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\usepackage{tensor}
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\usepackage[T1]{fontenc}

\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{GWU-1612}
\def\title{On Elves and Invariants}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/\thistalk}{http://drorbn.net/\thistalk/}}}
\def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}}
\def\titleA{{\title}}
\def\titleB{{\title}}
\def\titleC{{\title}}

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\def\blue{\color{blue}}
\def\mgray{\color{mgray}}
\def\morange{\color{morange}}
\def\red{\color{red}}
\def\cbox#1#2{{\setlength{\fboxsep}{0pt}\colorbox{#1}{#2}}}

\def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:\linebreak[0]#1}}}
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\def\ub#1{\underbracket[0.5pt][1pt]{#1}}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\def\ad{\operatorname{ad}}
\def\Ad{\operatorname{Ad}}
\def\aft{$\overrightarrow{\text{4T}}$}
\def\AS{\mathit{AS}}
\def\bbZZ{{\mathbb Z\mathbb Z}}
\def\CW{\text{\it CW}}
\def\diag{\operatorname{diag}}
\def\FL{\text{\it FL}}
\def\IHX{\mathit{IHX}}
\def\PvT{{\mathit P\!v\!T}}
\def\remove{\!\setminus\!}
\def\STU{\mathit{STU}}
\def\TC{\mathit{TC}}
\def\tr{\operatorname{tr}}
\def\vT{{\mathit v\!T}}

\def\bbE{{\mathbb E}}
\def\bbO{{\mathbb O}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\calA{{\mathcal A}}
\def\calD{{\mathcal D}}
\def\calG{{\mathcal G}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}
\def\calO{{\mathcal O}}
\def\calP{{\mathcal P}}
\def\calR{{\mathcal R}}
\def\calS{{\mathcal S}}
\def\calT{{\mathcal T}}
\def\calU{{\mathcal U}}
\def\frakb{{\mathfrak b}}
\def\frakg{{\mathfrak g}}
\def\frakh{{\mathfrak h}}
\def\tilE{\tilde{E}}

%%%

\def\credits{{\raisebox{0mm}{\parbox[t]{2in}{
Follows Rozansky \cite{Ro, Rozansky:Burau, Rozansky:U1RCC} and \text{Overbay}
\cite{Overbay:Thesis}, joint with van der Veen.
}}}}

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Abstract.} Whether or not you like the formulas on this page,
they describe the strongest truly computable knot invariant we know.
}}}}

\def\ThreeSteps{{\raisebox{2mm}{\parbox[t]{2.25in}{
{\red Three steps} to the computation of $\rho_1$:
\newline{\red 1.~Preparation.} Given $K$, results
\[ \langle\text{\sl long word}\|\,\text{\sl simple formulas}\rangle. \]
{\red 2.~Rewrite rules.} Make the {\sl word} simpler and the {\sl
formulas} more complicated, until the {\sl word} ``$\mathit{elf}$''
is reached.
\newline{\red 3.~Readout.} The invariant $\rho_1$ is read from the last {\sl formulas}.
}}}}

\def\prepres{$\langle \mathit{elf}\ldots\mathit{elf}\|\, \omega_0;L_0;Q_0;P_0\rangle$}
\def\endres{$\langle \mathit{elf}\|\, \omega;-;-;P\rangle$}
\def\rhoone{$\rho_1(K)=\rho_1(\omega,P)$}

\def\Preparation{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 8 0in 2.95in 0in 2.95in 0in 2.95in 0in 2.95in 0in 2.95in 0in 2.95in 0in 2.95in 0in 3.95in
{\red Preparation.} Draw $K$ using a 0-framed 0-rotation planar diagram $D$ where
all crossings are pointing up. Walk along $D$ labeling features by
$1,\ldots,m$ in order: over-passes, under-passes, and right-heading cups
and caps (``$\pm$-cuaps''). If $x$ is a xing, let $i_x$ and $j_x$ be
the labels on its over/under strands, and let $s_x$ be its sign. If $c$
is a cuap, let $i_c$ be its labels and $s_c$ be its sign. With $\epsilon^2=0$, form
\[
  \prod_{x:\text{ xing}}R^{s_x}_{i_xj_x} \prod_{c:\text{ cuap}}U^{s_c}_{i_c}
  = \omega_0^{-1}e^{L_0\log t+Q_0/\omega_0}(1+\epsilon\omega_0^{-4}P_0)
\]
where $(R^+_{ij}, R^-_{ij})=$
\[ \left(
    e^{l_j\log t+e_if_j+\epsilon(e_il_if_j+l_il_j+e_l^2f_j^2/4)}
    ,\,
    e^{-l_j\log t-e_if_j/t+\epsilon(e_il_if_j/t-l_il_j-e_l^2f_j^2/4t^2)}
  \right),
\]
where $U^\pm_i = t^{\mp 1/2}e^{\pm\epsilon l_it^{\mp 2}}$, and where $P_0$ collects all the terms
proportional to $\epsilon$ and $(\omega_0,L_0,Q_0)$ are $\epsilon$-free and $L_0$ has the $l_j$ terms in
the exponential and $Q_0$ the $e_if_j$ terms. This done, output
\[ \langle e_1l_1f_1e_2l_2f_2\cdots e_ml_mf_m \|\, \omega_0;L_0;Q_0;P_0\rangle. \]
}}}}

