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!
alt text

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


I just read the following paper, where an answer can be found:

Youngsik Huh, Seungsang Oh, An upper bound on stick number of knots, J. Knot Theory Ram. 20 (2011), no. 5, 741–747, doi: 10.1142/S0218216511008966, arXiv: 1512.03592.

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).

  • $\begingroup$ That $\frac{3}{2}$ bound is nice! Thanks for posting this update. $\endgroup$ – Joseph O'Rourke Nov 8 '11 at 12:33
  • $\begingroup$ Huh & Oh have posted their paper to the arXiv: arXiv:1512.03592. $\endgroup$ – Joseph O'Rourke Dec 14 '15 at 14:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.