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2006 Bottom tangles and universal invariants
Kazuo Habiro
Algebr. Geom. Topol. 6(3): 1113-1214 (2006). DOI: 10.2140/agt.2006.6.1113


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.


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Kazuo Habiro. "Bottom tangles and universal invariants." Algebr. Geom. Topol. 6 (3) 1113 - 1214, 2006.


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

zbMATH: 1130.57014
MathSciNet: MR2253443
Digital Object Identifier: 10.2140/agt.2006.6.1113

Primary: 57M27
Secondary: 18D10 , 57M25

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

Rights: Copyright © 2006 Mathematical Sciences Publishers


Vol.6 • No. 3 • 2006
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