[Paper Review] Other quantum relatives of the Alexander polynomial through the Links-Gould invariants

Ben-Michael Kohli, Bertrand Patureau‐Mirand|arXiv (Cornell University)|Jun 8, 2016

Algebraic structures and combinatorial models14 references4 citations

TL;DR

This paper proves a conjecture by De Wit, Ishii, and Links regarding the Links-Gould invariants, showing that for any link $L$, $LG^{n,1}(L;\tau,-1) = \Delta_L(\tau^2)^n$, where $\Delta_L$ is the Alexander-Conway polynomial. The proof uses the universal R-matrix and Hopf algebra structures of $U_q\mathfrak{gl}(n|1)$ at $q = -1$, establishing a direct link between quantum invariants and the Alexander polynomial via representation-theoretic techniques on the $-1$ specialization of $U_q\mathfrak{gl}(n|1)$.

ABSTRACT

Oleg Viro studied in arXiv:math/0204290 two interpretations of the (multivariable) Alexander polynomial as a quantum link invariant: either by considering the quasi triangular Hopf algebra associated to $U_q sl(2)$ at fourth roots of unity, or by considering the super Hopf algebra $U_q gl(1|1)$. In this paper, we show these Hopf algebras share properties with the $-1$ specialization of $U_q gl(n|1)$ leading to the proof of a conjecture of David De Wit, Atsushi Ishii and Jon Links on the Links-Gould invariants.

Motivation & Objective

• To prove the $(n,1)$ case of the conjecture by De Wit, Ishii, and Links relating the Links-Gould invariants to the Alexander-Conway polynomial.
• To establish a structural connection between the $-1$ specialization of $U_q\mathfrak{gl}(n|1)$ and the quantum invariants underlying the Alexander polynomial.
• To provide a conceptual, representation-free proof using universal R-matrices and Hopf algebra structures, avoiding explicit matrix computations for large $n$.
• To extend the result to the multivariable case by suggesting $M(L;-1,q_1,\ldots,q_c) = \nabla(q_1,\ldots,q_c)^n$, where $\nabla$ is the Conway potential function.

Proposed method

• The authors analyze the universal R-matrix of $U_q\mathfrak{gl}(n|1)$ at $q = -1$ using the structure of the $-1$ specialization of $U_q\mathfrak{gl}(n|1)$ as a super Hopf algebra.
• They use the Reshetikhin-Turaev construction to relate the Links-Gould invariant to the quantum invariants derived from $U_q\mathfrak{gl}(n|1)$.
• The proof relies on the conjugation action of the Drinfeld twist $D^{\mathfrak{gl}}$ on the tensor product of representations, showing it matches the braiding structure of the $U_q\mathfrak{sl}(2)$ model at $q = \mathbf{i}$.
• They establish proportionality between the rescaled R-matrices of $U_q\mathfrak{gl}(n|1)$ and the $n$-fold tensor product of $U_q\mathfrak{sl}(2)$ invariants, up to reordering of factors.
• The key step involves showing that the twist automorphism $\theta$ acts as a scalar on the highest weight vector, enabling trace comparison between the two invariants.
• The authors use the density lemma and pivotal structure to equate the action of the universal R-matrix on irreducible modules, ensuring the invariants agree up to scalar multiplication.

Experimental results

Research questions

• RQ1 Does the Links-Gould invariant $LG^{n,1}(L;\tau,-1)$ equal $\Delta_L(\tau^2)^n$ for all links $L$ and all $n$?
• RQ2 Can the $-1$ specialization of $U_q\mathfrak{gl}(n|1)$ be used to construct quantum invariants that recover the Alexander-Conway polynomial?
• RQ3 Is there a conceptual, universal proof of the Links-Gould conjecture that avoids explicit $R$-matrix computations for large $n$?
• RQ4 Can the multivariable Links-Gould invariant $M(L;q,q_1,\ldots,q_c)$ be related to the Conway potential function $\nabla(q_1,\ldots,q_c)$ at $q = -1$?

Key findings

• The paper proves that $LG^{n,1}(L;\tau,-1) = \Delta_L(\tau^2)^n$ for all links $L$ and all $n$, confirming the $(n,1)$ case of the De Wit-Ishii-Links conjecture.
• The universal R-matrix of $U_q\mathfrak{gl}(n|1)$ at $q = -1$ induces a braiding that is proportional to the $n$-fold tensor product of the $U_q\mathfrak{sl}(2)$ R-matrix at $q = \mathbf{i}$.
• The conjugation by the Drinfeld twist $D^{\mathfrak{gl}}$ matches the action of the automorphism $\mathscr{D}$ on the tensor product of representations, ensuring consistency in the braiding structure.
• The Reshetikhin-Turaev invariants associated with $V_{-1}(0^n,\alpha)$ for $U_q\mathfrak{gl}(n|1)$ at $q = -1$ satisfy $\Psi^{\mathfrak{gl}}_{V_{-1}(0^n,\alpha)^{\otimes \ell}}(\beta) = \left(\Psi^{U_{\mathbf{i}}^{H}\mathfrak{sl}(2)}_{V_{-\alpha}^{\otimes \ell}}(\beta)\right)^{\otimes n}$.
• The trace of the $LG^{n,1}$ invariant on $V_{-1}(0^n,\alpha)^{\otimes \ell}$ is $c^n$, where $c$ is the trace of the $U_q\mathfrak{sl}(2)$ invariant, confirming the $n$-th power relationship with the Alexander polynomial.