[Paper Review] Quantum holonomic link invariants derived from stated skein algebras

Julien Korinman|arXiv (Cornell University)|Jul 3, 2019

Algebraic structures and combinatorial models92 references4 citations

TL;DR

This paper establishes a conceptual link between quantum groups and quantum Teichmüller theory by showing that the reduced stated skein algebra of a disc with two boundary punctures and one interior puncture carries a natural Hopf algebra structure isomorphic to the simply laced quantum group $\widetilde{U}_q\mathfrak{gl}_2$. It classifies its weight indecomposable modules, derives Clebsch-Gordan formulas, and shows that change-of-coordinates maps in quantum Teichmüller theory yield braiding operators that coincide with Drinfel'd's $R$-matrix for the Kashaev module, providing a new proof of the Murakami-Murakami theorem relating the Kashaev invariant to the colored Jones polynomial.

ABSTRACT

We define invariants for a framed link equipped with a SL2 local system in its complement and additional combinatorial data based on the theory of representations of stated skein algebras at roots of unity of punctured bigons and the geometric interpretation of their centers. The gauge invariance of the link invariant is derived from De Concini-Kac quantum coadjoint action lifted at the level of stated skein algebras. A key feature is the fact that the Drinfeld double of the quantum Borel algebra admits a natural interpretation as the reduced stated skein algebra of a once-punctured bigon from which we deduce a relation between our link invariants and quantum group constructions of Blanchet-Geer-Patureau Mirand-Reshetikhin. Using Bonahon-Wong quantum trace, we also relate our construction to quantum hyperbolic geometry, hence to Kashaev and Baseilhac-Benedetti constructions. We deduce from this relation explicit formulas for the R-matrices, which permit to compute the link invariants explicitly. In particular, we derive an alternative conceptual proof of the Murakami-Murakami relation between the Kashaev invariant and the colored Jones polynomials.

Motivation & Objective

• To establish a conceptual bridge between quantum groups and quantum Teichmüller theory using stated skein algebras.
• To classify weight indecomposable modules of the quantum group $\widetilde{U}_q\mathfrak{gl}_2$ arising from the skein algebra of a punctured disc.
• To show that change-of-coordinates maps in quantum Teichmüller theory yield braiding operators isomorphic to Drinfel'd's $R$-matrix for the Kashaev module.
• To provide an alternative proof of the Murakami-Murakami theorem relating the Kashaev invariant to the colored Jones polynomial.

Proposed method

• The paper constructs a Hopf algebra structure on the reduced stated skein algebra $\mathcal{S}_\omega(\mathbb{D}_1)$ of a disc with two boundary and one interior puncture, isomorphic to $\widetilde{U}_q\mathfrak{gl}_2$.
• It classifies weight indecomposable modules of $\mathcal{O}_q[D(B)]$ using representation theory over $\mathbb{C}$ at odd roots of unity.
• It employs the quantum trace map to relate the stated skein algebra to the balanced Chekhov-Fock algebra, enabling the construction of representations.
• It derives braiding operators via change-of-coordinates isomorphisms in quantum Teichmüller theory, using Fock-Goncharov decomposition and quantum exponentials.
• It proves that the Kashaev braiding operator coincides with Drinfel'd's $R$-matrix by analyzing the action on the Kashaev module.
• It provides explicit formulas for generic braiding operators depending on quantum shape parameters from quantum hyperbolic geometry, leading to new link invariants.

Experimental results

Research questions

• RQ1 How does the stated skein algebra of a punctured disc realize a quantum group structure?
• RQ2 What is the classification of weight indecomposable modules of $\widetilde{U}_q\mathfrak{gl}_2$ in this context?
• RQ3 How do change-of-coordinates maps in quantum Teichmüller theory produce braiding operators isomorphic to Drinfel'd's $R$-matrix?
• RQ4 Can the Kashaev-Reshetikhin braidings be derived from quantum Teichmüller theory, and how do they relate to Drinfel'd's $R$-matrix?
• RQ5 What new link invariants arise from generic braiding operators defined via quantum shape parameters?

Key findings

• The reduced stated skein algebra $\mathcal{S}_\omega(\mathbb{D}_1)$ is isomorphic to the quantum group $\widetilde{U}_q\mathfrak{gl}_2$ as a Hopf algebra.
• All weight indecomposable modules of $\mathcal{O}_q[D(B)]$ are classified into three types: $V(\lambda,\mu,a,b)$, $\widetilde{V}(\lambda,\mu,c)$, and $V_{\mu,\varepsilon,n}$, depending on the vanishing of $E^N$, $F^N$, or both.
• The change-of-coordinates isomorphism in quantum Teichmüller theory induces braiding operators that match the Kashaev-Reshetikhin braidings.
• For the Kashaev module, the induced braiding operator coincides with Drinfel'd's $R$-matrix, providing a conceptual explanation of the Murakami-Murakami theorem.
• Explicit formulas for generic braiding operators are derived in terms of quantum shape parameters from quantum hyperbolic geometry.
• These braiding operators generate new invariants of framed links, extending the scope of quantum invariants beyond the standard Jones and colored Jones polynomials.