[Paper Review] Invariants of tangles with flat connections in their complements.I. Invariants and holonomy R-matrices

Rinat Kashaev, Nicolai Reshetikhin|ArXiv.org|Feb 20, 2002

Geometric and Algebraic Topology4 references6 citations

TL;DR

This paper introduces holonomy R-matrices to construct invariants of tangles equipped with flat G-bundle connections, where G is a simple complex algebraic group. It demonstrates that these invariants are gauge-invariant and, for links, descend to functions on the moduli space of flat connections, generalizing quantum group invariants via holonomy-based constructions.

ABSTRACT

The notion of holonomy $R$-matrices is introduced. It is shown how to define invariants of tangles with flat connections in a principle $G$-bundle of the complement of a tangle using holonomy $R$-matrices.

Motivation & Objective

• To define a new class of invariants for tangles with flat connections in the complement using holonomy R-matrices.
• To establish that these invariants are independent of gauge choice, ensuring physical and topological consistency.
• To generalize quantum group-based invariants (e.g., from RTT theory) by embedding them in a holonomy framework.
• To lay the foundation for extending these invariants to 3-manifolds with flat connections in the follow-up work.

Proposed method

• Introduces the concept of holonomy R-matrices as solutions to the Yang-Baxter equation that depend on holonomies around tangle components.
• Constructs a functor F from the category of G-colored tangles with flat connections to a category of representations, using non-degenerate and cross-nondegenerate holonomy R-matrices.
• Defines group presentations via Wirtinger-like relations on tangle diagrams, identifying the fundamental group of the tangle complement.
• Uses monodromy maps to associate group elements to edges in the tangle diagram, encoding path-ordered holonomies.
• Derives transformation laws for the functor F under gauge transformations, proving gauge invariance via explicit matrix identities.
• Applies recurrence relations involving conjugation by group elements to track how holonomy data evolves across tangle crossings and critical points.

Experimental results

Research questions

• RQ1 How can invariants of tangles with flat connections be constructed using holonomy data rather than quantum R-matrices directly?
• RQ2 What conditions on R-matrices ensure gauge invariance of the resulting tangle invariants?
• RQ3 How do holonomy R-matrices relate to standard quantum group R-matrices at roots of unity?
• RQ4 Can the invariant construction be extended to 3-manifolds with flat connections in future work?
• RQ5 What is the geometric meaning of the invariant when restricted to links, particularly in terms of the moduli space of flat connections?

Key findings

• The invariant F(t) associated with a tangle t is gauge-invariant under the action of the structure group G, as shown by explicit transformation formulas in equations (24) and (25).
• For links, the invariant F(t) is a scalar-valued function (or section of a line bundle) over the moduli space of flat G-connections in the complement of t.
• The construction relies on non-degenerate and cross-nondegenerate holonomy R-matrices, which generalize standard R-matrices used in quantum group theory.
• The holonomy R-matrices are shown to be compatible with the tangle category structure, preserving composition and tensor product under the functor F.
• The paper establishes a precise correspondence between the monodromy map and the fundamental group of the tangle complement, using a Wirtinger-type presentation.
• The recurrence relations for group elements (e.g., x_{i-1} = (α_{x_-}(y)_i)_{-}^{-ε_i} x_i (α_{x_-}(y)_i)_{-}^{ε_i}) allow consistent tracking of holonomy evolution across tangle components.