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\title{Certified ropelength bounds for torus knots and the tight knot constant}
\author{David Niedbala Giraudin\thanks{ORCID 0009-0009-1526-1178. Technical assistance: Claude (Anthropic).}}
\date{30 October 2026; this version 1 November 2026}

\begin{document}
\maketitle

\begin{abstract}
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 $T(\Pm,\Qm)$, with $\mathrm{Cr}=\Crm$, of
thickness at least $1$ and length at most $\Lm$. Hence $\alpha_0\le\Am$. 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 exact integer and
rational arithmetic. Two separately written verifiers and the certificate are supplied. Bounds for
$T(14,17)$ and $T(11,14)$, obtained in the same way, are also reported.
\end{abstract}

\section{Introduction}

For a closed $C^{1,1}$ curve $\gamma$ of thickness $\Delta(\gamma)$ (the infimum of the radii of
circles through three distinct points of $\gamma$), the ropelength of a knot type $K$ is
$L(K)=\inf\{\mathrm{length}(\gamma)/\Delta(\gamma)\}$ over curves $\gamma$ of type $K$ \cite{CKS}.
Buck and Simon \cite{BS} proved $L(K)\ge c\,\mathrm{Cr}(K)^{3/4}$ for an absolute $c>0$, and the
exponent is sharp \cite{CKS}. Let $\alpha_0$ be the best constant (entry 22a of
\cite{Registry}). The smallest published upper bound is $\alpha_0\le10.76$, from a numerically
tightened $T(25,26)$ \cite{KM}. The value $10.02$ for $T(24,19)$ appears in \cite{Registry}, attributed
to \cite{Klotz26} (where it is credited to an earlier online post of the author) and marked there as
not yet verified; no coordinates are available.

\begin{theorem}\label{thm:main}
There is a $C^{1,1}$ curve of knot type $T(\Pm,\Qm)$ with thickness $\ge1$ and length $\le\Lm$.
Consequently $L(T(\Pm,\Qm))\le\Lm$ and, since $\mathrm{Cr}(T(\Pm,\Qm))=\Crm$,
\[
\alpha_0\ \le\ \frac{\Lm}{\Crm^{3/4}}\ <\ \Am .
\]
In the same way $L(T(14,17))\le584.481$ and $L(T(11,14))\le416.652$ (their certificates are not part of
this deposit, because of its size, and are available from the author).
\end{theorem}

The crossing number of the torus knot $T(p,q)$ is $\min\{p(q-1),q(p-1)\}$ \cite{Murasugi}. Each
curve is obtained from an explicit polygon $F$ by rounding its corners;
Sections~\ref{sec:type}--\ref{sec:thick} prove that it has the required type, thickness and length.
All statements below refer to one pair $(p,q)$ at a time.

\section{Knot type}\label{sec:type}

\paragraph{The starting polygon.} Let $\Sigma$ be the polyhedral torus with vertices
$V(a,b)=\big((R+r c_b)c_a,\,(R+r c_b)s_a,\,r s_b\big)$, $a,b\in\mathbb Z/4$, where
$(c_k,s_k)=(\cos\frac{k\pi}{2},\sin\frac{k\pi}{2})$, $R=2q+6$, $r=q+2$, where $2\le p<q$ are coprime.
Since $R>r>0$, the faces $V(a,b)V(a+1,b)V(a+1,b+1)V(a,b+1)$ are planar convex trapezoids, and $\Sigma$ is
an embedded torus bounding an unknotted solid torus. On the chart $(\alpha,\beta)\in(\mathbb R/\mathbb Z)^2$
of $\Sigma$ (bilinear on each face, mapping the boundary of each grid square homeomorphically onto the
boundary of the face) consider the line $t\mapsto(pt,qt)$. It meets the grid corners only at
$t\in\frac14\mathbb Z$, at four distinct corners. Cutting it at the grid lines
$\alpha,\beta\in\frac14\mathbb Z$ gives points on the boundaries of the faces; two consecutive cut points
lie on the boundary of a common face, and $P_0$ is formed by the straight segments joining them, each
lying in that face. Distinct passes of the line through a grid square are parallel chords, so their
endpoints do not interleave on the boundary of the square, hence not on the boundary of the face; two
straight chords of a convex polygon with non-interleaving endpoints are disjoint. So $P_0$ is a simple
closed curve on $\Sigma$, isotopic on $\Sigma$ (face by face: two systems of disjoint arcs in a disk
with the same non-interleaving endpoints are isotopic rel boundary) to the image of the line, i.e.\ a curve of
class $(p,q)$, and therefore a $(p,q)$ torus knot \cite[\S3.C]{Rolfsen}. Each edge of $P_0$ is
subdivided into $16$ equal parts. All vertices are rational; in the unit $1/U$, with $U=16pq\kappa$
($\kappa=16,32,64$ for $T(19,24),T(14,17),T(11,14)$), they are integers. The verifiers recompute $P_0$
exactly and reject any other choice of $R$, $r$, the subdivision or $(p,q)$.

