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)?