I've been trying to be more diligent about reading motivic homotopy theory, and have been reading Levine's 'An Overview of Motivic Homotopy Theory.' I think the subject is fascinating, and I've developed a question during my reading.


Levine defines $\mathcal{X}:\mathbf{Sm}/S^{\text{op}}\rightarrow\mathbf{Spc}$, where the source is the category of smooth, separated $S$-schemes of finite type.


There are three suspension functors, 
$$
\Sigma\_{S^1}\mathcal{X}\simeq\mathcal{X}\wedge S^1\quad\Sigma\_{\mathbb{G}\_m}\mathcal{X}\simeq\mathcal{X}\wedge\mathbb{G}\_m\quad\Sigma\_{\mathbb{P^1}}\mathcal{X}\simeq\mathcal{X}\wedge\mathbb{P}^1
$$


Assume two smooth embeddings ("motivic knots") $f,f^\*$
$$
S^{a+b,b}\xrightarrow{f\>\wedge\text{ id}}\Sigma\_{S^1}^2S^{a+b,b}
\quad
S^{a+b,b}\xrightarrow{\text{id}\>\wedge\>f^\*}\Sigma\_{\mathbb{G}\_m}^2S^{a+b,b}
$$
but Levine states that $\mathbb{P}^1$ is the preferred object to stabilize the category of motivic spectra about, which suggests to me as a 'more natural' setting for the preferred motivic knot
$$
S^{a+b,b}\xrightarrow{f\wedge f^\*}\Sigma\_{\mathbb{P}^1}^2S^{a+b,b}
$$


**Question**: Is there any research on a "motivic knots" (in any broad sense encompassing interesting behavior around $f,f^\*$ or some combination)?



Motivic knot embedding
Apparently, [there is now](https://arxiv.org/abs/2210.11048)! Clémentine Lemarié-Rieusset, a PhD student of Frédéric Déglise and Adrien Dubouloz at Université de Bourgogne is developing the theory.


