Rasmussen invariant,\\ Slice-Bennequin inequality,\\ and sliceness of knots
Alexander Shumakovitch
We use recently introduced Rasmussen invariant to find knots that are topologically locally-flatly slice but not smoothly slice. We note that this invariant can be used to give a combinatorial proof of the Slice-Bennequin inequality. Finally, we compute the Rasmussen invariant for quasipositive knots and show that most of our examples of non-slice knots are not quasipositive and were to the best of our knowledge unknown before.
16n_809057-homology-notorsion-small.tex
\def\DSp{\kern 0.2em}%
\def\emptyline{&&&&&&&&&&&&&&&&&}%
\def\numcolumns{19}%

\setbox\tablebox\vbox{\small\let\hline\tableline\offinterlineskip\ialign{%
\vrule\TSp\vrule #\relax&\DSp\hfil #\DSp\vrule\TSp\vrule&
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16n_864894-homology-notorsion-small.tex
\def\DSp{\kern 0.2em}%
\def\emptyline{&&&&&&&&&&&&&&&&&}%
\def\numcolumns{19}%

\setbox\tablebox\vbox{\small\let\hline\tableline\offinterlineskip\ialign{%
\vrule\TSp\vrule #\relax&\DSp\hfil #\DSp\vrule\TSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&
\DSp\hfil\bf #\hfil\DSp\vrule&#\TSp\vrule\cr
%
\dblhline
height 11pt depth 4pt&&
\rm\DSp-9\DSp&
\rm\DSp-8\DSp&
\rm\DSp-7\DSp&
\rm\DSp-6\DSp&
\rm\DSp-5\DSp&
\rm\DSp-4\DSp&
\rm\DSp-3\DSp&
\rm\DSp-2\DSp&
\rm\DSp-1\DSp&
\rm\DSp0\DSp&
\rm\DSp1\DSp&
\rm\DSp2\DSp&
\rm\DSp3\DSp&
\rm\DSp4\DSp&
\rm\DSp5\DSp&
\rm\DSp6\DSp&\cr\dblhline
%
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\box\tablebox

KhoHo-tables.tex
% Components of the table: spaces and braces
\let\TSp\thinspace
\def\DSp{\thinspace\thinspace}
\def\lbr{\raise 1pt\hbox{[}}
\def\rbr{\raise 1pt\hbox{]}}

% Components of the table: lines of all kinds
\def\gobble#1{}
\def\tableline{\multispan\numcolumns\hrulefill\cr}
\def\dblhline{\hline height 0.16667em\gobble&\emptyline\cr\hline}

\def\inline#1{\leaders\hrule\hskip 0.33em plus 1fill%
  \hbox{\vbox to 0pt{\vss\hbox{\TSp #1\TSp}\vss}}%
  \leaders\hrule\hskip 0.33em plus 1fill\vrule
}

% The table will usually be put into this box
\newbox\tablebox

% -------------------------------------------------------------------------- %
%                                                                            %
%    PostScript circles, squares and diagonals in different arrangements     %
%                                                                            %
% -------------------------------------------------------------------------- %

\newbox\emptybox
\newbox\circlebox
\newbox\graycirclebox
\newbox\squarebox
\newbox\graysquarebox
\newbox\diagbox

\setbox\emptybox\null

\def\PSpreamble#1{%
  gsave /scaleFactor {Resolution 72 div} def
  currentpoint translate #1 scaleFactor mul dup dup dup neg translate scale
  0.4 #1 div setlinewidth
}
\def\PStrailer#1{%
  gsave #1 grestore 0 setgray stroke grestore
}

%
% Puts a circle of diameter #2 and filled using PS directive #3 into the box #1
\def\makecircle#1#2#3{%
  \global\setbox#1\hbox to #2bp{\vbox to #2bp{\vss
    \special{ps: \PSpreamble{#2 2 div} newpath 0 0 1 0 360 arc \PStrailer{#3}}%
  }\hss}%
}
\makecircle\circlebox9{1 setgray eofill}
\makecircle\graycirclebox9{0.75 setgray eofill}

%
% Puts a square of size #2 and filled using PS directive #3 into the box #1
\def\makesquare#1#2#3{%
  \global\setbox#1\hbox to #2bp{\vbox to #2bp{\vss
    \special{ps: \PSpreamble{#2 2 div}
      newpath -1 -1 moveto 1 -1 lineto 1 1 lineto -1 1 lineto closepath
      \PStrailer{#3}}%
  }\hss}%
}
\makesquare\squarebox{11}{1 setgray eofill}
\makesquare\graysquarebox{11}{0.75 setgray eofill}

%
% Puts a diagonal into the box #1 of width #2 and height #3
% that goes from top-left to bottom-right if #4 is '-',
% and from bottom-left to top-right otherwise.
\newdimen\tempPSdima \newdimen\tempPSdimb
\newcount\tempPScnta \newcount\tempPScntb
\def\makediagonal#1#2#3#4{%
  \tempPSdima #2\tempPScnta\tempPSdima
  \tempPSdimb #3\tempPScntb\tempPSdimb
  \ifx#4-\relax
    \def\tempPS{dup 0 exch translate 1 -1 scale}%
  \else
    \def\tempPS{}%
  \fi
  \global\setbox#1\hbox to #2{\vbox to #3{\vss
    \special{ps: gsave /scaleFactor {Resolution 72 div} def
      currentpoint translate
      \the\tempPScnta\space  72 mul 72.27 div 65536 div scaleFactor mul
      \the\tempPScntb\space -72 mul 72.27 div 65536 div scaleFactor mul
      \tempPS\space newpath 0 0 moveto rlineto
      0 setgray 0.4 scaleFactor mul setlinewidth stroke grestore}%
  }\hss}%
}

