\parshape 4 0pt\hsize 0pt\hsize 0pt\hsize 0.65in 3in
{\red Weaker Categorification.} Do the same in the category of vector
spaces: ``3'' becomes $V$ s.t.\ $\dim V=3$, or better,
$V^\bullet=(\cdots V^{r-1}\to V^r\to V^{r+1}\cdots)$ s.t.\ $d^2=0$ and
$\chi(V^\bullet):=\sum(-1)^r\dim V^r=3=\sum(-1)^r\dim H^r$. Equalities
become homotopies between complexes.
}}}}

\def\vert{\rule{0mm}{2mm}}
\def\qrg{{\raisebox{0mm}{\resizebox{8in}{!}{\parbox[t]{6.625in}{

\par\noindent {\red Khovanov:} $K(L)$ is a chain complex of graded
$\bbZ$-modules;
\newline\null\quad$\displaystyle
  V=\Span\langle v_+,v_-\rangle;
  \quad \deg v_\pm=\pm 1;
  \quad \qdim V=q+q^{-1};$
\newline$\displaystyle
  K(\bigcirc^k) = V^{\otimes k};
  \quad
  K(\overcrossing) =
  \text{Flatten}\left(\!
    0\to\underset{\text{height }0}{K(\smoothing)\{1\}}
    \to\underset{\text{height }1}{K(\hsmoothing)\{2\}}
    \to 0
  \right);$
\newline\null\qquad$\displaystyle
  K(\undercrossing) =
  \text{Flatten}\left(\!
    0\to\underset{\text{height }-1}{K(\hsmoothing)\{-2\}}
    \to\underset{\text{height }0}{K(\smoothing)\{-1\}}
    \to 0
  \right);$

\vskip -2mm
\noindent$\displaystyle
  \parbox{5in}{\begin{align*}
    \Big(\begin{array}{c}
      \includegraphics[width=1.5in]{figs/JoinRule.eps}
    \end{array}\Big)
    & \longrightarrow \big(V\otimes V\overset{m}{\rightarrow}V\big)
    &\quad& m:\begin{cases}
      v_+\otimes v_-\mapsto v_- &
      v_+\otimes v_+\mapsto v_+ \\
      v_-\otimes v_+\mapsto v_- &
      v_-\otimes v_-\mapsto 0
    \end{cases}
  \\
    \Big(\begin{array}{c}
      \includegraphics[width=1.5in]{figs/SplitRule.eps}
    \end{array}\Big)
    & \longrightarrow \big(V\overset{\Delta}{\rightarrow}V\otimes V\big)
    &\quad& \Delta:\begin{cases}
      v_+ \mapsto v_+\otimes v_- + v_-\otimes v_+ &\\
      v_- \mapsto v_-\otimes v_- &
    \end{cases}
  \end{align*}}
$

