A universal Vassiliev invariant over the real numbers is a formal-series-valued knot invariant from which every real-valued Vassiliev invariant can be recovered by a linear functional of a finite-degree truncation. For oriented unframed knots, its values lie in the graded completion of the unframed algebra of chord diagrams over the real numbers.
More precisely, if 
is universal and 
is a Vassiliev invariant
of order at most 
, there is a linear functional 
into 
such that
 |
where 
retains the homogeneous components of degrees 0 through 
. The Kontsevich integral
provides such a universal invariant over the real numbers
(Bar-Natan 1995, Kneissler 1997). This formal invariant need not itself have any
fixed finite order as a scalar-valued invariant.
Universality does not by itself imply that Vassiliev invariants distinguish every pair of distinct knots.
