Certified ropelength bounds for torus knots and the tight knot constant
The tight knot constant $\alpha_0$ is the largest constant with $L(K)\ge\alpha_0\,\mathrm{Cr}(K)^{3/4}$ for every knot $K$, where $L$ is the ropelength (unit-radius rope) and $\mathrm{Cr}$ the crossing number. We exhibit an explicit $C^{1,1}$ embedding of the torus knot $T(19,24)$, with $\mathrm{Cr}=432$, of thickness at least $1$ and length at most $939.984$. Hence $\alpha_0\le 9.9200$. The published bound is $10.76$. The proof is computer-assisted: the knot type is certified by an explicit piecewise-linear path of embedded polygons starting from a polygon on a torus, and the thickness by…
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