# Motivating quantum groups from knot invariants

Quantum groups are useful for making knot/link invariants: for example, $$U_q(\mathfrak{sl}_2$$) you get the Jones polynomial. This boils down to the fact that $$\mathcal C = \operatorname{rep }U_q(\mathfrak{sl}_2)$$ is a braided monoidal category, which is not symmetric, hence gives us interesting knot invariants.

A deformation theoretic path to the quantum group $$U_q(\mathfrak g)$$ from a Lie algebra $$\mathfrak g$$ is as follows. Fixing a Casimir element $$\Omega\in\mathfrak g\otimes \mathfrak g$$ endows the category $$\mathcal C=\operatorname{rep }\mathfrak g$$ with the structure of an infinitesimally braided symmetric monoidal category. We can infinitesimally deform $$\mathcal C$$ to an honest braided monoidal category $$\mathcal C[\hbar]/(\hbar^2)$$ with braiding given by $$\sigma\circ(1+\hbar\Omega/2)$$. The next step is usually to use a Drinfeld associator to integrate this infinitesimal deformation to a formal one $$\mathcal C[[\hbar]]$$ with braiding given by $$\sigma\circ\exp(\hbar\Omega/2)$$. After some twisting/scrootching/variable replacement we get $$\mathcal C[[\hbar]]=\operatorname{ rep}U_q(\mathfrak g)$$.

Question From the vantage point of knot invariants, is there any benefit to integrating the infinitesimal deformation $$\mathcal C[\hbar]/(\hbar^2)$$ to the formal one? The infinitesimal deformation is already a non-symmetric braided monoidal category.

Let $$\mathcal{C}$$ be the category of finite-dimensional representations of a semisimple Lie algebra $$\mathfrak{g}$$ and $$\mathcal{C}[\![\hbar]\!]$$ the ribbon category you mention (which depends on the choice of a Drinfeld associator).
Given an irreducible $$\mathfrak{g}$$-representation $$V$$, the corresponding knot invariant obtained from $$\mathcal{C}[\![\hbar]\!]$$ is the Kontsevich integral evaluated using the weight system coming from $$\mathfrak{g}$$, $$\Omega$$ and $$V$$; this is Theorem 10 in Lê--Murakami's The universal Vassiliev-Kontsevich invariant for framed oriented links'' (https://arxiv.org/abs/hep-th/9401016).
The $$n$$-th Taylor coefficient (with respect to the $$\hbar$$ expansion) of the Kontsevich integral is a finite type (Vassiliev) invariant of degree less than $$n + 1$$. But the first nontrivial finite type invariant of oriented knots has degree 2 and so requires working at least with $$\mathcal{C}[\hbar]/\hbar^3$$.
So, you could consider oriented knot invariants working modulo $$\hbar^2$$, but they are all independent of the knot.
• You actually don't even need to rely on this thm by Le-Murakami. It's fairly easy to show directly that, after setting $q=e^h$, any knot invariant associated with a representation of a quantum group (or more generally, with any object in a ribbon category obtaind by any deformation of an infinitesimal symmetric monoidal category) has the property that its degree $n$ coefficient is a finite type invariant of degree $n$. This is an invariant of framed knot, but there's only one degree 1 invariant, the self-linking number. Mar 25 at 16:43