A bottom tangle is a tangle in a cube consisting only of arc components,
each of which has the two endpoints on the bottom line of the
cube, placed next to each other. We introduce a subcategory
of the category
of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of
generators of ,
which provides an especially convenient way to generate all the bottom tangles, and
hence all the framed, oriented links, via closure. We also define a kind of “braided
Hopf algebra action” on the set of bottom tangles.
Using the universal invariant of bottom tangles associated to each ribbon Hopf algebra
, we define a
braided functor
from to the
category of left
–modules. The functor
, together with the
set of generators of ,
provides an algebraic method to study the range of quantum invariants of
links. The braided Hopf algebra action on bottom tangles is mapped
by
to the standard braided Hopf algebra structure for
in
.
Several notions in knot theory, such as genus, unknotting number, ribbon knots,
boundary links, local moves, etc are given algebraic interpretations in the setting involving
the category .
The functor
provides a convenient way to study the relationships between these notions and
quantum invariants.
