A holonomic knot is a knot in 3–space which arises as the 2–jet extension of a
smooth function on the circle. A holonomic knot associated to a generic
function is naturally framed by the blackboard framing of the knot diagram
associated to the 1–jet extension of the function. There are two classical
invariants of framed knot diagrams: the Whitney index (rotation number)
and the self linking number
. For a framed holonomic
knot we show that
is bounded above by the negative of the braid index of the knot, and that the sum of
and
is bounded
by the negative of the Euler characteristic of any Seifert surface of the knot. The invariant
restricted to framed
holonomic knots with ,
is proved to split into ,
where is the largest
natural number with ,
integer invariants. Using this, the framed holonomic isotopy classification of framed
holonomic knots is shown to be more refined than the regular isotopy classification of
their diagrams.
Keywords
framing, holonomic knot, Legendrian knot, self-linking
number, Whitney index