%
% Box of zero width and height with its content in the middle
\def\zerobox#1{\hbox to 0pt{\hss\vbox to 0pt{\vss #1\vss}\hss}}

%
% Creates a box of width #1 and height #2 with boxes #3 and #4 in the
% left and right corners that are joined by a diagonal going from top-left
% to bottom-right if #5 is '-', and from bottom-left to top-right otherwise.
\def\joincorners#1#2#3#4#5{%
  \makediagonal\diagbox{#1}{#2}{#5}%
  \vbox{\offinterlineskip
    \copy\diagbox
    \ifx#5-\relax\kern -\ht\diagbox\fi
      \hbox to\wd\diagbox{\zerobox{\copy #3}\hfill}%
    \ifx#5-\relax\kern \ht\diagbox\else\kern -\ht\diagbox\fi
      \hbox to\wd\diagbox{\hfill\zerobox{\copy #4}}%
    \ifx#5-\relax\else\kern \ht\diagbox\fi
  }%
}

% -------------------------------------------------------------------------- %
%                                                                            %
%          All kinds of Knight Move pairs made of PostScript pieces          %
%                                                                            %
% -------------------------------------------------------------------------- %

\newbox\tablerow
\newbox\KMtable
\newbox\tmpKMbox

\newcount\rowcount

\newdimen\rowheight
\newdimen\botheight
\newdimen\lastentrywidth
\newdimen\leftKMskip
\newdimen\rightKMskip
\newdimen\KMwidth
\newdimen\entrymargin	% will be determined for each table individually
\newdimen\Tentrymargin	% depending on how large \DSp is defined to be

\def\showKMpairs#1{%
  \edef\KMpairs{#1}%
  % unpack the \tablebox, take a sample row from it (the 4th from the bottom,
  % since the horizontal lines should also be counted), and repack it back
  \setbox\tmpKMbox\vbox{}%
  \setbox\tablebox\vbox{%
    \offinterlineskip
    \unvbox\tablebox
    \rowcount 4\setbox\tablebox\vbox{}%
    \loop
      \advance\rowcount -1\relax
      \global\setbox\tablerow\lastbox\unskip
      \global\setbox\tmpKMbox\vbox{\copy\tablerow\unvbox\tmpKMbox}%
    \ifnum\rowcount>0\repeat
    \unvcopy\tmpKMbox
  }%

  % height of the row include the thickness of the horizontal line
  \rowheight.4pt\advance\rowheight\ht\tablerow \advance\rowheight\dp\tablerow
  \botheight-\rowheight\advance\botheight.2pt
  \advance\botheight\ht\tmpKMbox \advance\botheight\dp\tmpKMbox

  % set \entrymargin and \Tentrymargin according to the current value of \DSp
  \setbox\tmpKMbox\hbox{\DSp}\entrymargin\wd\tmpKMbox\Tentrymargin\entrymargin
  % take into account the width of the most common entry (with minor correction)
  \setbox\tmpKMbox\hbox{1\kern.4pt}\advance\entrymargin .5\wd\tmpKMbox
  \setbox\tmpKMbox\hbox{\lbr1\rbr\kern.2pt}\advance\Tentrymargin .5\wd\tmpKMbox

  % \KMtable is going to contain all the Knight Move pairings
  % and will be superimposed under the original table
  \setbox\KMtable\hbox{\vrule width0pt height\ht\tablebox depth\dp\tablebox}%
  \hbox{%
    \global\lastentrywidth 0pt\relax
    \unhbox\tablerow\unskip
    \loop
      \setbox\tmpKMbox\lastbox\unskip
    \ifhbox\tmpKMbox
      \global\setbox\KMtable\hbox{%
        \hbox to\wd\tmpKMbox{\expandafter\processKHpair\KMpairs\theend}%
        \unhbox\KMtable
      }%
      \global\lastentrywidth\wd\tmpKMbox
    \repeat
  }%
}

\def\empty{}
\def\processKHpair#1,#2:#3\theend{%
  \gdef\KMpairs{#3}%
  \def\firstargument{#1}%
  \ifx\firstargument\empty\else
    \vbox{\offinterlineskip
      \hbox{\csname KM#1\endcsname}%
      \kern #2.5\rowheight\kern\botheight\relax
    }\hss
  \fi
}

\def\setKMsizes{%
  \KMwidth\wd\tmpKMbox \advance\KMwidth\lastentrywidth
  \advance\KMwidth -\leftKMskip \advance\KMwidth -\rightKMskip

  % shift to the left by half the width of the vertical line
  \kern\leftKMskip\kern -.2pt
}


\def\simplepair{%
  \setKMsizes
  \joincorners\KMwidth{2\rowheight}\circlebox\circlebox{}%
}

% the same as \simplepair, but twice in adjacent diagonals
\def\doublepair{%
  \vbox{\offinterlineskip
    \hbox{\simplepair}\kern -\rowheight\hbox{\simplepair}%
  }%
}

\def\verticalpair{%
  \lastentrywidth 0pt\setKMsizes
  \joincorners\KMwidth\rowheight\graycirclebox\graycirclebox-%
}

\def\specialVpair{%
  \advance\lastentrywidth\wd\tmpKMbox\setKMsizes
  \joincorners\KMwidth\rowheight\graycirclebox\graycirclebox{}%
}

