Learning to Unknot
- Gukov, Sergei et al
- arXiv:2010.16263CALT-2020-046CERN-TH-2020-179
| Examples of knots. From left to right: unknot ($0_1$), trefoil ($3_1$), figure-eight ($4_1$), $5_1$, and $5_2$. |
| Examples of knots. From left to right: unknot ($0_1$), trefoil ($3_1$), figure-eight ($4_1$), $5_1$, and $5_2$. |
| Examples of knots. From left to right: unknot ($0_1$), trefoil ($3_1$), figure-eight ($4_1$), $5_1$, and $5_2$. |
| Examples of knots. From left to right: unknot ($0_1$), trefoil ($3_1$), figure-eight ($4_1$), $5_1$, and $5_2$. |
| Examples of knots. From left to right: unknot ($0_1$), trefoil ($3_1$), figure-eight ($4_1$), $5_1$, and $5_2$. |
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| : Reidemeister moves. |
| Obtaining new knots as the sum of prime knots. This knot is the sum of the the knot $5_2$ (left) and the trefoil $3_1$ (right). |
| A braid $\sigma_1 \sigma_2^{-1} \sigma_1 \sigma_2^{-1}$ (left) and its closure (right). |
| A braid $\sigma_1 \sigma_2^{-1} \sigma_1 \sigma_2^{-1}$ (left) and its closure (right). |
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| Distribution on the number of crossings induced by our prior, and also by a uniform distribution on all knots of $9$ or fewer crossings. |
| Drawing $6455$ $N=9$ braids from our prior yields knots with $9$ or fewer crossings, $4664$ of which are prime. Plotted are the number of occurences of knots in the Rolfsen table for knots with $3$ through $9$ crossings, with mirrors counted for knots that are not self-mirror. |
| Drawing $6455$ $N=9$ braids from our prior yields knots with $9$ or fewer crossings, $4664$ of which are prime. Plotted are the number of occurences of knots in the Rolfsen table for knots with $3$ through $9$ crossings, with mirrors counted for knots that are not self-mirror. |
| Drawing $6455$ $N=9$ braids from our prior yields knots with $9$ or fewer crossings, $4664$ of which are prime. Plotted are the number of occurences of knots in the Rolfsen table for knots with $3$ through $9$ crossings, with mirrors counted for knots that are not self-mirror. |
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| Performance comparison between Reformer and feedforward network for different braid lengths $N$. Shaded regions are confidence intervals associated to hyperparameters listed in Table \ref{tab:decision_parameters} that are not displayed on the plots. |
| Performance comparison between Reformer and feedforward network for different braid lengths $N$. Shaded regions are confidence intervals associated to hyperparameters listed in Table \ref{tab:decision_parameters} that are not displayed on the plots. |
| Performance comparison between Reformer and feedforward network for different braid lengths $N$. Shaded regions are confidence intervals associated to hyperparameters listed in Table \ref{tab:decision_parameters} that are not displayed on the plots. |
| Performance comparison between Reformer and feedforward network for different braid lengths $N$. Shaded regions are confidence intervals associated to hyperparameters listed in Table \ref{tab:decision_parameters} that are not displayed on the plots. |
| A study of Reformer outputs for $N=12$ knots. \emph{Top Left:} Output distribution for knots. \emph{Top Right:} Output distribution for unknots. \emph{Bottom:} Correlation between network outputs and the maximum absolute value of Jones polynomial degrees. |
| A study of Reformer outputs for $N=12$ knots. \emph{Top Left:} Output distribution for knots. \emph{Top Right:} Output distribution for unknots. \emph{Bottom:} Correlation between network outputs and the maximum absolute value of Jones polynomial degrees. |
| A study of Reformer outputs for $N=12$ knots. \emph{Top Left:} Output distribution for knots. \emph{Top Right:} Output distribution for unknots. \emph{Bottom:} Correlation between network outputs and the maximum absolute value of Jones polynomial degrees. |
| Performance comparison of the TRPO, A3C and RW algorithms. Left: Fraction of unknots whose braid words could be reduced to the empty braid word as a function of initial braid word length $N$. Right: Average number of actions necessary to reduce the input braid word to the empty braid word as a function of $N$. |
| Comparison of the moves that are picked by an untrained agent vs a trained agent. |
| Knot or not? Five and ten crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Five and ten crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Five and ten crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Five and ten crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Five and ten crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Five and ten crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Five and ten crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Five and ten crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Five and ten crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Five and ten crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Five and ten crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Five and ten crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Fifteen and twenty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Fifteen and twenty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Fifteen and twenty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Fifteen and twenty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Fifteen and twenty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Fifteen and twenty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Fifteen and twenty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Fifteen and twenty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Fifteen and twenty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Fifteen and twenty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Fifteen and twenty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Fifteen and twenty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Twenty-five and thirty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Twenty-five and thirty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Twenty-five and thirty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Twenty-five and thirty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Twenty-five and thirty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Twenty-five and thirty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Twenty-five and thirty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Twenty-five and thirty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Twenty-five and thirty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Twenty-five and thirty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Twenty-five and thirty crossing in rows $1$-$2$ and $3$-$4$, respectively. |
| Knot or not? Twenty-five and thirty crossing in rows $1$-$2$ and $3$-$4$, respectively. |