Jones polynomials from matrix elements of tangles in a pseudounitary representation

In these notes, we review the calculation of Jones polynomials using a matrix representation of the braid group and Temperley–Lieb algebra. The pseudounitary representation that we consider allows constructing “states” from the group/algebra matrices and computing the knot invariants as matrix elements, rather than traces. In comparison with a more standard way of computing the invariants through traces, the matrix element method is more interesting and complete from the point of view of applications. In particular, we expect it to be more efficient, in comparison with the trace methods, and serve as a natural framework for quantum computation algorithms. As a byproduct of the discussion we prove a general formula for pretzel knots.