Arcadian Functor

A while back we looked at associating the number of strands in number theoretic braids with the size of the matrix operators in the Fourier transform, which is the same as the number of points on the circle (3 for mass). In the two strand case, the braids are easy to classify: copies of the only generator, . In other words, an integer labels all possible knots.

The homflypt polynomial for the torus knot is

for two parameters and . Specialisations include which results (effectively) in the Jones polynomial

Since the endpoints of braid diagrams lie on a circle, there are two circles bounding a diagram. For the 3-strand case, there are two sets of three points defining the boundary, which thus looks like a 6 point torus.

In quantum computation [1] the Fourier transform for $2^{n}$ basis states on $n$ qubits is implemented with Hadamard gates and unitary gates $G_{k}$ of the form

$1$ $0$
$0$ $\textrm{exp} \frac{2 \pi i}{2^{k}}$

which add a phase factor to $| 1 >$. A ternary analogue of such gates would be a $3 \times 3$ diagonal matrix with entries $1$, $\textrm{exp} \frac{2 \pi i}{3^{k}}$ and $\textrm{exp} \frac{4 \pi i}{3^{k}}$, responsible for adding phases to each of the three basis states. For example, for two ternary objects ($n = 2$) the central phase factor in $G_{2}$ is $\textrm{exp} \frac{2 \pi i}{9}$. Any number from 1 to 9 is expressable in base 3 as a combination of 1, 3 and 9, so the product representation of the qubit transform has a ternary analogue. This ability to switch from an additive expression to a product expansion in superpositions of qutrits is quite reminiscent of Euler’s relation for zeta functions. It is no surprise then that the quantum qubit transform is used in the algorithm for integer factorization.

[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge (2000)