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Bottom tangles and universal invariants

Kazuo Habiro

TL;DR

The paper develops a comprehensive algebraic framework for studying quantum link invariants via bottom tangles, introducing the braided subcategory ${\mathsf B}$ of tangles and its action on bottom tangles. It defines the universal invariant J for bottom tangles using ribbon Hopf algebras, and establishes a braided functor to left $H$-modules that encodes invariant data. Core contributions include a finite generating set for ${\mathsf B}$, a concrete Hopf-algebra action on bottom tangles, and a detailed analysis of how local moves, clasper moves, and GV filtrations translate into algebraic statements about $J$ and its closures. The work also develops refinements like band-reembedding and a downstream treatment of bottom knots, connecting topological moves to algebraic operations and enabling systematic study of the range and behavior of quantum invariants across broad classes of links and tangles.

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.

Bottom tangles and universal invariants

TL;DR

The paper develops a comprehensive algebraic framework for studying quantum link invariants via bottom tangles, introducing the braided subcategory of tangles and its action on bottom tangles. It defines the universal invariant J for bottom tangles using ribbon Hopf algebras, and establishes a braided functor to left -modules that encodes invariant data. Core contributions include a finite generating set for , a concrete Hopf-algebra action on bottom tangles, and a detailed analysis of how local moves, clasper moves, and GV filtrations translate into algebraic statements about and its closures. The work also develops refinements like band-reembedding and a downstream treatment of bottom knots, connecting topological moves to algebraic operations and enabling systematic study of the range and behavior of quantum invariants across broad classes of links and tangles.

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.

Paper Structure

This paper contains 79 sections, 54 theorems, 275 equations, 46 figures.

Table of Contents

  1. Introduction
  2. Bottom tangles
  3. An approach to quantum link invariants using universal invariants of bottom tangles
  4. Universal link invariants associated to ribbon Hopf algebras
  5. Universal invariant of bottom tangles and their closures
  6. Reduction to the colored link invariant
  7. The category ${\mathsf B}$ acting on bottom tangles
  8. The braided functor ${\mathsf J} \colon {\mathsf B} \rightarrow {\mathsf{Mod}}_H$
  9. Generators of the braided category ${\mathsf B}$
  10. Hopf algebra action on bottom tangles
  11. Local moves
  12. Other applications
  13. Organization of the paper
  14. Braided categories and Hopf algebras
  15. Monoidal and braided categories
  16. ...and 64 more sections

Key Result

Theorem 1

As a braided subcategory of ${\mathcal{T}}$, ${\mathsf B}$ is generated by the object ${\mathsf b}$ and the morphisms

Figures (46)

  • Figure 1: (a) A $3$--component bottom tangle $T=T_1\cup T_2\cup T_3$. (b) The closure $\operatorname{cl}(T)=L_1\cup L_2\cup L_3$ of $T$.
  • Figure 2: (a) A morphism $T\in {\mathsf B} (3,2)$. (b) A bottom tangle $T'\in \operatorname{\sf BT}_3$. (c) The composition $TT'\in \operatorname{\sf BT}_2$.
  • Figure 3: The Borromean tangle $B$
  • Figure 4: A tangle $T\colon \downarrow \otimes \downarrow \otimes \uparrow \rightarrow \uparrow \otimes \downarrow \otimes \downarrow$
  • Figure 5: Here each string may be replaced with parallel strings.
  • ...and 41 more figures

Theorems & Definitions (120)

  • Remark 1.1
  • Theorem 1: thm:2
  • Theorem 2
  • Remark 1.2
  • Theorem 3: Consequence of lem:2 and thm:5
  • Theorem 4: Part of thm:24
  • Remark 1.3
  • Theorem 4.1
  • Lemma 4.1
  • proof
  • ...and 110 more