\documentclass[11pt,notitlepage]{article} \usepackage{amsmath,graphicx,amssymb,xcolor,stmaryrd,datetime,amscd,txfonts,picins} % Following http://tex.stackexchange.com/a/847/22475: \usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks, linkcolor={green!50!black}, citecolor={green!50!black}, urlcolor=blue } \def\sheeturl{{\url{http://drorbn.net/AcademicPensieve/2014-07/}}} \def\green{\color{green}} \def\red{\color{red}} \def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}} \def\diag{{\operatorname{diag}}} \def\remove{\!\setminus\!} \def\yellow#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{#1}}} \paperwidth 8in \paperheight 10.5in \textwidth 8in \textheight 10.5in \oddsidemargin -0.75in \evensidemargin \oddsidemargin \topmargin -0.75in \headheight 0in \headsep 0in \footskip 0in \parindent 0in \setlength{\topsep}{0pt} \pagestyle{empty} \dmyydate % Following http://tex.stackexchange.com/questions/23521/tabular-vertical-alignment-to-top: \def\imagetop#1{\vtop{\null\hbox{#1}}} \begin{document} \setlength{\jot}{0ex} \setlength{\abovedisplayskip}{0.5ex} \setlength{\belowdisplayskip}{0.5ex} \setlength{\abovedisplayshortskip}{0ex} \setlength{\belowdisplayshortskip}{0ex} {\LARGE\bf Cheat Sheet $\beta$}\hfill \parbox[b]{5.2in}{\tiny \null\hfill verification at {\tt 2014-06/CheatSheetBeta-Verification.nb}, {\tt 2014-05/GoodFormulas/Demo.nb} \newline\null\hfill \sheeturl; initiated 24/3/13; continues \href{http://drorbn.net/AcademicPensieve/2014-06/}{2014-06}; continued \href{http://drorbn.net/AcademicPensieve/2015-04/}{2015-04}; modified \today, \ampmtime } \vskip -10pt \rule{\textwidth}{1pt} \textbf{$\sigma$ calculus.} \hfill $\sigma_1\ast\sigma_2=\sigma_1\cup\sigma_2$, \quad $tm^{uv}_w=({T_u,T_v\to T_w})$, \quad $hm^{xy}_z\colon\sigma\mapsto(\sigma\remove\{x,y\})\cup(z\to\sigma_x\sigma_y),$ \quad $tha^{ux}=I,$ \quad $R^\pm_{ux}\mapsto T_u^{\pm 1}$ \vskip 1mm \textbf{$\beta$-calculus.} \hfill Constraints. $\bullet$ Sum of column $x$ is $\sigma_x-1$. \quad $\bullet$ At $T_\ast=1$, $\omega=1$ and $A=0$. \vskip 1mm $\displaystyle \begin{array}{c|c}\omega_1&H_1\\\hline T_1&A_1\end{array} \ast \begin{array}{c|c}\omega_2&H_2\\\hline T_2&A_2\end{array} \underset{\beta}{=} \begin{array}{c|cc} \omega_1\omega_2 & H_1 & H_2 \\ \hline T_1 & A_1 & 0 \\ T_2 & 0 & A_2 \end{array} $ \hfill $\displaystyle \begin{CD} \begin{array}{c|c} \omega & H \\ \hline u & \alpha \\ v & \beta \\ T & \Xi \end{array} @>tm^{uv}_w>\beta> \left(\!\begin{array}{c|c} \omega & H \\ \hline w & \alpha+\beta \\ T & \Xi \end{array}\!