\vskip -3mm
\par\noindent Example:\hfill
$ = \quad q + q^3 + q^5 - q^9.$
\[
  \hspace{-3mm}\begin{array}{c} \xymatrix@R=1.697cm@C=2cm{
    q^3(q+q^{-1})^2
        \ar@{}[r]|{-}
        \ar@{}[dr]|{\hspace{-6mm}
          \includegraphics[width=0.7in]{figs/Trefoil.eps}
        }
      & 3q^4(q+q^{-1})
        \ar@{}[r]|{+}
      & 3q^5(q+q^{-1})^2
        \ar@{}[r]|{-}
      & q^6(q+q^{-1})^3 \\
    & \S{100}{q^4(q+q^{-1})}{V\{4\}}
        \ar@{o->}[r]^{\C{1S0}}_{d_{1\star 0}}
        \ar@{o->}[rd]|\hole^(0.36){\C{10S}}_(0.6){d_{10\star}\neg}
        \ar@{.>}@<1.5ex>[u]
        \ar@{.}@<-1.5ex>[d]_+
        \ar@{.}@<1.5ex>[d]^\oplus
      & \S{110}{q^5(q+q^{-1})^2}{V^{\otimes 2}\{5\}}
        \ar[rd]^{\neg
          \overset{\substack{\text{that's a}\\ \text{cobordism!}\\\ }}{\C{11S}}
        }_{d_{11\star}\neg}
        \ar@{.>}@<1.5ex>[u]
        \ar@{.}@<-1.5ex>[d]_+
        \ar@{.}@<1.5ex>[d]^\oplus
      & \\
    \S{000}{q^3(q+q^{-1})^2}{V^{\otimes 2}\{3\}}
        \ar[ru]^{\C{S00}\neg}_{d_{\star 00}}
        \ar[r]^{\C{0S0}}_{d_{0\star 0}}
        \ar[rd]^{\C{00S}}_{d_{00\star}}
        \ar@{.>}[uu]^{\begin{sideways} \scriptsize
          go up for Jones
        \end{sideways}}
        \ar@{.>}[dd]_{\begin{sideways} \scriptsize
          go down for Khovanov
        \end{sideways}}
      & \S{010}{q^4(q+q^{-1})}{V\{4\}}
        \ar[ru]^(0.3){\C{S10}\neg}_(0.66){d_{\star 10}}
        \ar@{o->}[rd]^(0.36){\C{01S}}_(0.6){d_{01\star}\neg}
        \ar@{.}@<-1.5ex>[d]_+
        \ar@{.}@<1.5ex>[d]^\oplus
      & \S{101}{q^5(q+q^{-1})^2}{V^{\otimes 2}\{5\}}
        \ar@{o->}[r]^{\C{1S1}}_{d_{1\star 1}}
        \ar@{.}@<-1.5ex>[d]_+
        \ar@{.}@<1.5ex>[d]^\oplus
      & \S{111}{q^6(q+q^{-1})^3}{V^{\otimes 3}\{6\}}
        \ar@{.>}[uu]
        \ar@{.>}[dd] \\
    \ar@{}[r]_\ ="1"
      & \S{001}{q^4(q+q^{-1})}{V\{4\}}
        \ar[ru]|\hole^(0.3){\C{S01}\neg}_(0.66){d_{\star 01}}
        \ar[r]^{\C{0S1}}_{d_{0\star 1}}="3"
        \ar@{.>}@<1.5ex>[d]
      & \S{011}{q^5(q+q^{-1})^2}{V^{\otimes 2}\{5\}}
        \ar[ru]^{\C{S11}\neg}_{d_{\star 11}}
        \ar@{.>}@<1.5ex>[d]
        \ar@{}[r]_\ ="5" & \\
    K(\righttrefoil)^0
        \ar[r]^{d^0}="2"
      & K(\righttrefoil)^1
        \ar[r]^{d^1}="4"
      & K(\righttrefoil)^2
        \ar[r]^{d^2}="6"
      & K(\righttrefoil)^3
    \ar@{.>}"1";"2"|{\underset{|\xi|=0}{\sum}(-1)^\xi d_\xi}
    \ar@{.>}"3";"4"|{\underset{|\xi|=1}{\sum}(-1)^\xi d_\xi}
    \ar@{.>}"5";"6"|{\underset{|\xi|=2}{\sum}(-1)^\xi d_\xi}
  } \end{array}
\]
\qquad(here $(-1)^\xi:=(-1)^{\sum_{i<j}\xi_i}$ if $\xi_j=\star$)
\hfill $=\quad K(\righttrefoil).$

\par\noindent{\bf Theorem 1. } The graded Euler characteristic of
$K(L)$ is $J(L)$.

\par\noindent{\bf Theorem 2. } The homology $\Kh(L)$ of $K(L)$ is a
link invariant.

\par\noindent{\bf Theorem 3. } $\Kh(L)$ is strictly stronger than
$J(L)$: $J(\bar{5}_1)=J(10_{132})$ yet $\Kh(\bar{5}_1)\neq\Kh(10_{132})$.

\par\noindent{\bf Theorem 4. } $\Kh$ extends to a functor defined on the category of 2D cobordisms in $\bbR^4$ between links in
$\bbR^3$.

\par\noindent{\bf References. } Khovanov's \arXiv{math.QA/9908171} and
\arXiv{math.QA/0103190} and my\newline
\rightline{\Magenta\url{http://www.math.toronto.edu/~drorbn/papers/Categorification/}}.

}}}}}

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