\def\torsionpair{%
  \vbox{\offinterlineskip
    \hbox{%
      \leftKMskip\wd\tmpKMbox \advance\leftKMskip .5\lastentrywidth
      \setKMsizes\joincorners\KMwidth\rowheight\circlebox\emptybox-%
    }%
    \hbox{\setKMsizes\joincorners\KMwidth\rowheight\circlebox\graysquarebox{}}%
  }%
}

\def\KMcc{\leftKMskip .5\wd\tmpKMbox\rightKMskip .5\lastentrywidth\simplepair}
\def\KMdbl{\leftKMskip .5\wd\tmpKMbox\rightKMskip .5\lastentrywidth\doublepair}

\def\KMVcc{\leftKMskip .5\wd\tmpKMbox\rightKMskip\leftKMskip\verticalpair}
\def\KMVlc{\leftKMskip\entrymargin\rightKMskip .5\wd\tmpKMbox\verticalpair}
\def\KMVsp{\leftKMskip .5\wd\tmpKMbox\rightKMskip\leftKMskip\specialVpair}

\def\KMTcc{\leftKMskip .5\wd\tmpKMbox\rightKMskip .5\lastentrywidth\torsionpair}
\def\KMTcr{\leftKMskip .5\wd\tmpKMbox\rightKMskip\Tentrymargin\torsionpair}
\def\KMTlr{\leftKMskip\entrymargin\rightKMskip\Tentrymargin\torsionpair}

s-slice.tex
\documentclass[reqno]{amsart} \usepackage{s-slice}
\usepackage{graphicx}

\input KhoHo-tables

\floatsep=5pt plus 1pt
\textfloatsep=2\floatsep \intextsep=2\floatsep

\def\KhoHo{{\tt KhoHo}\space}

\begin{document}
\title[Rasmussen invariant and sliceness of knots]{Rasmussen invariant,\\
Slice-Bennequin inequality,\\ and sliceness of knots}
\author[A.~Shumakovitch]{Alexander Shumakovitch}
\address{Department of Mathematics, Dartmouth College, 6188 Bradley Hall,
Hanover, NH~03755-3551}
\email{Shurik@Dartmouth.edu}
\subjclass{57M25, 57M27}
\keywords{Rasmussen invariant, Slice-Bennequin inequality, slice genus,
slice knots, quasipositive knots}
\begin{abstract}
We use recently introduced Rasmussen invariant to find knots that are
topologically locally-flatly slice but not smoothly slice. We note that this
invariant can be used to give a combinatorial proof of the Slice-Bennequin
inequality.  Finally, we compute the Rasmussen invariant for quasipositive
knots and show that most of our examples of non-slice knots are not
quasipositive and were to the best of our knowledge unknown before.
\end{abstract}
\maketitle

\section{Rasmussen invariant and the Slice-Bennequin inequality}
In~\cite{Rasmussen-s} Jacob Rasmussen used the theory of knot (co)homology
developed by Mikhail Khovanov~\cite{Khovanov-categor} and results of Eun
Soo Lee~\cite{Lee-alt-links} to introduce a new invariant $s$ of knots in
$S^3$. This invariant takes values in even integers. Its main properties are
summarized as follows.

% \begin{thm}[{\cite[Theorems~1--4 and Corollary~1]{Rasmussen-s}}]
\begin{thm}[{Rasmussen~\cite[Theorems~1--4]{Rasmussen-s}}]
\label{thm:Rasmussen-main}
Let $K$ be a knot in $S^3$. Then
\begin{enumerate}
\item $s$ gives a lower bound on the slice ($4$-dimensional) genus $g_4(K)$
of $K$:
\begin{equation}\label{eq:s-g-bound}
|s(K)|\le2g_4(K);
\end{equation}
\item $s$ induces a homomorphism from $Conc(S^3)$, the concordance
group of knots in $S^3$, to $\Z$;
\item If $K$ is alternating, then $s(K)=\Gs(K)$, where $\Gs(K)$ is the
classical knot signature of $K$.
\item If $K$ can be represented by a positive diagram $D$, then
\begin{equation}\label{eq:s-positive-knots}
s(K)=2g_4(K)=2g(K)=n(D)-O(D)+1,
\end{equation}
where $n(D)$ and $O(D)$ are the number of crossings and Seifert circles of
$D$, respectively, and $g(K)$ is the ordinary genus of $K$;
% \item If $K$ is the $(p,q)$-torus knot with $p,q>0$, then
% \begin{equation}
% s(K)=(p-1)(q-1).
% \end{equation}
\end{enumerate}
\end{thm}

\begin{cor}[{\cite[Corollary~4.3]{Rasmussen-s}}]\label{cor:s-Xing-change}
Let $K_+$ and $K_-$ be two knots that are different at a single crossing that
is positive in $K_+$ and negative in $K_-$. Then
\begin{equation}\label{eq:s-change-Xing}
s(K_-)\le s(K_+)\le s(K_-)+2.
\end{equation}
\end{cor}

Equality~\eqref{eq:s-positive-knots} can easily be generalized to the case of
arbitrary knots. It becomes an inequality then.

