Recently Vaughan Jones showed that the R. Thompson group F encodes in a natural way all knots and links in md5=7d61cdbf8523b68be316608182562497" title="Click to view the MathML source">R3, and a certain subgroup md5=9600d3fdff0d917cea640675eac1dcc7"> of F encodes all oriented knots and links. We answer several questions of Jones about md5=9600d3fdff0d917cea640675eac1dcc7">. In particular we prove that the subgroup md5=9600d3fdff0d917cea640675eac1dcc7"> is generated by md5=4030a76762529c76cdef2f96cba665e6" title="Click to view the MathML source">x0x1, md5=4ddd595c51acc61ad3f3bdacb24f5b33" title="Click to view the MathML source">x1x2, md5=21efd567dc7a00799a40f3f671eec573" title="Click to view the MathML source">x2x3 (where md5=341d6a87b69736dc3469259e9ecfb8d2" title="Click to view the MathML source">xi, md5=4f7aeb42098acd4b4278ed06a0943379" title="Click to view the MathML source">i∈N are the standard generators of F ) and is isomorphic to md5=d2033ad416883e85f51b4d146c54f426" title="Click to view the MathML source">F3, the analog of F where all slopes are powers of 3 and break points are 3-adic rationals. We also show that md5=9600d3fdff0d917cea640675eac1dcc7"> coincides with its commensurator. Hence the linearization of the permutational representation of F on md5=e2e643d7b1a3688eef794be55cfaead5"> 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.