The linearity of the ropelengths of Conway algebraic knots in terms of their crossing numbers (Q2900334)

Given a smooth knot or link \(K\) in \(\mathbb R^3\), the thickness of \(K\) is the maximal radius of the normal tubular neighborhoods of \(K\). A knot type \(\mathcal K\) is a collection of knots or links that are equivalent in \(\mathbb R^3\). Then the ropelength of \(\mathcal K\), denoted by \(L(\mathcal K)\), is defined to be the infimum of the length of \(K\) taken over all \(K\) in \(\mathcal K\) with unit thickness. The existence of \(L(\mathcal K)\) for each \(\mathcal K\) has been shown in [\textit{J. Cantarella, R. B. Kusner} and \textit{J. M. Sullivan}, Invent. Math. 150, No. 2, 257--286 (2002; Zbl 1036.57001)]. In the present paper, the authors study the relationship between \(L(\mathcal K)\) and the minimal crossing number \(Cr(\mathcal K)\) of \(\mathcal K\) for Conway algebraic knots. They show that there exists a constant \(a > 0\) such that if a knot of \(\mathcal K\) admits a Conway algebraic knot diagram with \(n\) crossings, then \(L(\mathcal K) \leq a\cdot n\), and further if the diagram is alternating (but not necessarily reduced), then \(L(\mathcal K) \leq a\cdot Cr(\mathcal K)\).