\begin{lem}\label{lem:s-inequality}
Let $K$ be a knot represented by a diagram $D$ and let $w(D)$ be the {\em
writhe number} of $D$, that is, the number of positive crossings of $D$ minus
the number of negative ones:
$\displaystyle w(D)=\#\left\{
\vcenter{\hbox{\includegraphics[scale=0.55]{pos_Xing-black}}}\right\}-\#\left\{
\vcenter{\hbox{\includegraphics[scale=0.55]{neg_Xing-black}}}\right\}$.
Then 
\begin{equation}\label{eq:s-writhe-Seifert}
s(K)\ge w(D)-O(D)+1.
\end{equation}
\end{lem}
\begin{proof}
If all crossings of $D$ are positive, the Lemma follows
from~\eqref{eq:s-positive-knots}. The inequality~\eqref{eq:s-writhe-Seifert}
is an equality then. When a positive crossing of $D$ is changed into a
negative one, the right hand side of~\eqref{eq:s-writhe-Seifert} decreases by
$2$, while the left hand side decreases by at most $2$, because
of~\eqref{eq:s-change-Xing}. Hence, the inequality is preserved.
\end{proof}

Lemma~\ref{lem:s-inequality} implies the Slice-Bennequin inequality that
was originally proved by Rudolph~\cite{Rudolph-quasipositivity} using the
gauge theory. The theory developed by Khovanov, Lee, and Rasmussen provides
the first purely combinatorial proof of this inequality.

\begin{cor}[Slice-Bennequin Inequality, cf. \cite{Rudolph-quasipositivity}]
\label{cor:slice-bennequin}
Let $\Gb$ be a braid with $k$ strands and let $\widehat\Gb$ be its closure
that is a knot. Then
\begin{equation}\label{eq:slice-bennequin}
g_4(\widehat\Gb)\ge \frac{w(\Gb)-k+1}2.
\end{equation}
\end{cor}
\begin{proof}
Follows from~\eqref{eq:s-writhe-Seifert} and~\eqref{eq:s-g-bound}. 
% If $\widehat\Gb$ is a knot, then~\eqref{eq:slice-bennequin} follows
% from~\eqref{eq:s-writhe-Seifert} and~\eqref{eq:s-g-bound}. Otherwise ... {\bf
% % !!!!!!!!!!!!!!!!!!!!!!!!!!!
% TBW}
\end{proof}

The Slice-Bennequin inequality leads to a formula for the Rasmussen invariant
of (strongly) quasipositive knots. We use this formula in
section~\ref{sec:sliceness}.

\begin{defins}
1.~A knot $K$ is said to be {\em quasipositive} if it is the closure of a braid
that has the form $(w_1\Gs_{j_1}w_1^{-1})(w_2\Gs_{j_2}w_2^{-1})\cdots
(w_p\Gs_{j_p}w_p^{-1})$, where $\Gs_i$ are the standard generators of the
corresponding braid group, and $w_i$ are braid words.

\noindent
2.~A knot $K$ is said to be {\em strongly quasipositive} if it is the closure
of a braid that has the form
$\Gs_{{i_1,j_1}}\Gs_{{i_2,j_2}}\cdots\Gs_{{i_p,j_p}}$, where
$\Gs_{i,j}=(\Gs_i\Gs_{i+1}\cdots\Gs_{j-2})\Gs_{j-1}
(\Gs_i\Gs_{i+1}\cdots\Gs_{j-2})^{-1}$ for $j\ge i+2$ and
$\Gs_{i,i+1}=\Gs_i$.
\end{defins}

\begin{figure}
\centerline{\hbox{%
\vbox{\halign{\hfil#\hfil\cr
\includegraphics[scale=0.50]{pretzel-n3-5-7}\cr a.\cr}}%
\qquad\qquad
\vbox{\halign{\hfil#\hfil\cr
\includegraphics[scale=0.60]{pretzel-n3-5-7-braid}\cr b.\cr}}$
\qquad\qquad
\vbox{\halign{\hfil#\hfil\cr
\includegraphics[scale=0.60]{quasipositive-seifert}\cr c.\cr}}$
}}
\caption{The $(-3,5,7)$-pretzel knot, its braid representation as a strongly
quasipositive braid
$\Gs_1\,\Gs_2\,\Gs_{2,4}\,\Gs_{3,6}\,\Gs_{1,4}\,\Gs_5\,\Gs_{2,5}$, and
the corresponding Seifert surface.} \label{fig:3-5-7-pretzel}
\end{figure}

The $(-3,5,7)$-pretzel knot depicted in Figure~\ref{fig:3-5-7-pretzel}.a is a
strongly quasipositive knot. It is the closure of the braid
$\Gs_1\,\Gs_2\,\Gs_{2,4}\,\Gs_{3,6}\,\Gs_{1,4}\,\Gs_5\,\Gs_{2,5}$ (see
Figure~\ref{fig:3-5-7-pretzel}.b).

\begin{propos}\label{prop:s-for-quasipositive}
Let $K$ be a knot that can be represented as the closure of a quasipositive
braid $\Gb$ with $k$ strands, such that $\Gb$ is a product of $b$ factors of
the form $w\Gs_i w^{-1}$. Then
\begin{equation}\label{eq:s-for-quasipositive}
s(K)=2g_4(K)=b-k+1.
\end{equation}
If, moreover, $\Gb$ is strongly quasipositive, then
\begin{equation}\label{eq:s-for-strongly-quasipositive}
s(K)=2g(K)=2g_4(K)=b-k+1.
\end{equation}
\end{propos}
\begin{proof}
It is clear that $w(K)=b$ and $O(K)=k$. Hence,
\begin{equation*}
b-k+1\le s(K)\le|s(K)|\le2g_4(K)\le2g(K)
\end{equation*}
by~\eqref{eq:s-writhe-Seifert} and~\eqref{eq:s-g-bound}. On the other hand, for a (strongly)
quasipositive knot one can explicitly construct an embedded surface $S$ in
$D^4$ (respectively, $S^3$), such that $\partial S=K$ and the Euler
characteristic $\chi(S)$ of $S$ is $k-b$ (see Figure~\ref{fig:3-5-7-pretzel}.c
that illustrates this construction in the strongly quasipositive case). If
follows that $g_4(K)$ (respectively, $g(K)$) does not exceed $\frac{b-k+1}2$.
This finishes the proof.
\end{proof}

