Recently Vaughan Jones showed that the R. Thompson group F encodes in a natural way all knots and links in 608182562497" title="Click to view the MathML source">R3, and a certain subgroup 600d3fdff0d917cea640675eac1dcc7"> of F encodes all oriented knots and links. We answer several questions of Jones about 600d3fdff0d917cea640675eac1dcc7">. In particular we prove that the subgroup 600d3fdff0d917cea640675eac1dcc7"> is generated by x0x1, x1x2, 0799a40f3f671eec573" title="Click to view the MathML source">x2x3 (where xi, i∈N are the standard generators of F ) and is isomorphic to F3, the analog of F where all slopes are powers of 3 and break points are 3-adic rationals. We also show that 600d3fdff0d917cea640675eac1dcc7"> coincides with its commensurator. Hence the linearization of the permutational representation of F on is irreducible. 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. Finally we analyze Jones' construction and deduce that the Thompson index of a link is linearly bounded in terms of the number of crossings in a link diagram.