2. Reverse parallel links of alternating knots

Let K be a knot in $\textbf{S}^3$ . The regular neighbourhood of K on any orientable surface M embedded in $\textbf{S}^3$ on which K lies is an annulus. Let A be such an annulus. The link formed by the boundaries $K^\prime$ , $K^{\prime\prime}$ of A is called a reverse parallel link of K if $K^\prime$ , $K^{\prime\prime}$ are assigned opposite orientations. The linking number f between $K^\prime$ and $K^{\prime\prime}$ when they are assigned parallel orientations is called the framing of K [ Reference Nutt12, Reference Rudolph13 ]. The framing f is independent of the orientation of K and the ambient isotopy class of A in $S^3$ depends only on K and the framing. Furthermore, the reverse parallel links of K are characterised by their framing numbers. That is, two reverse parallel links of K are ambient isotopic if and only if they have the same linking number. We shall denote a reverse parallel link of K with framing f by $\mathbb{K}_f$ following [ Reference Diao and Morton9 ].

Let K be an alternating knot with a reduced diagram D. Let c(D) be the number of crossings in D and w(D) the writhe of D. It is known that $c(D)=Cr(K)$ is the crossing number of K. Also, all crossings of D are positive with respect to one checkerboard shading of D and are all negative with respect to the other checkerboard shading of D. Let $v_+(D)$ be the number of regions in the shading with respect to which all crossings are positive, and $v_-(D)$ be the number of regions in the complementary shading with respect to which all crossings are negative, then $v_+(D)+v_-(D)=c(D)+2$ [ Reference Kauffman10 ]. The following results provide the key to solving the ropelength conjecture for alternating knots.

Theorem 2·1. [ Reference Diao and Morton9 ] Let K be an alternating knot and D a reduced diagram of K. Let c(D), w(D), $v_+(D)$ and $v_-(D)$ be as defined above, then the braid index of $\mathbb{K}_f$ , denoted by $\mathbf{b}(\mathbb{K}_f)$ , is given by the following formula:

\begin{equation}\mathbf{b}(\mathbb{K}_f)=\left\{\begin{array}{ll}c(D)+2+a(D)-f, &\ {\rm if}\ f \lt a(D),\\[5pt] c(D)+2, &\ {\rm if}\ a(D)\le f \le b(D),\\[5pt] c(D)+2-b(D)+f, &\ {\rm if}\ f \gt b(D),\\ \end{array}\right.\end{equation}

where $a(D)=-v_-(D)+w(D)$ and $b(D)=v_+(D)+w(D)$ . In particular, $\mathbf{b}(\mathbb{K}_f)\ge c(D)+2$ for any f.

3. The ropelengths of alternating knots

Now, the set $\{\textbf{x}+t(({1}/{2}, {1}/{2}, {1}/{2}): \textbf{x}\in K_c,\ 0\le t\le 1\}$ is an embedding of the annulus $\{1\le x^2+y^2\le 2:\ x, y\in \mathbb{R}\}$ into $\mathbb{R}^3$ with $K_c$ and $K_c^\prime=(({1}/{2}, {1}/{2}, {1}/{2}))+K_c$ as its boundary curves. Assigning $K_c$ and $K_c^\prime$ opposite orientations yields a reverse parallel link $L_c$ of K. Let $\mathbb{K}_c$ and $\mathbb{K}^\prime_c$ be the lattice knots obtained from $K_c$ and $K_c^\prime$ by scaling up with a scaling factor of 2, and let $\mathbb{L}_c$ be the link formed by them. Notice that $\ell(\mathbb{L}_c)=4\ell(K_c)$ . By Theorem 2·1, we have $\mathbf{b}(\mathbb{L}_c)\ge Cr(K)+2$ . Then by Theorem 2·2, we have $\ell(K_c)\ge (Cr(K)+2)/4$ . Therefore $R(K) \gt ({1}/{14})\ell(K_c)\ge (Cr(K)+2)/56$ .