\begin{rem}
Charles Livingston~\cite{Livingston-tau} used the Ozsv\'ath-Szab\'o knot
invariant $\Gt$~\cite{Ozsvath-Szabo-tau} to give new proofs to several
results of Lee Rudolph~\cite{Rudolph-quasipositivity, Rudolph-sliceness} on
the slice genus, including the Slice-Bennequin Inequality. $s$ and $\Gt$ share
many of their main properties and our approach is similar to the Livingston's
one. The key difference is that the Rasmussen invariant is defined
combinatorially, while the Ozsv\'ath-Szab\'o invariant is based on the theory
of knot Floer homology. Rasmussen conjectured that $s(K)=2\Gt(K)$ for every
knot $K$.
\end{rem}

\begin{rem}
Relation between the Rasmussen invariant and the Slice-Bennequin
inequality was independently observed by several other authors, including
Olga Pla\-me\-nevskaya~\cite{Plamenevskaya-transverse} and
Alexander Stoimenow~\cite{Stoimenow-sliceness}.
\end{rem}

\section{Sliceness of knots}\label{sec:experiments}
\label{sec:sliceness}

In many cases one can easily compute $s(K)$ if one knows the Khovanov homology
of $K$. For a given knot $K$, let
$h^{i,j}(K)=\dim_\Q(\CalH^{i,j}(K)\otimes\Q)$ be the ranks of its homology and
$Kh(K)(t,q)=\sum_{i,j}t^iq^jh^{i,j}(K)$ be the corresponding Poincar\'e
polynomial in variables $t$ and $q$. Denote by $hw(K)$ the {\em homological
width} of $K$, that is, the minimal number of diagonals $2i-j=const$ that
support the homology of $K$.

In was shown by Rasmussen~\cite[Proposition 5.2]{Rasmussen-s} that for all
knots $K$ with $hw(K)\le 3$, one has
\begin{equation}\label{eq:s-equation}
Kh(K)=q^{s(K)-1}\big(1+q^2+(1+tq^4)Kh'(K)\big),
\end{equation}
where $Kh'(K)$ is some (Laurent) polynomial in $t$ and $q$ with non-negative
coefficients. In fact, Rasmussen's arguments can be applied to a more general
case.
\begin{prop}\label{prop:s-from-Kh}
Assume that $h^{i,j}(K)h^{i+1,j+8}(K)=0$ for all $i$ and $j$ (this is
automatically the case for all $K$ with $hw(K)\le 3$).
Then~\eqref{eq:s-equation} holds true.
\end{prop}
\begin{proof}
Construction of the Rasmussen invariant is based on a spectral sequence
structure on the Khovanov chain complex that is due to
Lee~\cite{Lee-alt-links}. The differential $d_n$ in this spectral sequence
has bidegree $(1, 4(n-1))$. The condition on $h^{i,j}$ implies that $d_n$ is
trivial for all $n\ge 3$. The rest of the arguments is the same as
in~\cite[Proposition 5.2]{Rasmussen-s}.
\end{proof}

It is possible for a knot to have homological width $4$, but still satisfy the
condition of~\ref{prop:s-from-Kh} (see Table~\ref{tbl:16n_809057-knot}). On
the other hand, the knot $16n_{864894}$\footnote{The enumeration for knots
that we use is the one from Knotscape~\cite{Knotscape}. For example, the knot
$13n_{1496}$ is a non-alternating knot number 1496 with 13 crossings.} may
theoretically have $d_3^{-1,-7}\not=0$, since $h^{-1,-7}=h^{0,1}=1$ (see
Table~\ref{tbl:16n_864894-knot}). We can not conclude whether the Rasmussen
invariant is $0$ or $-2$ in this case. In order to
satisfy~\eqref{eq:s-equation}, it should be $0$.

\begin{table}[t]
\centerline{\input 16n_809057-homology-notorsion-small}
\medskip
\caption{Homology ranks of the knot $16n_{809057}$. The homological width is
4, but the differential $d_3$ of bidegree $(1,8)$ is trivial.}
\label{tbl:16n_809057-knot}
\end{table}

\begin{table}[t]
\centerline{\input 16n_864894-homology-notorsion-small}
\medskip
\caption{Homology ranks of the knot $16n_{864894}$. Encircled entries show
that $d_3^{-1,-7}$ can possibly be non-trivial.}
\label{tbl:16n_864894-knot}
\end{table}

\begin{rem}
It was conjectured~\cite{BarNatan-categor} that~\eqref{eq:s-equation} holds
true for all knots for {\em some} integer $s(K)$. It is not currently known
whether this integer always equals the Rasmussen invariant or not. The knot
$16n_{864894}$ might be a counter-example.
\end{rem}

Given a knot $K$, denote by $\GD_K(t)$ its Alexander polynomial. M.~Freedman
proved~\cite{Freedman-concordance} that if $\GD_K(t)=1$, then $K$ is
topologically locally-flatly slice. We used Knotscape~\cite{Knotscape} to
list all knots with up to 16 crossings that have $\GD=1$. In total, there are
$699$ such knots. The first one of them has 11 crossings (see
Table~\ref{tbl:num-non-slice}).

