The Ropelength of Special Alternating Knots

Abstract

A long standing open conjecture states that if a link $\mathcal{K}$ is alternating, then its ropelength $L(\mathcal{K})$ is at least of the order $O(Cr(\mathcal{K}))$. A recent result shows that the maximum braid index of a link bounds the ropelength of the link from below. Thus in the case an alternating link has a maximum braid index proportional to its minimum crossing number, such as the $T(2,2n)$ torus link, then the ropelength of the link is bounded below by a constant multiple of its minimum crossing number. However if the maximum braid index of a link is small compared to its crossing number, then there are no known results about whether its ropelength is bounded below by a constant multiple of its crossing number. For example, the $T(2,2n+1)$ torus knot has a minimum knot diagram that looks almost identical to that of the $T(2,2n)$ torus link, yet whether its ropelength is bounded below by a constant multiple of $n$ remains open to date. In this paper, we provide a first such result, and in fact for a large class of alternating knots. Specifically, we prove that if an alternating knot (namely a link with only one component) has a reduced alternating knot diagram in which the crossings are either all positive or all negative (such a knot is called a special alternating knot), then its ropelength is bounded below by a constant multiple of its crossing number.