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)?



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