\begin{table}[ht]
\def\arraystretch{1.2}
$$
\begin{tabular}{|l||*{6}{c|}}
\hline
Number of crossings&11&12&13&14&15&16 \\
\hline\hline
Number of non-alternating knots&185&888&5110&27436&168030&1008906 \\
\hline
\quad among them with $\GD=1$&2&2&15&36&145&499 \\
\hline
\qquad among them with $s\not=0$&0&0&1&1&15&65(67??) \\
\hline
\end{tabular}
$$
\caption{Number of knots that have Alexander polynomial 1 and those among them
that have non-zero Rasmussen invariant.}
\label{tbl:num-non-slice}
\end{table}

Next, we used {\tt KhoHo}, a program for computing and studying Khovanov
homology~\cite{Me-KhoHo}, to find the homology of all the knots with
$\GD=1$. Fortunately for us, most of the knots considered (in particular, all
knots with at most $15$ crossings) have homological width $3$.  42 knots with
16 crossings have homological width $4$, but satisfy the condition
of~\ref{prop:s-from-Kh}. Hence, we can deduce the Rasmussen invariant of these
knots from their homology using~\eqref{eq:s-equation}.

There are only two knots with 16 crossings and $\GD=1$, for
which~\ref{prop:s-from-Kh} can not be applied. They are $16n_{864894}$ and
$16n_{925408}$ (see Table~\ref{tbl:16n_864894-knot}). Their Rasmussen
invariant can either be $0$ or $2$. Hence, the last entry in
Table~\ref{tbl:num-non-slice} can range from 65 to 67.

\begin{table}[ht]
$$
\begin{tabular}{|l|c|c|c|c||l|c|c|c|c|}
\hline
knot $K$& $\!s(K)\!$& $\!hw(K)\!$& $\!e(K)\!$& $\!E(K)\!$&
knot $K$& $\!s(K)\!$& $\!hw(K)\!$& $\!e(K)\!$& $\!E(K)\!$\\
\hline\hline
$13n_{1496}$  &  $2$& $3$&  $0$& $8$& $14n_{7708}$  &  $2$& $3$&  $0$& $8$\\
\hline
$15n_{28998}$ &  $2$& $3$& $-4$& $8$& $15n_{87941}$ &  $2$& $3$& $-2$& $8$ \\
$15n_{40132}$ &  $2$& $3$&  $0$& $8$& $15n_{89822}$ & $-2$& $3$& $-8$& $0$ \\
$15n_{52282}$ &  $2$& $3$&  $0$& $6$&
$\mathbf{15n_{113775}}$\!&  $\mathbf2$& $3$&  $\mathbf2$& $12$\\
$15n_{54221}$ & $-2$& $3$& $-8$& $2$& $15n_{132396}$&  $2$& $3$&  $0$& $6$ \\
$15n_{58433}$ &  $2$& $3$& $-2$& $8$& $15n_{139256}$&  $2$& $3$&  $0$& $10$\\
$15n_{58501}$ &  $2$& $3$& $-2$& $8$& $15n_{145981}$& $-2$& $3$& $-8$& $0$ \\
$15n_{65084}$ &  $2$& $3$&  $0$& $8$& $15n_{165398}$&  $2$& $3$&  $0$& $6$ \\
$15n_{65980}$ & $-2$& $3$& $-8$& $2$&               &     &    &     &     \\
\hline
$\mathbf{16n_{412372}}$\!& $\mathbf{-2}$& $3$&$-12$&$\mathbf{-2}$&
$\mathbf{16n_{955859}}$\!&  $\mathbf2$& $3$&  $\mathbf2$&$12$ \\
\hline
\end{tabular}
$$
\caption{All knots with at most 15 crossings that have Alexander polynomial 1
and non-zero Rasmussen invariant. 2 such knots with 16 crossings that can be
strongly quasipositive or a mirror image thereof.}
\label{tbl:non-slice}
\end{table}

As it turns out, 82 (possibly 84) knots with up to 16 crossings have Alexander
polynomial 1 and non-zero Rasmussen invariant. Theorem~\ref{thm:Rasmussen-main}
implies the following.
\begin{propos}\label{prop:not-slice}
All knots with up to 15 crossings from Table~\ref{tbl:non-slice} as well as 65
knots with 16 crossings are topologically locally-flatly slice, but not
smoothly slice.
\end{propos}

These 65 knots with 16 crossings have numbers 2601, 4787, 10734, 15919, 35456,
38567, 54888, 55405, 63905, 64312, 85435, 88272, 95001, 100099, 146445,
196836, 201101, 205822, 211749, 213930, 225414, 231486, 233317, 247683,
247710, 249903, 253331, 271353, 281590, 287865, 322069, 345376, 355871,
359271, 367431, 380325, 383790, 412372, 418128, 432810, 446116, 464148,
470729, 487352, 499458, 528093, 538818, 542632, 548142, 591990, 596477,
637428, 644951, 696243, 707728, 738706, 740175, 762813, 762859, 768960,
809057, 817682, 842714, 884475, and 955859.

At least three of these examples were already known to be non-slice. The knots
$13n_{1496}$ and $16n_{955859}$ were considered by Alexander
Stoimenow in~\cite[Example~1\footnote{Stoimenow arranges all the knots with a
given number of crossings into a single list. Hence, the knot $13n_{1496}$ has
number $1496+4878=6374$ in~\cite{Stoimenow-example}, where 4878 is the number
of alternating knots with 13 crossings.}]{Stoimenow-example}
and~\cite[Example~4.1]{Stoimenow-sliceness}. The knot $15n_{113775}$ is the
$(-3,5,7)$-pretzel knot. It was shown to be non-slice by
Rudolph~\cite{Rudolph-quasipositivity}. Another his example, the untwisted
positive double of the trefoil, has 19 crossings. Hence, it is not in the
table.

