Multivariable Knot Polynomials from Braided Hopf Algebras with Automorphisms

Abstract

We construct knot invariants from solutions to the Yang–Baxter equation associated to appropriately generalized left/right Yetter–Drinfel’d modules over a braided Hopf algebra with an automorphism. When applied to Nichols algebras, our method reproduces known knot polynomials and naturally produces multivariable polynomial invariants of knots. We analyze in detail the Nichols algebra of rank $1$, from which we recover the ADO and the colored Jones polynomials of a knot, and a Nichols algebra of rank $2$ from which we obtain two sequences of knot invariants. One sequence starts with the product of two Alexander polynomials, and continues conjecturally with the Harper polynomial. The second sequence starts with the Links–Gould invariant (conjecturally), and then continues with a new 2-variable knot polynomial that detects chirality and mutation, and whose degree gives sharp bounds for the genus for a sample of 30 computed knots.