\def\InFormulas{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red In {\sl formulas},} $L$ is always $\bbZ$-linear in $\{l_i\}$, $Q$
is an $R$-linear combination of $\{e_if_j\}$ where $R\coloneqq\bbZ[t^{\pm
1}]$, and $P$ is an $R$-linear combination of $\{1,\, l_i,\, l_il_j,\,
e_if_j,\, e_il_jf_k,\, e_ie_jf_kf_l\}$.
}}}}

\def\RewriteRules{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Rewrite Rules.} Manipulate
$\langle\text{\sl word}\|\,\text{\sl formulas}\rangle$ expressions
using the rewrite rules below, until you come to the form
${\langle e_1l_1f_1\|\, \omega;-;-;P\rangle}$. Output $(\omega,P)$.

{\red Rule 1, Deletions.} If a letter appears in {\sl word} but not in {\sl formulas}, you can delete it.

{\red Rule 2, Merges.} In {\sl word}, you can replace {\em adjacent}
$v_iv_j$ with $v_k$ (for $v\in\{e,l,f\}$) while making the same changes in {\sl formulas} (provided $k$
creates no naming clashes). E.g.,
\[ \langle\ldots e_ie_j\ldots \|\, Z\rangle \to \langle\ldots e_k\ldots\|\, Z|_{e_i,e_j\to e_k}\rangle. \]

{\red Rule 3, $le$ Corrections.} Provided $k$ introduces no
clashes, given $\langle\ldots l_je_i\ldots\|\,\omega;L;Q;P\rangle$,
decompose $L=\lambda l_j+L'$, $Q=\alpha e_i+Q'$, write $P=P(e_i,l_j)$
(with messy coefficients), set $q=e^\gamma\beta e_k+\gamma l_k$, and output
\[
  \left\langle
    \ldots e_kl_k\ldots\|\,\omega; L|_{l_j\to l_k}; t^\lambda\alpha e_k+Q';
    e^{-q}P(\partial_\beta,\partial_\gamma)e^q|_{\beta\to\alpha/\omega,\,\gamma\to\lambda\log t}
  \right\rangle.
\]

{\red Rule 4, $fl$ Corrections.} Provided $k$ introduces no
clashes, given $\langle\ldots f_il_j\ldots\|\,\omega;L;Q;P\rangle$,
decompose $L=\lambda l_j+L'$, $Q=\alpha f_i+Q'$, write $P=P(f_i,l_j)$
(with messy coefficients), set $q=e^\gamma\beta f_k+\gamma l_k$, and output
\[
  \left\langle
    \ldots l_kf_k\ldots\|\,\omega; L|_{l_j\to l_k}; t^\lambda\alpha f_k+Q';
    e^{-q}P(\partial_\beta,\partial_\gamma)e^q|_{\beta\to\alpha/\omega,\,\gamma\to\lambda\log t}
  \right\rangle.
\]
}}}}

