Compression lower bound from supertrunk for knot types

Let $K$ be a knot type. For a representative $\gamma$ of $K$, define

$$\operatorname{strunk}(\gamma)=\max_{v\in S^2}\operatorname{trunk}_v(\gamma),$$

and let

$$\operatorname{strunk}(K)=\inf_{\gamma\in\mathcal{H}(K)}\operatorname{strunk}(\gamma).$$

Let $\operatorname{CRad}_R(K)$ denote the scale-free compression radius of $K$.

Compression lower bound from supertrunk. There is a universal constant $c>0$ such that for every knot type $K$,

$$\operatorname{CRad}_R(K)\geq c\sqrt{\operatorname{strunk}(K)}.$$

Here $\operatorname{strunk}(K)$ is an appropriate supertrunk-type invariant.

The bound is motivated by packing intersection disks in planar cross-sections of a tubular neighborhood, but the preceding heuristic does not establish the required disjointness or control of intersection angles. A proof would likely require transversality or an averaged coarea estimate.