===== recycled on Tue 31 Dec 2024 06:54:09 AM EST by drorbn on Ubuntu-on-X2 ====== \includegraphics[width=0.19\linewidth]{figs/PP300.png} \includegraphics[width=0.19\linewidth]{figs/PP301.png} \includegraphics[width=0.19\linewidth]{figs/PP302.png} \includegraphics[width=0.19\linewidth]{figs/PP303.png} \includegraphics[width=0.19\linewidth]{figs/PP304.png} \includegraphics[width=0.19\linewidth]{figs/PP305.png} \includegraphics[width=0.19\linewidth]{figs/PP306.png} \includegraphics[width=0.19\linewidth]{figs/PP307.png} \includegraphics[width=0.19\linewidth]{figs/PP308.png} \includegraphics[width=0.19\linewidth]{figs/PP309.png} \includegraphics[width=0.19\linewidth]{figs/PP310.png} \includegraphics[width=0.19\linewidth]{figs/PP311.png} \includegraphics[width=0.19\linewidth]{figs/PP312.png} \includegraphics[width=0.19\linewidth]{figs/PP313.png} \includegraphics[width=0.19\linewidth]{figs/PP317.png} ===== recycled on Tue 31 Dec 2024 07:31:33 AM EST by drorbn on Ubuntu-on-X2 ====== The formulas for $\theta$ depend on three fixed polynomials $F_1(c)$, $F_2(c_0,c_1)$ and $F_3(\varphi,k)$ in the $g_{\nu\alpha\beta}$'s, which we admit, are rather ugly. So we prefer to assert their existance and postpone displaying them to a few paragraphs later. ===== recycled on Thu 02 Jan 2025 04:08:35 AM EST by drorbn on Ubuntu-on-X2 ====== \begin{multline} \label{eq:F1} F_1(c) = s \left[ 1/2 - g_{3ii} + T_2^s g_{1ii} g_{2ji} - T_2^s g_{3jj} g_{2ji} - (T_2^s-1) g_{3ii} g_{2ji} \right. \\ \left. + (T_3^s-1) g_{2ji} g_{3ji} - g_{1ii} g_{2jj} + 2 g_{3ii} g_{2jj} + g_{1ii} g_{3jj} - g_{2ii} g_{3jj} \right] \\ + \frac{s}{T_2^s-1} \left[ (T_1^s-1)T_2^s \left( g_{3jj} g_{1ji} - g_{2jj} g_{1ji} + T_2^s g_{1ji} g_{2ji} \right) \right. \\ + (T_3^s-1) \left( g_{3ji} - T_2^s g_{1ii} g_{3ji} + g_{2ij} g_{3ji} + (T_2^s-2) g_{2jj} g_{3ji} \right) \\ \left. - (T_1^s-1) (T_2^s+1) (T_3^s-1) g_{1ji} g_{3ji} \right] \end{multline} \begin{equation} \label{eq:F2} F_2(c_0,c_1) = \frac{s_1 (T_1^{s_0}-1) (T_3^{s_1}-1) g_{1j_1i_0} g_{3j_0i_1}}{T_2^{s_1}-1} \left(T_2^{s_0} g_{2i_1i_0}+g_{2j_1j_0} - T_2^{s_0} g_{2j_1i_0}-g_{2i_1j_0} \right) \end{equation} \begin{equation} \label{eq:F3} F_3(\varphi,k) = \varphi(g_{3kk}-1/2) \end{equation} ===== recycled on Sat 11 Jan 2025 02:07:36 PM EST by drorbn on Ubuntu-on-X2 ====== \begin{technicality} \label{tech:nonseq} Some Reidemeister moves create or lose an edge and to avoid the need for renumbering it is beneficial to also allow labelling the edges with non-consecutive labels. Hence we allow that, and write $\ip$ for the successor of the label $i$ along the knot, and $\ipp$ for the successor of $\ip$ (these are $i+1$ and $i+2$ if the labelling is by consecutive integers). Also, by convention ``$1$'' will always refer to the label of the first edge, and ``$2n+1$'' will always refer to the label of the last. \endpar{\ref{tech:nonseq}} \end{technicality} ===== recycled on Sat 11 Jan 2025 02:07:55 PM EST by drorbn on Ubuntu-on-X2 ====== \begin{clarification} In Theorem~\ref{thm:RelativeInvariant}, $\alpha$ and $\beta$ stand for {\em edges} and not for their serial numbers, which may change if edges are numbered sequencialy when a Reidemeister move that changes the overall number of crossings is performed. It is for this reason that we've introdcued Technicality~\ref{tech:nonseq}. \end{clarification} ===== recycled on Sat 11 Jan 2025 02:07:59 PM EST by drorbn on Ubuntu-on-X2 ====== \begin{proviso} A further minor issue arrises if $\alpha$ and $\beta$ are distinct before a Reidemeister move is performed but become the same after the move (see an illustration on the right). \end{proviso} ===== recycled on Fri 24 Jan 2025 01:39:26 PM EST by drorbn on Ubuntu-on-X2 ====== Indeed Reidemeister moves may changes the indexing $\alpha$ and $\beta$ of edges even when those edges are far from the move location, while it makes sense to keep points like $a$ and $b$ in place when the moves are away. Furthemore and more importantly, the distinction between $ab$ when $a$ and $b$ are on the same edge matters below. ===== recycled on Thu 21 Aug 2025 02:14:11 PM EDT by drorbn on Ubuntu-on-X2 ====== The setup leading to the definition of $\Theta$ is the same as the setup leading to the definition of the invariant $\rho_1$ of~\cite{APAI}, and hence we copy a few relevant paragraphs from~\cite{APAI} nearly verbatim, with only a few modifications. ===== recycled on Fri 22 Aug 2025 11:57:18 AM EDT by drorbn on Ubuntu-on-X2 ====== $\begin{pmatrix} 1 & -1/2 \\ 0 & \sqrt{3}/2 \end{pmatrix}\begin{pmatrix} n_1 \\ n_2 \end{pmatrix}$ ===== recycled on Mon 25 Aug 2025 06:33:18 AM EDT by drorbn on Ubuntu-on-X2 ====== \[ g'_{\alpha\beta} = \begin{array}{|c|ccc|} \hline & \beta=j & \beta=k & \beta\not\in\{j,k\} \\ \hline \alpha=j & g_{ii} & g_{ii} & g_{i\beta} \\ \alpha=k & g_{ii}-1 & g_{ii} & g_{i\beta} \\ \alpha\not\in\{j,k\} & g_{\alpha i} & g_{\alpha i} & g_{\alpha\beta} \\ \hline \end{array}. \] \[ \def\tif{{\text{if }}} g'_{\alpha\beta} = \left\{\begin{array}{cccl} \tif \beta=j & \tif \beta=k & \tif \beta\not\in\{j,k\} & \\ g_{ii} & g_{ii} & g_{i\beta} & \tif \alpha=j \\ g_{ii}-1 & g_{ii} & g_{i\beta} & \tif \alpha=k \\ g_{\alpha i} & g_{\alpha i} & g_{\alpha\beta} & \tif \alpha\not\in\{j,k\} \\ \end{array}\right. \] ===== recycled on Mon 25 Aug 2025 08:48:19 AM EDT by drorbn on Ubuntu-on-X2 ====== \noindent{\em Proof.} The proof is essentially contained in the caption of Figure~\ref{fig:UprightRMoves}. A more detailed version is in~\cite{BecerraVanHelden:RotationalReidemeister}. \qed ===== recycled on Thu 28 Aug 2025 02:24:15 AM EDT by drorbn on Ubuntu-on-X2 ====== \needspace{34mm} % 33mm is not enough. \parpic[r]{\resizebox{0.43\linewidth}{!}{\input{figs/GST48-Marked.pdf_t}}} ===== recycled on Thu 28 Aug 2025 08:34:29 AM EDT by drorbn on Ubuntu-on-X2 ====== \begin{theorem}[The Main Theorem, proof in Section~\ref{sec:Proof}] \label{thm:Main} The following is a knot invariant: \begin{equation} \label{eq:Main} \theta(D) \coloneqq \Delta_1\Delta_2\Delta_3 \left(\sum_{c\in X} F_1(c) + \sum_{c_0,c_1\in X} F_2(c_0,c_1) + \sum_{\text{edges }k}F_3(k)\right). \end{equation} \end{theorem} ===== recycled on Thu 28 Aug 2025 11:15:22 AM EDT by drorbn on Ubuntu-on-X2 ====== By the invariance of the Alexander polynomial, the pre-factor $\Delta_1\Delta_2\Delta_3$ is the same for $\theta(D^l)$ and for $\theta(D^r)$ (see Equation~\eqref{eq:Main}). ===== recycled on Fri 29 Aug 2025 09:17:47 AM EDT by drorbn on Ubuntu-on-X2 ====== \begin{conjecture} \label{conj:D6} $\theta$ has hexagonal symmetry. That is, for any knot $K$, we have that %$\theta(T_1,T_2) = \theta(T_1,T_1^{-1}T_2^{-1})$ $\theta = \theta|_{T_1\to T_1,T_2\to T_1^{-1}T_2^{-1}}$ (``the QR code is invariant under reflection about a horizontal line''), and %$\theta(T_1,T_2) = \theta(T_1T_2,T_2^{-1})$ $\theta = \theta|_{T_1\to T_1T_2, T_2\to T_2^{-1}}$ (``the QR code is invariant under reflection about the line of slope $30^\circ$''). \end{conjecture}