\right)_{T_u,T_v\to T_w} \end{CD} $ \hfill $\displaystyle \begin{CD} \begin{array}{c|ccc} \omega & x & y & H \\ \hline T & \alpha & \beta & \Xi \end{array} @>{hm^{xy}_z}>\beta> \begin{array}{c|cc} \omega & z & H \\ \hline T & \alpha+\sigma_x\beta & \Xi \end{array} \end{CD} $ \hfill$\displaystyle \begin{CD} \begin{array}{c|cc} \omega & x & H \\ \hline u & \alpha & \theta \\ T & \phi & \Xi \end{array} @>{tha^{ux}}>\beta\atop\nu\coloneqq 1+\alpha> % \begin{array}{c|cc} % \omega\epsilon & x & H \\ % \hline % u & \alpha(1+\langle\gamma\rangle/\epsilon) % & \beta(1+\langle\gamma\rangle/\epsilon) \\ % T & \gamma/\epsilon & \delta-\gamma\beta/\epsilon % \end{array} = \begin{array}{c|cc} \nu\omega & x & H \\ \hline u & \sigma_x\alpha/\nu & \sigma_x\theta/\nu \\ T & \phi/\nu & \Xi-\phi\theta/\nu \end{array} \end{CD} $ \hfill $\displaystyle \rho^\pm_{ux} \underset{\beta}{=} \begin{array}{c|c} 1 & x \\ \hline u & T_u^{\pm 1}-1 \end{array} $\hfill\null \rule{\textwidth}{1pt} \textbf{Gassner calculus $\Gamma$.} \hfill Preserves $C_1\coloneqq[\text{col sum}=1]$ ($\Leftrightarrow$OC) and $\green\checkmark$ $C_2\coloneqq[\forall a,b,\, (T_a-1)\mid(A_{ab}-\delta_{ab}\sigma_b)]$ \hfill $\displaystyle \begin{CD} \left(\!\begin{array}{c|cc} \nu\omega & c & S \\ \hline c & \beta+\alpha\delta/\nu & \theta+\alpha\epsilon/\nu \\ S & \psi+\delta\phi/\nu & \Xi+\epsilon\phi/\nu \end{array}\!\right)_{T_a,T_b\to T_c} \hspace{-7mm} @<{m^{ba}_{c}}<\nu\coloneqq 1-\gamma< \begin{array}{c|ccc} \omega & a & b & S \\ \hline a & \alpha & \beta & \theta \\ b & \gamma & \delta & \epsilon \\ S & \phi & \psi & \Xi \end{array} @>{m^{ab}_{c}}>\mu\coloneqq 1-\beta> \left(\!\begin{array}{c|cc} \mu\omega & c & S \\ \hline c & \gamma+\alpha\delta/\mu & \epsilon+\delta\theta/\mu \\ S & \phi+\alpha\psi/\mu & \Xi+\psi\theta/\mu \end{array}\!\right)_{T_a,T_b\to T_c} \hspace{-8mm} \end{CD} $ \hfill$R^{\pm}_{ab} \underset{\gamma}{=} \begin{array}{c|cc} 1 & a & b \\ \hline a & 1 & 1-T_a^{\pm 1} \\ b & 0 & T_a^{\pm 1} \end{array}$ $\displaystyle \begin{CD} \begin{array}{c|cc} \omega & a & S \\ \hline a & \alpha & \theta \\ S & \phi & \Xi \end{array} @>q\Delta^a_{bc}>{\mu\coloneqq T_a-1\atop\nu\coloneqq \alpha-\sigma_a}> \left(\begin{array}{c|ccc} \omega & b & c & S \\ \hline b & (\sigma_a-\alpha T_a-\nu T_c)/\mu & (T_b-1)T_c\nu/\mu & (T_b-1)T_c\theta/\mu \\ c & (T_c-1)\nu/\mu & (\alpha-\sigma_a T_a-\nu T_c)/\mu & (T_c-1)\theta/\mu \\ S & \phi & \phi & \Xi \end{array}\right)_{T_a\mapsto T_bT_c} \end{CD} $ \hfill\parbox{2.1in}{Satisfies: $\green\checkmark$ $R^+_{13}\act