## Algebraic & Geometric Topology

### Bottom tangles and universal invariants

Kazuo Habiro

#### 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 $ModH$ 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 $ModH$.

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.

#### Article information

Source
Algebr. Geom. Topol., Volume 6, Number 3 (2006), 1113-1214.

Dates
Accepted: 4 May 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796574

Digital Object Identifier
doi:10.2140/agt.2006.6.1113

Mathematical Reviews number (MathSciNet)
MR2253443

Zentralblatt MATH identifier
1130.57014

#### Citation

Habiro, Kazuo. Bottom tangles and universal invariants. Algebr. Geom. Topol. 6 (2006), no. 3, 1113--1214. doi:10.2140/agt.2006.6.1113. https://projecteuclid.org/euclid.agt/1513796574

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