Extending knot polynomials of braided Hopf algebras to links

TL;DR

This work develops a unified, tangle-based framework to extend knot polynomials arising from rigid $R$-matrices to link invariants via enhanced $R$-matrices within the Reshetikhin–Turaev paradigm. It proves concrete identifications between the extended invariants and classical link polynomials: $ extLambda_{1,L}(t_0,t_1)= ext{Δ}_L(t_0) ext{Δ}_L(t_1)$ and $ extLambda_{-1,L}(t^{-2},s^{-2})= ext{Δ}_{ ext{sl}_3,L}(t,s)$, while developing a method to relate the $V_1$ and $ ext Lambda$ families to established invariants. By exploiting rotated tangles, weak conjugacy, and tensor-product constructions, the authors connect the Garoufalidis–Kashaev/ Nichols-algebra framework to Alexander-type and $ ext{sl}_3$ invariants and set the stage for further identifications with Links–Gould invariants. The results illuminate how multivariable knot polynomials from braided Hopf algebras fit into classical topology, enabling efficient tangle-based computations and revealing deeper interplays among quantum and classical invariants.

Abstract

Recently, a plethora of multivariable knot polynomials were introduced by Kashaev and one of the authors, by applying the Reshetikhin-Turaev functor to rigid $R$-matrices that come for braided Hopf algebras with automorphisms. We study the extension of these knot invariants to links, and use this to identify some of them with known link invariants, as conjectured in that same recent work.

Figures (6)

• Figure 1: The left and right closures $T_1$ and $T_2$ of a $(2,2)$-tangle $T$.
• Figure 2: Elementary oriented tangles.
• Figure 3: A graphical definition of the Reshetikhin-Turaev functor on oriented elementary tangle diagrams.
• Figure 4: A tangle isotopy.
• Figure 5: More tangle isotopies.

Theorems & Definitions (23)

• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Lemma 2.6