Abstract: This is a joint work with Gili Golan. Recently Vaughan Jones showed that Thompson's group $F$ encodes in a natural way all knots and links in $\mathbb R^3$, and a certain subgroup $\overrightarrow F$ of $F$ encodes all oriented knots. We answer several questions of Jones about $\overrightarrow F$. In particular we prove that the subgroup $\overrightarrow F$ is generated by $x_0x_1, x_1x_2, x_2x_3$ (where $x_i,i\in \mathbb N$ are the standard generators of $F$) and is isomorphic to $F_3$, the analog of $F$ where all slopes are powers of $3$ and break points are $3$-adic rationals. We also show that $\overrightarrow F$ coincides with its commensurator. Hence the linearization of the permutational representation of $F$ on $F/\overrightarrow F$ is irreducible. Finally we show how to replace $3$ in the above results by an arbitrary $n$, and to construct a series of irreducible representations of $F$ defined in a similar way.