q\Delta^1_{12}=R^+_{23}\#R^+_{13}$. \newline$\green\checkmark$ $R^-_{13}\act q\Delta^1_{12}=R^-_{13}\#R^-_{23}$. \newline$\green\checkmark$ $q\Delta^a_{a_1a_2}\act q\Delta^b_{b_1b_2}\act m^{a_1b_1}_{c_1}\act m^{a_2b_2}_{c_2}$ \newline\null\quad$=m^{ab}_c\act q\Delta^c_{c_1c_2}$. } $\displaystyle \begin{CD} \begin{array}{c|cc} \omega & a & S \\ \hline a & \alpha & \theta \\ S & \phi & \Xi \end{array} @>dS^a>> \left(\begin{array}{c|cc} \alpha\omega/\sigma_a & a & S \\ \hline a & 1/\alpha & \theta/\alpha \\ S & -\phi/\alpha & (\alpha\Xi-\phi\theta)/\alpha \end{array}\right)_{T_a\to T_a^{-1}} \end{CD} $\hfill\parbox{4in}{Satisfies: $\green\checkmark$ $R^\pm_{12}\act dS^{1\text{ or }2}=R^\mp_{12}$. \hfill$\green\checkmark$ $dm^{ab}_c\act dS^c=dS^a\act dS^b\act dm^{ba}_c$. \newline$\green\checkmark$ $dS^a\act dS^a=I$. \hfill$\green\checkmark$ $q\Delta^a_{bc}\act dS^b\act dS^c=dS^a\act q\Delta^a_{cb}$. \newline$\green\checkmark$ Assuming $C_2$, $d\eta^a\act d\epsilon_a=q\Delta^a_{bc}\act dS^c\act dm^{bc}_a$ (also 3 variants). } The map (tangle $T$ $\mapsto$ matrix $A$) is anti-multiplicative. \hfill The MVA mod units: $L\mapsto(\omega,A) \mapsto \omega\det'(A-I)/(1-T')$\ ${\green\checkmark}$ \vskip -2mm \rule{\textwidth}{0.5pt} \vskip -1mm {\bf Burau.} On $b\in uB_n$, $Bu\colon\sigma_i^{\pm 1}\mapsto U_i^{\pm 1}$. \hfill{\bf Unitarity.} With $U=Bu(b)$, $\bar{U}\Omega_nU^T=\Omega_n$. {\bf Thm.} $\Gamma(b)=\begin{array}{c|ccc} 1&s_{b(1)}&s_{b(2)}&\cdots\\ \hline s_1&&&\\ s_2 &&\hspace{-3mm}Bu(b)^T\hspace{-3mm}&\\ \vdots&&& \end{array}$. \hfill $U_i=\begin{pmatrix} I_i&&& \\ &1-t&t& \\ &1&0& \\ &&&I_{n-i-1}\end{pmatrix}$, $U_i^{-1}=\begin{pmatrix} I_i&&& \\ &0&1& \\ &\bar{t}&1-\bar{t}& \\ &&&I_{n-i-1}\end{pmatrix}$, $\Omega_n=\begin{pmatrix} 1&0&\cdots&0 \\ 1-t&1&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 1-t&1-t&\cdots&1 \end{pmatrix}$ \vskip 0mm \rule{\textwidth}{1pt} Some matrices: $\begin{pmatrix} 1-t_i & 1 \\ t_i & 0 \end{pmatrix}$, $\left( \begin{array}{ccc} \frac{1}{1-t_1} & 0 & 0 \\ 1 & \frac{1}{1-t_2} & 0 \\ 1 & 1 & \frac{1}{1-t_3} \\ \end{array} \right)$, $\begin{pmatrix} 1-t_j & 1 \\ t_i & 0 \end{pmatrix}$, $\left( \begin{array}{ccc} -\frac{t_1-1}{t_1} & 0 & 0 \\ \frac{\left(t_1-1\right) \left(t_2-1\right)}{t_2} & -\frac{t_2-1}{t_2} & 0 \\ \frac{\left(t_1-1\right) \left(t_3-1\right)}{t_3} & \frac{\left(t_2-1\right) \left(t_3-1\right)}{t_3} & -\frac{t_3-1}{t_3} \\ \end{array} \right)$ \vfill {\bf To do.