\begin{lemma}[linear steps]\label{lem:step}
Let $C,C'$ be polygons with the same number of vertices, $\delta\ge\max_v|C'_v-C_v|$, and for each
$s\in[0,1]$ let $C_s=(1-s)C+sC'$. Suppose (i) every two non-adjacent edges of $C$ are at distance
$>2\delta$, and (ii) at each vertex, with adjacent edges $e_0,e$ and turning angle $\theta$,
$2\delta<\min(|e_0|,|e|)\cos(\theta/2)$. Then every $C_s$ is an embedded polygon; in particular $C$
and $C'$ are ambient isotopic.
\end{lemma}

\begin{proof}
Points of an edge move by at most $\delta$, so non-adjacent edges of $C_s$ stay at positive
distance. An edge vector moves by at most $2\delta<|e|$, so its direction turns by less than
$\arcsin(2\delta/|e|)$; the turning angle between two adjacent edges therefore stays below
$\theta+2\arcsin(\cos(\theta/2))=\pi$, and adjacent edges meet only at their common vertex. So every
$C_s$ is embedded. By compactness there is $\eta>0$ such that, for all $s$, non-adjacent edges of
$C_s$ are at distance $>\eta$, edges are longer than $\eta$ and turning angles are below $\pi-\eta$.
Cut $[0,1]$ into intervals on which every vertex moves by less than $\varepsilon$, and on each of them
move the vertices one at a time. For $\varepsilon$ small in terms of $\eta$, the two triangles swept by
one such move (moving $v$ to $v'$ between its neighbours $u,w$: the triangles $uvv'$ and $v'vw$)
meet the polygon only along the two edges being moved, so each move is an elementary
($\Delta$-) move; hence $C$ and $C'$ are equivalent knots \cite[Ch.~1]{BZ}.
\end{proof}

In verifier (i) both conditions are checked exactly, in integers. Write $D$ for the largest squared
displacement of a vertex (so $\delta^2=D$), $d_U=\lfloor\sqrt D\rfloor+1>\delta$ and
$\ell_i=\lfloor\sqrt{|e_i|^2}\rfloor+1>|e_i|$. Condition (i) follows from
$\mathrm{dist}(e_i,e_j)\ge|m_i-m_j|-\frac12(|e_i|+|e_j|)$ ($m$ the midpoints) when
$4|m_i-m_j|^2>(4d_U+\ell_i+\ell_j)^2$, and otherwise from the exact rational distance between the
two segments; pairs whose midpoints are farther apart are excluded by an exact integer grid.
For (ii), with $n_0=|e_0|^2$, $n=|e|^2$, $m=\min(n_0,n)$ and $P=\lfloor\sqrt{n_0n}\rfloor$, the
condition $4\delta^2<m(1+\cos\theta)/2$ is implied by the integer inequality
$8D(P+1)<m(P+e_0\cdot e)$. Coordinates are asserted to be below $2^{24}$.

The certificate lists integer polygons $X_0=P_0,X_1,\dots,X_K$ (the states of a tightening process)
and the final polygon $F$. In the certificate for $T(\Pm,\Qm)$ every edge of $X_K$ is then split at its
midpoint, which leaves the polygon unchanged and is exact in the unit $1/(2U)$, and the path continues
in that unit. Each step $X_k\to X_{k+1}$ (and $X_K\to F$) is subdivided by
the integer points $X_k+\lfloor j(X_{k+1}-X_k)/m\rfloor$, and every sub-step satisfies
Lemma~\ref{lem:step}. Hence $F$ is a $(p,q)$ torus knot.

