Stick number

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Stick number is a way to measure knots. It is the smallest number of straight sticks joined end to end that can form a given knot in 3D.

- Equilateral stick number is the smallest number of sticks of equal length needed to form the knot. It’s not known whether this is always the same as the regular stick number for every knot.

- The smallest possible stick number for any nontrivial knot is 6.

- For some torus knots T(p, q), people have found their stick numbers when p and q are not too far apart. A related result was found independently by another group as well.

- Some knots have their stick numbers known exactly because their upper and lower bounds meet. Examples include: knot 31 (stick number 6), knot 41 (stick number 7), all 5- and 6-crossing knots, and all 7-crossing knots.

- For 8-crossing knots (numbers 16–21 in the usual notation) and certain 9-crossing knots (numbers 29, 34, 35, 39–49) and knot 10124 (a torus knot), the stick numbers are known as well.

- There are 19 additional non-alternating knots with 11 or 13 crossings that have stick number exactly 10.

- When you form the connected sum of two knots K1 and K2, their stick numbers satisfy:
stick(K1 # K2) ≤ stick(K1) + stick(K2) − 3.

- The stick number c(K) relates to the crossing number c(K) by these bounds:
1/2 [7 + sqrt(8 c(K) + 1)] ≤ stick(K) ≤ 3/2 [c(K) + 1].
These bounds are exact for the trefoil knot (crossing number 3, stick number 6).

- Note: The upper bound does not apply to the unknot, which has crossing number 0 but stick number 3.