In general, all knots that are either strongly quasipositive or the mirror
image of a strongly quasipositive knot are not slice~\cite{Rudolph-sliceness}.
Let us show that only two more knots with up to 16 crossings can (potentially)
fall into this category.

Let $P_K(v,z)$ be a HOMFLY polynomial of $K$, that is, the polynomial in
variables $v$ and $z$ determined by the skein relations
\begin{equation}\label{eq:HOMFLY-skein}
v^{-1}P_{\includegraphics[scale=0.45]{pos_Xing-black}}
-vP_{\includegraphics[scale=0.45]{neg_Xing-black}}
=zP_{\includegraphics[scale=0.45]{smooth_Xing-black}};
\qquad
P_{\includegraphics[scale=0.45]{circle-black}}=1.
\end{equation}
Let $e(K)$ and $E(K)$ be the lowest and highest exponents of $v$ in $P_K$,
respectively.

\begin{lem}[{\cite[Theorem 1]{Morton-HOMFLY}}]\label{lem:Morton-degree}
Let $K$ be a knot represented by a diagram $D$. Then
\begin{equation}
w(D)-O(D)+1\le e(K)\le E(K)\le w(D)+O(D)-1.
\end{equation}
\end{lem}

\begin{lem}[cf.~\cite{Stoimenow-example}, Lemma 1]
1.~If $K$ is a (strongly) quasipositive knot, then
\begin{equation}
0\le s(K)=2g_4(K)\le e(K).
\end{equation}
\noindent
2.~If $K$ is the mirror image of a (strongly) quasipositive knot, then
\begin{equation}
E(K)\le s(K)\le 0.
\end{equation}
\end{lem}
\begin{proof}
Part 1 follows from~\ref{prop:s-for-quasipositive} and~\ref{lem:Morton-degree}
immediately. For part 2 notice, that $w(\overline D)=-w(D)$, $O(\overline
D)=O(D)$, where $\overline D$ is the mirror image of $D$, and that
$s(\overline K)=-s(K)$ (see~\cite[Proposition~3.11]{Rasmussen-s}).
\end{proof}

\begin{cor}
Among all knots from Proposition~\ref{prop:not-slice}, only knots
$15n_{113775}$ and $16n_{955859}$ can be strongly quasipositive and only
$16n_{412372}$ can be the mirror image of a strongly quasipositive knot (see
Table~\ref{tbl:non-slice}). $15n_{113775}$ is the $(-3,5,7)$-pretzel knot and
is indeed strongly quasipositive (see Figure~\ref{fig:3-5-7-pretzel}).
\end{cor}

\subsection*{Acknowledgments}
I am very grateful to Mikhail Khovanov for bringing to my attention the remark
by Justin Roberts and Peter Teichner that the Rasmussen invariant shows that
certain knots with trivial Alexander polynomial are not smoothly
slice~\cite{Roberts-Teichner}. I started to work on this paper in an attempt
to understand their remark. I also thank Sebastian Baader for teaching me
quasipositive braids and knots.

% To avoid names being indecently stretched
\def\bibbox{\leavevmode\hbox}
\raggedright

\begin{thebibliography}{99999}
\bibitem{BarNatan-categor}
\bibbox{D. Bar-Natan,} {\sl On Khovanov's categorification of the Jones
    polynomial}, Algebr. Geom. Topol. {\bf 2} (2002), 337--370;
    arXiv:math.QA/0201043.

\bibitem{Freedman-concordance}
\bibbox{M. Freedman,} {\sl A surgery sequence in dimension four; the relations
    with knot concordance}, Invent. Math. {\bf 68} (1982), no. 2, 195-226.

\bibitem{Knotscape}
\bibbox{J. Hoste and M. Thistlethwaite,} {\tt Knotscape} --- a program for
    studying knot theory and providing convenient access to tables of
    knots, {\tt http://www.math.utk.edu/\~{}morwen/knotscape.html}

\bibitem{Khovanov-categor}
\bibbox{M. Khovanov,} {\sl A categorification of the Jones polynomial},
    Duke Math. J. {\bf 101} (2000), no. 3, 359--426; arXiv:math.QA/9908171.

\bibitem{Lee-alt-links}
\bibbox{E. S. Lee,} {\sl On Khovanov invariant for alternating links},
    arXiv:math.GT/0210213.

\bibitem{Livingston-tau}
\bibbox{C. Livingston,} {\sl Computations of the Ozsv\'ath-Szab\'o knot
    concordance invariant}, Geom. Topol. {\bf 8} (2004), 735--742;
    arXiv:math.GT/0311036.

\bibitem{Morton-HOMFLY}
\bibbox{H. Morton,} {\sl Seifert circles and knot polynomials}, Math. Proc.
    Camb. Phil. Soc. {\bf 99} (1986), 107--109;

\bibitem{Ozsvath-Szabo-tau}
\bibbox{P. Ozsv\'ath and Z. Szab\'o,} {\sl Knot Floer homology and the
    four-ball genus}, Geom. Topol. {\bf 7} (2003), 615--639;
    arXiv:math.GT/0301149.

\bibitem{Plamenevskaya-transverse}
\bibbox{O. Plamenevskaya,} {\sl Transverse knots and Khovanov homology},
    preprint, 2004.

\bibitem{Rasmussen-s}
\bibbox{J. Rasmussen,} {\sl Khovanov homology and the slice genus},
    arXiv:math.GT/0402131.

