Bottom tangles and universal invariants.(English)Zbl 1130.57014

Summary: 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 $$\text{Mod}_H$$.
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.

MSC:

 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010) 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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References:

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