\section{Thickness and length}\label{sec:thick}

Let $F$ have vertices $v_i$, edges $e_i=v_{i+1}-v_i$, turning angles $\theta_i<\pi/2$. Put
$t_i=\tan(\theta_i/2)$ and suppose $\mathrm{MinRad}(v_i)=\min(|e_{i-1}|,|e_i|)/(2t_i)\ge1$ for all
$i$. Replace the corner at $v_i$ by the arc of the unit circle tangent to $e_{i-1}$ and $e_i$; the
tangency points lie at distance $t_i\le\frac12\min(|e_{i-1}|,|e_i|)$ from $v_i$, so the arcs do not
overlap. The result is a $C^{1,1}$ curve $\gamma$ with curvature $\le1$ and
$\mathrm{length}(\gamma)\le\mathrm{length}(F)$. Every point of the arc at $v_i$ is within
$\tau_i=\sec(\theta_i/2)-1\le t_i^2/2$ of the two half-edges at $v_i$ (since
$\sec x-1=\tan^2x/(\sec x+1)$).

\begin{lemma}\label{lem:crit}
Let $\gamma$ be a $C^{1,1}$ curve parametrized by arclength with curvature $\le1$, and let $s<t$ with
$t-s<\pi$. Then $\langle\gamma'(s),\gamma(t)-\gamma(s)\rangle>0$. In particular a pair of points
at which the chord is orthogonal to both tangents is separated by arc length $\ge\pi$ in both
directions.
\end{lemma}

\begin{proof}
The unit tangent moves on the sphere with speed $\le1$, so the angle between $\gamma'(s)$ and
$\gamma'(x)$ is at most $x-s$, and
$\langle\gamma'(s),\gamma(t)-\gamma(s)\rangle=\int_s^t\langle\gamma'(s),\gamma'(x)\rangle\,dx
\ge\int_0^{t-s}\cos y\,dy=\sin(t-s)>0$.
\end{proof}

For a $C^{1,1}$ closed curve the thickness equals the minimum of the minimal radius of curvature
and half the minimal distance between doubly critical pairs (pairs at which the chord is orthogonal
to both tangents) \cite[Lemma~1]{CKS}; see also \cite{LSDR}. By Lemma~\ref{lem:crit},
$\Delta(\gamma)\ge1$ as soon as every two points of $\gamma$ at arc distance $\ge\pi$ (in both
directions) are at distance $\ge2$.

\begin{proposition}\label{prop:thick}
Suppose that every edge is shorter than $\pi$, that $\mathrm{MinRad}(v_i)\ge1$ for all $i$, and that
$\mathrm{dist}(e_i,e_j)\ge 2+\bar\tau_i+\bar\tau_j$ for every pair of distinct edges containing points at
polygonal arc distance $\ge\pi$ in both directions, where $\bar\tau_i=\max(\tau_i,\tau_{i+1})$. Then
$\Delta(\gamma)\ge1$.
\end{proposition}

\begin{proof}
Let $x,y\in\gamma$ be at arc distance $\ge\pi$ along $\gamma$ in both directions. If $x$ lies on a
straight part of $\gamma$, put $x^*=x$. If $x$ lies on the arc at $v_i$, at angle $\varphi$ from the
nearer tangency point, let $x^*$ be its radial projection from the centre of the arc onto the
half-edge on that side; then $|x-x^*|=\sec\varphi-1\le\tau_i$, and the piece of arc of length
$\varphi$ is replaced by a segment of length $\tan\varphi\ge\varphi$. Define $y^*$ in the same way.
Then $x^*\in e_i$, $y^*\in e_j$ for edges with $|x-x^*|\le\bar\tau_i$, $|y-y^*|\le\bar\tau_j$. Between
$x^*$ and $y^*$, each complete arc of $\gamma$ is replaced by its two tangent segments, which are
longer, and the partial arcs at the ends are replaced by longer paths (through the nearer tangency point
an arc of length $\varphi$ becomes a segment of length $\tan\varphi$; through the other one an arc of
length $\theta_i-\varphi$ becomes a path of length $2\tan(\theta_i/2)-\tan\varphi$, and
$2\tan(\theta_i/2)-\tan\varphi-(\theta_i-\varphi)$ decreases in $\varphi\in[0,\theta_i/2]$ to
$\tan(\theta_i/2)-\theta_i/2\ge0$); hence the polygonal arc
distance between $x^*$ and $y^*$ is at least $\pi$ in both directions. As edges are shorter than
$\pi$, the edges $e_i$ and $e_j$ are distinct, and
$|x-y|\ge\mathrm{dist}(e_i,e_j)-\bar\tau_i-\bar\tau_j\ge2$. In particular $\gamma$ is injective:
points at arc distance $<\pi$ are distinct by Lemma~\ref{lem:crit}, and the others are at distance
$\ge2$; so $\gamma$ is an embedded curve, to which \cite[Lemma~1]{CKS} applies.
\end{proof}

The verifiers check the distance condition on larger sets of pairs: verifier (i) uses the threshold
$\pi-\frac25$ and edges shorter than $\pi-\frac25$, verifier (ii) the threshold $\pi-2\max_it_i$.

