Bottom tangles and universal invariants

Abstract

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 $B$ of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of $B$, 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 $H$, we define a braided functor $J$ from $B$ to the category ${Mod}_H$ of left $H$–modules. The functor $J$, together with the set of generators of $B$, provides an algebraic method to study the range of quantum invariants of links. The braided Hopf algebra action on bottom tangles is mapped by $J$ to the standard braided Hopf algebra structure for $H$ in ${Mod}_H$.

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 $B$. The functor $J$ provides a convenient way to study the relationships between these notions and quantum invariants.