In summary, we have introduced and studied the following objects, spaces, and maps:

**Objects: **
is the space of all framed oriented knots in
an oriented
.
More precisely, it is the free
-module
generated by framed oriented knots in an oriented
.

**The ***n***-Cubes: **
is the free
-module generated
by framed oriented knots in an oriented
,
that have
precisely *n* double point singularities
as in
Figure 2, modulo the co-differentiability relation
of Definition 1.2,

**The Co-Derivative: ** The co-derivative
is the map
defined by

**The Cube Ladder and Finite
Type Invariants: **

**The ***n***-Symbols: ** The space of *n*-symbols
is the space
of *n*-chord diagrams, as in
Figure 3.

**The Relator Ladder: ** The relator ladder is the ladder

(see Equation 3), of singular knots with exactly one ``Topological Relator'' singularity as in Figure 6.

**The Primary
Integrability Constraints: ** The primary integrability constraints are the images of
the relators via the map *b*; that is, they are the Topological 4-Term
relations of Figure 4.

**The Relator Symbols and
the Symbol-Level Relations: ** The relator symbols are diagrams of the kind appearing in
Figure 7.

**The Once-Reduced Symbol
Space and Once Integrable Weight Systems: **

**The Inductive
Problem: **

**The Lifting Problem: **

**Generic Symbol-Level
Redundencies: **

**The Object-Level
Redundencies: **

**The Redundency
Problem: **