\def\RuleFive{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Rule 5, $fe$ Corrections.} Provided $k$ introduces no
clashes, given $\langle\ldots f_ie_j\ldots\|\,\omega;L;Q;P\rangle$,
decompose $Q=Q_{fe}f_ie_j+Q_ff_i+Q_ee_j+Q'$
write $P=P(f_i,e_j)$ (with messy coefficients),
set $\mu=1+(t-1)\delta$ and $q=((1-t)\alpha\beta+\beta e_k+\alpha f_k+\delta e_kf_k)/\mu$, and output
\[
  \left.\left\langle
    \ldots e_kf_k\ldots
    \left\|\,{
      \mu\omega; L;\,\mu\omega q+\mu Q';\atop
      \omega^4\Lambda_k+e^{-q}P(\partial_\alpha,\partial_\beta)(e^q)
    }\right.
  \right\rangle\right|_{\alpha\to Q^f/\omega,\,\beta\to Q^e/\omega,\atop\delta\to Q^{fe}/\omega},
\]
where $\Lambda_k$ is the {\greektext L'ogos}, ``a principle of order and knowledge'':
\begin{align*} \Lambda_k = \frac{t+1}{4}\bigg(&
  -\delta  (\mu +1) \left(\beta ^2 e_k^2+\alpha ^2 f_k^2\right)
    -\delta^3 (3 \mu +1)e_k^2f_k^2 \\
  & -2 \left(\beta  e_k+\alpha  f_k\right)\left(
      \alpha\beta+2\delta\mu+\delta^2(2\mu+1)e_kf_k+2\delta\mu^2l_k
    \right) \\
  & -4 (\alpha  \beta +\delta  \mu ) \left(\delta  (\mu +1) e_k f_k+\mu ^2 l_k\right)
    -4 \delta ^2 \mu ^2 e_k f_k l_k \\
  & +(t-1) \left(2 (\alpha  \beta +\delta  \mu )^2-\alpha ^2 \beta
   ^2\right)\bigg).
\end{align*}

}}}}

\def\Readout{{\raisebox{2mm}{\parbox[t]{3.25in}{
{\red Readout.} Given $\langle\mathit{elf}\|\,\omega;-;-;P\rangle$, output
\[ \rho_1(K)\coloneqq \frac{t(t\omega'\omega^3-P|_{e,l,f\to 0})}{(t-1)^2\omega^2}. \]
{\footnotesize ($\omega$ is the Alexander polynomial, $L$ and $Q$ are not interesting).}
}}}}

\def\ExperimentalA{{\raisebox{3mm}{\parbox[t]{3.95in}{
{\red Experimental Analysis} (\web{Exp}). Log-log plot of computation
time (sec) vs.\ crossing number, for all knots with up to 12 crossings
(mean times) and for all torus knots with up to 48 crossings:
}}}}

\def\To12Times{{\includegraphics[width=2in]{../UNC-1610/To12Times.pdf}}}
\def\TKTimes{{\includegraphics[width=2in]{../UNC-1610/TKTimes.pdf}}}

\def\ExperimentalB{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Power.} On the 250 knots with at most 10 crossings, the pair $(\omega,\rho_1)$
attains 250 distinct values, while (Khovanov, HOMFLYPT) attains only
249 distinct values. To 11 crossings the numbers are (802, 788, 772)
and to 12 they are (2978, 2883, 2786).

{\red Genus.} Up to 12 xings, always $\rho_1$ is symmetric under $t\leftrightarrow t^{-1}$. With
$\rho_1^+$ denoting the positive-degree part of $\rho_1$, always $\deg \rho_1^+\leq 2g-1$, where $g$
is the 3-genus of $K$ (equallity for 2530 knots). This gives a lower
bound on $g$ in terms of $\rho_1$ (conjectural, but undoubtedly true). This
bound is often weaker than the Alexander bound, yet for 10 of the 12-xing
Alexander failures it does give the right answer.
}}}}

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\begin{document} \latintext
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\null\vfill\input{Elves1.pdftex_t}\vfill\null\eject
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\eject
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\def\cellscale{0.65}

\begin{multicols}{2} \raggedcolumns

{\large\red Demo Programs.}\hfill\web{Demo}

{\red What we didn't say.}


{\red References.}{\footnotesize
%\par\vspace{-3mm}
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\begin{thebibliography}{}
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.3ex}
\input{refs.tex}
\end{thebibliography}}

\end{multicols}

\def\N{\ding{56}}
\def\gY{\textcolor{ForestGreen}{\ding{52}}}
\def\oY{\textcolor{Orange}{\ding{52}}}
\def\headcell{
  diagram & \parbox{3in}{{\blue $n^t_k$}\quad Alexander's $A_+$
    \hfill genus / \textcolor{ForestGreen}{ribbon} \newline
  {\red Today's / Rozansky's $\rho_1^+$}
    \hfill unknotting number / \textcolor{Orange}{amphicheiral}}
}
\def\rolcell#1#2#3#4#5#6#7#8{
  \raisebox{-3pt}{\includegraphics[height=26.5pt]{../UNC-1610/KnotFigs/#1.pdf}} & \parbox[b]{3in}{
    {\blue $#2$}\quad $#3$\hfill $#5$ / #7 \\
    {\red $#4$} \hfill $#6$ / #8
}}

{\footnotesize \begin{longtable}{|cl|cl|cl|}
\hline \headcell & \headcell \\ \endhead
\hline
%\input table.tex
\end{longtable}}

\end{document}

\endinput