def non_zero_multiplicities(knot_multiplicities):
    """
    Returns a list of the non-zero multiplicities in a list of knot
    multiplicities, as well as the corresponding knot configuration.
    """
    non_zero_m = []
    knot_confg = []
    for i, m in enumerate(knot_multiplicities):
        if m != 0:
            non_zero_m.append(m)
            knot_confg.append(i)
    return non_zero_m, knot_confg
