Algebraic & Geometric Topology

Bottom tangles and universal invariants

Kazuo Habiro

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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

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

Received: 20 December 2005
Accepted: 4 May 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]

knots links tangles braided categories ribbon Hopf algebras braided Hopf algebras universal link invariants transmutation local moves Hennings invariants bottom tangles claspers


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

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