\paragraph{Isotopy of the smoothing.} Since $\Delta(\gamma)\ge1$, the closed normal disks of radius
$\frac12$ of $\gamma$ are pairwise disjoint and fill a solid torus $N$ with core $\gamma$. On straight
parts $F=\gamma$. Near $v_i$, the two half-edges lie in the plane of the arc, within $\tau_i<\frac12$ (in fact $\tau_i\le0.003$ here)
of it, and the radial projection from the centre of the arc maps them homeomorphically onto the arc;
this projection is the nearest-point projection to $\gamma$. Hence $F\subset N$ meets each normal disk
in exactly one point, and the homotopy moving each point of $F$ along the segment to its projection
stays in its own normal disk, so it is an isotopy from $F$ to $\gamma$ inside $N$. So $\gamma$ is a
$(p,q)$ torus knot.

\paragraph{Exact checks.} With integer coordinates all hypotheses of Proposition~\ref{prop:thick}
reduce to integer inequalities: $t_i$ is bounded above by
$(\lfloor\sqrt{|e_{i-1}\times e_i|^2}\rfloor+1)/(\lfloor\sqrt{|e_{i-1}|^2|e_i|^2}\rfloor+e_{i-1}\cdot e_i)$,
edge lengths are bracketed by integer square roots, arc distances by sums of these brackets (with
$\pi$ replaced by $333/106<\pi$), and the squared distance between two segments is computed exactly
as a rational number.

\section{The certificates and their verification}

\begin{table}[h]
\centering
\begin{tabular}{llll}
\toprule
 & $T(\Pm,\Qm)$ & $T(14,17)$ & $T(11,14)$\\
\midrule
vertices of $F$ & \Nm & 1920 & 1536\\
path steps / certified sub-steps & \Km & 365 / 2137 & 189 / 1759\\
$\max_i t_i$ & \Tm & 0.0811 & 0.0676\\
$\mathrm{length}(F)\le$ & \Lm & 584.481 & 416.652\\
$\mathrm{Cr}$ & \Crm & 221 & 140\\
$\mathrm{length}/\mathrm{Cr}^{3/4}\le$ & \Am & 10.1971 & 10.2372\\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Verification.} The certificate for $T(\Pm,\Qm)$ is the set of files
\texttt{anc/C22a\_T19\_24\_*.csv}: exact integer coordinates, one file per array, the first line giving
its shape; the two sequences of polygons are stored as the first polygon followed by the differences
between consecutive polygons, and are split into parts \texttt{\_p01}, \texttt{\_p02}, \dots{} (files of
at most 4\,MiB). (i)
\texttt{anc/verify\_C22a.py}, with \texttt{certlib.py} and \texttt{p0.py} (Python~3 with numpy):
every test is an exact integer or rational test, as described above; floating point only chooses how
finely a step is subdivided. (ii) \texttt{anc/verify2\_C22a.py} (numpy, scipy, python-flint), written
separately: a KD-tree search for candidate pairs; for the path, double precision with explicit margins
and exact rational distances for close pairs; for the thickness, Arb balls \cite{Arb} for lengths,
angles and arc distances, and exact rational segment distances. The two programs share only the
definition of $P_0$ (\texttt{p0.py}); they implement the same reductions, Lemma~\ref{lem:step} and
Proposition~\ref{prop:thick}, so their agreement guards against programming errors, not against an
error in these lemmas. Both run without argument, in a few minutes. Verifier (ii) gives the length bounds \LmTwo, $584.4727$ and $416.6468$. On the certificate for
$T(\Pm,\Qm)$, both verifiers reject the modified files obtained by shrinking the final polygon by
$0.1\%$, by moving one vertex of one intermediate state (of the first or of the refined part of the
path) onto another strand, by exchanging $(p,q)$, and a file for $T(2,3)$ with other torus
parameters.

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\bibitem{CKS} J.~Cantarella, R.~B.~Kusner, J.~M.~Sullivan, \emph{On the minimum ropelength of knots and links}, Invent.\ Math.\ \textbf{150} (2002), 257--286.
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\bibitem{Klotz26} A.~R.~Klotz, \emph{Tight bounds for tight links: ropelength of $T(Q,Q)$ torus links}, arXiv:2603.02416v2 (13 March 2026).
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\end{thebibliography}
\end{document}