} $\bullet$ Full verification program. $\bullet$ R1? $\bullet$ Precise relation with Burau/Gassner. $\bullet$ Concordance. $\bullet$ Unitarity. $\bullet$ Planarity. $\bullet$ A depth-mirror property for u-objects. $\bullet$ Mutations? $\bullet$ Link relations? $\bullet$ Behaviour of A/MVA under mirror/strand reversal? \newpage \textbf{$\beta$-better calculus.} \hfill Constraints. $\bullet$ Sum of column $x$ is $(\sigma_x-1)w$. \quad $\bullet$ $\omega^{k-1}\mid\Lambda^kA$. \quad $\bullet$ At $T_\ast=1$, $\omega=1$ and $A=0$. \vskip 1mm $\displaystyle \begin{array}{c|c}\omega_1&H_1\\\hline T_1&A_1\end{array} \ast \begin{array}{c|c}\omega_2&H_2\\\hline T_2&A_2\end{array} \underset{\beta_b}{=} \begin{array}{c|cc} \omega_1\omega_2 & H_1 & H_2 \\ \hline T_1 & \omega_2A_1 & 0 \\ T_2 & 0 & \omega_1A_2 \end{array} $ \hfill $\displaystyle \begin{CD} \begin{array}{c|c} \omega & H \\ \hline u & \alpha \\ v & \beta \\ T & \gamma \end{array} @>tm^{uv}_w>\beta_b> \left(\begin{array}{c|c} \omega & H \\ \hline w & \alpha+\beta \\ T & \gamma \end{array}\right)_{T_u,T_v\to T_w} \end{CD} $ \hfill $\displaystyle \rho^\pm_{ux} \underset{\beta_b}{=} \begin{array}{c|cc} 1 & x \\ \hline u & T_u^{\pm 1}-1 \end{array} $ \vskip 1mm $\displaystyle \begin{CD} \begin{array}{c|ccc} \omega & x & y & H \\ \hline T & \alpha & \beta & \gamma \end{array} @>{hm^{xy}_z}>\beta_b> \begin{array}{c|cc} \omega & z & H \\ \hline T & \alpha+\sigma_x\beta & \gamma \end{array} \end{CD} $ \hfill $\displaystyle \begin{CD} \begin{array}{c|cc} \omega & x & H \\ \hline u & \alpha & \beta \\ T & \gamma & \delta \end{array} @>{tha^{ux}}>\beta_b> \begin{array}{c|cc} \omega+\alpha & x & H \\ \hline u & \sigma_x\alpha & \sigma_x\beta \\ T & \gamma & \delta+\frac{\alpha\delta-\gamma\beta}{\omega} \end{array} =: \begin{array}{c|c} \cdot & - \\ \hline \mid & \begin{pmatrix}\sigma_x&0\\0&1\end{pmatrix}\cdot A^{ux} \end{array} \end{CD} $ \[ \begin{CD} \begin{array}{c|ccc} \omega & a & b & S \\ \hline a & \alpha & \beta & \theta \\ b & \gamma & \delta & \epsilon \\ S & \phi & \psi & \Xi \end{array} @>{m^{ab}_{c}}>\beta_b\ {\green\checkmark}> \left(\begin{array}{c|cc} \omega+\beta & c & S \\ \hline c & \gamma+\sigma_a\delta+\sigma_b(\alpha+\sigma_a\beta) +\frac{\beta\gamma-\alpha\delta}{\omega} & \epsilon+\sigma_b\theta+\frac{\beta\epsilon-\delta\theta}{\omega} \\ S & \phi+\sigma_a\psi+\frac{\beta\phi-\alpha\psi}{\omega} & \Xi+\frac{\beta\Xi-\psi\theta}{\omega} \end{array}\right)_{T_a,T_b\to T_c} \end{CD} \] The MVA (mod units):\hfill $ n\text{-component }L\mapsto(\sigma,\omega,A) \mapsto\omega^{2-n}\det'(A-\omega\diag((\sigma_i-1))/(1-T') $ \hfill${\green\checkmark}$ \vskip -2mm \rule{\textwidth}{0.5pt} Note. $A^{ux}=\begin{pmatrix} \alpha & \beta \\ \gamma & \delta+\frac{\alpha\delta-\gamma\beta}{\omega} \end{pmatrix} = \begin{pmatrix} \alpha & \beta \\ \gamma & \frac{(\omega+\alpha)\delta-\gamma\beta}{\omega} \end{pmatrix} = \frac{1}{\omega}\left[ (\omega+\alpha)\begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} - \begin{pmatrix}\alpha\\\gamma\end{pmatrix} \begin{pmatrix}\alpha&\beta\end{pmatrix} \right] =\frac{1}{\omega}\left[(\omega+a_{ux})A-a_{\ast x}a_{u\ast}\right]. $ \vskip 1mm {\bf Claim.} $\omega^{k-1}\mid \Lambda^kA$ and $\omega^k\mid \Lambda^{k+1}A$ implies $(\omega+\alpha)^{k-1}\mid \Lambda^kA^{ux}$, with $\alpha=a_{ux}$. {\bf Proof.} With $\bar{u}\in T^k$ and $\bar{x}\in H^k$, $\omega^k$ divides $\left|\!\begin{array}{cc} \omega & 0 \\ 0 & a_{\bar{u}\bar{x}}\end{array}\!\right|$ and $\left|\!\begin{array}{cc} a_{ux} & a_{u\bar{x}} \\ a_{\bar{u}x} & a_{\bar{u}\bar{x}} \end{array}\!\right|$ and hence their sum, $\left|\!\begin{array}{cc} \omega+\alpha & a_{u\bar{x}} \\ a_{\bar{u}x} & a_{\bar{u}\bar{x}} \end{array}\!\right|=$\linebreak[5] $(\omega+\alpha)\left|\!\begin{array}{cc} 1 & 0 \\ 0 & a_{\bar{u}\bar{x}}-\frac{1}{\omega+\alpha}a_{\bar{u}x}a_{u\bar{x}} \end{array}\!\right|$ $=\frac{1}{(\omega+\alpha)^{k-1}}\left|(\omega+\alpha)a_{\bar{u}\bar{x}}-a_{\bar{u}x}a_{u\bar{x}}\right|$. So $\frac{1}{(\omega+\alpha)^{k-1}} \left|\frac{1}{\omega}\left[(\omega+\alpha)a_{\bar{u}\bar{x}}-a_{\bar{u}x}a_{u\bar{x}}\right]\right|$ is integral. \quad$\Box$ That is, with $A_{\bar{u};\bar{x}}$ denoting minors, if $\omega^{k-1}\mu_{\bar{u};\bar{x}}=A_{\bar{u};\bar{x}}$ and $\omega^k\mu_{u\bar{u};x\bar{x}}=A_{u\bar{u};x\bar{x}}$, then $(\omega+\alpha)^{k-1} (\mu_{\bar{u};\bar{x}}+\mu_{u\bar{u};x\bar{x}}) = A^{ux}_{\bar{u};\bar{x}}$. \vskip -8pt \rule{\textwidth}{1pt} \textbf{$\Lambda$-calculus.} $\Lambda(T;H)=R(T)\otimes\left(\Lambda(T)\otimes\Lambda(H)\right)_=$, with $R(T)$ Laurent polynomials in $\{T_u\}_{u\in T}$. \hfill$\lambda_1\ast\lambda_2 = \lambda_1(\wedge\otimes\wedge)\lambda_2$ $tm^{uv}_w:\ u,v\to w,\,T_u,T_v\to T_w$ \hfill$hm^{xy}_z:\ x\to z,\, y\to\sigma_xz$ \hfill$tha^{ux}:\lambda\mapsto(1+i_u\otimes i_x)\lambda\act(u\to\sigma_xu)$ \hfill$\rho^\pm_{ux}=1+(T_u^{\pm 1}-1)ux$ \rule{\textwidth}{1pt} {\bf Relations.} $\bullet$ $\rho^+_{ux}\rho^-_{vy}\act tm^{uv}_w\act hm^{xy}_z=t\epsilon_w h\epsilon_z$. \quad$\bullet$ $ \rho^{s_1}_{ux}\rho^{s_2}_{vy}\rho^{s_2}_{wz} \act tm^{vw}_v\act hm^{xy}_x\act tha^{ux} = \rho^{s_2}_{vx}\rho^{s_2}_{wz}\rho^{s_1}_{uy} \act tm^{vw}_v\act hm^{xy}_x. $ \end{document} \endinput