Let K be a 3-stranded knot (or link), and let L denote the number of crossings in of K. Let ε1 and ε2 be two
positive real numbers such that ε2≤1.
In this paper, we create two algorithms for computing the value of the Jones polynomial VK(t) at all points
t=exp(iφ) of the unit circle in the complex plane such that |φ|≤2π/3.
The first algorithm, called the classical 3-stranded braid (3-SB) algorithm, is a classical deterministic
algorithm that has time complexity O(L). The second, called the quantum 3-SB algorithm, is a quantum
algorithm that computes an estimate of VK (exp (iφ)) within a precision of ε1 with a probability of success
bounded below by 1-ε2. The execution time complexity of this algorithm is O(nL), where n is the ceiling
function of (ln(4/ε2))/2ε21
. The compilation time complexity, i.e., an asymptotic measure of the amount
of time to assemble the hardware that executes the algorithm, is O(L).