Part 1: calibration on 4-plat knot diagrams with at most 10 crossings
  170 knot diagrams: Fox det = Goeritz det (both colourings) = continued-fraction numerator;
  Gordon-Litherland sigma agrees for both colourings and is even; |sigma| = |floor-sum formula|.
  the other choice of type II gives an odd or colouring-dependent value on 58 of them (rejected).
  |sigma(T(2,m))| = m - 1 for m = 3,5,7,9,11; sigma(4_1) = 0; positive trefoil has sigma = -2.
Part 2: Table 1 (prime knots from the Conway notation; connected sums by additivity)
  n=  5  5_1       Conway 5            crossings  5  det   5  sigma +4
  n=  9  3_1#3_1   Conway 3 # 3        crossings  6  det   9  sigma +4
  n= 13  7_3       Conway 4 3          crossings  7  det  13  sigma +4
  n= 17  7_5       Conway 3 2 2        crossings  7  det  17  sigma +4
  n= 17  8_2       Conway 5 1 2        crossings  8  det  17  sigma +4
  n= 21  9_4       Conway 5 4          crossings  9  det  21  sigma -4
  n= 25  5_1#4_1   Conway 5 # 2 2      crossings  9  det  25  sigma +4
  n= 29  9_7       Conway 3 4 2        crossings  9  det  29  sigma +4
  n= 33  9_10      Conway 3 3 3        crossings  9  det  33  sigma -4
  n= 33  9_11      Conway 4 1 2 2      crossings  9  det  33  sigma +4
  n= 37  9_13      Conway 3 2 1 3      crossings  9  det  37  sigma +4
  n= 41  9_18      Conway 3 2 2 2      crossings  9  det  41  sigma -4
  n= 41  9_20      Conway 3 1 2 1 2    crossings  9  det  41  sigma +4
  n= 45  9_23      Conway 2 2 1 2 2    crossings  9  det  45  sigma +4
  n= 49  5_2#5_2   Conway 3 2 # 3 2    crossings 10  det  49  sigma -4
Part 3: S(2857,652) and S(33049,9229)
  S(2857,652): Conway 4 2 1 1 1 1 1 1 1 3 3 evaluates to 2857/652 (Q = q^(+-1) or -q^(+-1) mod p);
    reduced alternating diagram with 19 crossings; det (Fox) = det (Goeritz, both colourings) = 2857;
    sigma (Gordon-Litherland, both colourings) = -4; floor-sum formula gives -4
    examined all 40 fractions 2857/q (0<q<2857) with crossing number <= 19:
    least crossing number with |sigma| = 4 is 19 (e.g. q = 652) -> claim confirmed
  S(33049,9229): Conway 3 1 1 2 1 1 2 2 1 1 2 3 1 1 2 evaluates to 33049/9229 (Q = q^(+-1) or -q^(+-1) mod p);
    reduced alternating diagram with 24 crossings; det (Fox) = det (Goeritz, both colourings) = 33049;
    sigma (Gordon-Litherland, both colourings) = +4; floor-sum formula gives +4
    examined all 40 fractions 33049/q (0<q<33049) with crossing number <= 24:
    least crossing number with |sigma| = 4 is 24 (e.g. q = 9229) -> claim confirmed
Part 4: the primes 2857 and 33049
  2857 and 33049 are primes = 1 (mod 24) (Miller-Rabin and trial division)
ALL CHECKS PASSED (0.1 s)
