# Which knots' stick numbers are twice their crossing numbers?

Looking at a table of minimum stick numbers for knots (table here), it seems the known upper bound of $2 c(K)$ in terms of the knot crossing number $c(K)$ is realized by the trefoil $3_1$—it requires 6 sticks (see image below) and its crossing number is 3—but not by any other small knot, at least through cursory inspection. Whence the question in the title: Are there other knots whose minimal stick number reaches the upper bound of twice its crossing number? This is probably well-known (perhaps well-known to be unknown), in which case a reference would suffice. Thanks!

Addendum. I found a 12-year old answer to my question in a paper by Eric Furstenberg, Jie Lie, and Jodi Schneider [FLS]:

"Thus far, the trefoil is the only knot to realize Negami’s upper bound of $2c[K]$ on the stick number. Do other such knots exist, and if so, what are their similarities to the trefoil?"

If anyone knows of more recent information, I would appreciate hearing of it. Thanks!

[FLS] "Stick Knots." Eric Furstenberg, Jie Lie, and Jodi Schneider. Chaos, Solitons & Fractals, Vol. 9, No. 4-5, pp. 561-568, 1998. Elsevier link

There it is shown that the trefoil is the only knot whose stick number equals twice its crossing number. This is a consequence of the authors' main result (Thm. 1.1), which states that any nontrivial knot $K$ satisfies $s(K)\leq \frac{3}{2}(c(K)+1)$ (thus improving Negami's upper bound).
• That $\frac{3}{2}$ bound is nice! Thanks for posting this update. – Joseph O'Rourke Nov 8 '11 at 12:33