\bibitem{Roberts-Teichner}
\bibbox{J. Roberts and P. Teichner,} UCSD Quantum Topology in Dimension Four
    seminar, Spring 2004,
    {\tt http://math.ucsd.edu/\~{}justin/khovseminar.html}

\bibitem{Rudolph-quasipositivity}
\bibbox{L. Rudolph,} {\sl Quasipositivity as an obstruction to sliceness},
    Bull Amer. Math. Soc. (N.S.) {\bf 29} (1993), no. 1, 51--59;
    arXiv:math.GT/9307233.

\bibitem{Rudolph-sliceness}
\bibbox{L. Rudolph,} {\sl An obstruction to sliceness via contact geometry
    and ``classical'' gauge theory}, Invent. Math. {\bf 119} (1995), no. 1,
    155-163.

% \bibitem{Rudolph-pretzels}
% \bibbox{L. Rudolph,}  {\sl Quasipositive pretzels}, Top. Appl. {\bf 115}
%     (2001), no. 1, 115--123; arXiv:math.GT/9908028.

\bibitem{Me-KhoHo}
\bibbox{A. Shumakovitch,} \KhoHo --- a program for computing and studying
    Khovanov homology, {\tt http://www.geometrie.ch/KhoHo}

\bibitem{Stoimenow-example}
\bibbox{A. Stoimenow,} {\sl On polynomials and surfaces of variously
    positive links}, arXiv:math.GT/0202226.

\bibitem{Stoimenow-sliceness}
\bibbox{A. Stoimenow,} {\sl Some examples related to knot sliceness},
    preprint, 2004.

\end{thebibliography}

\end{document}

math0411643.extracted_bib
\begin{thebibliography}{99999}
\bibitem{BarNatan-categor}
\bibbox{D. Bar-Natan,} {\sl On Khovanov's categorification of the Jones
    polynomial}, Algebr. Geom. Topol. {\bf 2} (2002), 337--370;
    arXiv:math.QA/0201043.

\bibitem{Freedman-concordance}
\bibbox{M. Freedman,} {\sl A surgery sequence in dimension four; the relations
    with knot concordance}, Invent. Math. {\bf 68} (1982), no. 2, 195-226.

\bibitem{Knotscape}
\bibbox{J. Hoste and M. Thistlethwaite,} {\tt Knotscape} --- a program for
    studying knot theory and providing convenient access to tables of
    knots, {\tt http://www.math.utk.edu/\~{}morwen/knotscape.html}

\bibitem{Khovanov-categor}
\bibbox{M. Khovanov,} {\sl A categorification of the Jones polynomial},
    Duke Math. J. {\bf 101} (2000), no. 3, 359--426; arXiv:math.QA/9908171.

\bibitem{Lee-alt-links}
\bibbox{E. S. Lee,} {\sl On Khovanov invariant for alternating links},
    arXiv:math.GT/0210213.

\bibitem{Livingston-tau}
\bibbox{C. Livingston,} {\sl Computations of the Ozsv\'ath-Szab\'o knot
    concordance invariant}, Geom. Topol. {\bf 8} (2004), 735--742;
    arXiv:math.GT/0311036.

\bibitem{Morton-HOMFLY}
\bibbox{H. Morton,} {\sl Seifert circles and knot polynomials}, Math. Proc.
    Camb. Phil. Soc. {\bf 99} (1986), 107--109;

\bibitem{Ozsvath-Szabo-tau}
\bibbox{P. Ozsv\'ath and Z. Szab\'o,} {\sl Knot Floer homology and the
    four-ball genus}, Geom. Topol. {\bf 7} (2003), 615--639;
    arXiv:math.GT/0301149.

\bibitem{Plamenevskaya-transverse}
\bibbox{O. Plamenevskaya,} {\sl Transverse knots and Khovanov homology},
    preprint, 2004.

\bibitem{Rasmussen-s}
\bibbox{J. Rasmussen,} {\sl Khovanov homology and the slice genus},
    arXiv:math.GT/0402131.

\bibitem{Roberts-Teichner}
\bibbox{J. Roberts and P. Teichner,} UCSD Quantum Topology in Dimension Four
    seminar, Spring 2004,
    {\tt http://math.ucsd.edu/\~{}justin/khovseminar.html}

\bibitem{Rudolph-quasipositivity}
\bibbox{L. Rudolph,} {\sl Quasipositivity as an obstruction to sliceness},
    Bull Amer. Math. Soc. (N.S.) {\bf 29} (1993), no. 1, 51--59;
    arXiv:math.GT/9307233.

\bibitem{Rudolph-sliceness}
\bibbox{L. Rudolph,} {\sl An obstruction to sliceness via contact geometry
    and ``classical'' gauge theory}, Invent. Math. {\bf 119} (1995), no. 1,
    155-163.

% \bibitem{Rudolph-pretzels}
% \bibbox{L. Rudolph,}  {\sl Quasipositive pretzels}, Top. Appl. {\bf 115}
%     (2001), no. 1, 115--123; arXiv:math.GT/9908028.

\bibitem{Me-KhoHo}
\bibbox{A. Shumakovitch,} \KhoHo --- a program for computing and studying
    Khovanov homology, {\tt http://www.geometrie.ch/KhoHo}

\bibitem{Stoimenow-example}
\bibbox{A. Stoimenow,} {\sl On polynomials and surfaces of variously
    positive links}, arXiv:math.GT/0202226.

\bibitem{Stoimenow-sliceness}
\bibbox{A. Stoimenow,} {\sl Some examples related to knot sliceness},
    preprint, 2004.

\end{thebibliography}