(* As extracted from her email of February 9, 2019 *)
{Knot[3, 1] -> 1 - T^(-4) + T^(-2) + T^2 - T^4, 
 Knot[4, 1] -> 5 + T^(-4) - 4/T^2 - 4*T^2 + T^4, 
 Knot[5, 1] -> 61 + T^(-16) - 4/T^14 + 8/T^12 - 12/T^10 + 13/T^8 - 18/T^6 + 
   32/T^4 - 49/T^2 - 49*T^2 + 32*T^4 - 18*T^6 + 13*T^8 - 12*T^10 + 8*T^12 - 
   4*T^14 + T^16, Knot[5, 2] -> 80 + 3/T^8 - 14/T^6 + 31/T^4 - 59/T^2 - 
   59*T^2 + 31*T^4 - 14*T^6 + 3*T^8, 
 Knot[6, 1] -> -70 - T^(-8) + 10/T^6 - 32/T^4 + 57/T^2 + 57*T^2 - 32*T^4 + 
   10*T^6 - T^8, Knot[6, 2] -> 87 - T^(-12) + 14/T^10 - 63/T^8 + 135/T^6 - 
   131/T^4 + 2/T^2 + 2*T^2 - 131*T^4 + 135*T^6 - 63*T^8 + 14*T^10 - T^12, 
 Knot[6, 3] -> 405 - T^(-12) + 4/T^10 + 2/T^8 - 52/T^6 + 177/T^4 - 332/T^2 - 
   332*T^2 + 177*T^4 - 52*T^6 + 2*T^8 + 4*T^10 - T^12, 
 Knot[7, 1] -> 438 + 3/T^24 - 12/T^22 + 27/T^20 - 48/T^18 + 72/T^16 - 
   96/T^14 + 114/T^12 - 141/T^10 + 189/T^8 - 255/T^6 + 333/T^4 - 402/T^2 - 
   402*T^2 + 333*T^4 - 255*T^6 + 189*T^8 - 141*T^10 + 114*T^12 - 96*T^14 + 
   72*T^16 - 48*T^18 + 27*T^20 - 12*T^22 + 3*T^24, 
 Knot[7, 2] -> 1593 + 42/T^8 - 248/T^6 + 705/T^4 - 1294/T^2 - 1294*T^2 + 
   705*T^4 - 248*T^6 + 42*T^8, Knot[7, 3] -> 3849 + 23/T^16 - 130/T^14 + 
   372/T^12 - 740/T^10 + 1193/T^8 - 1793/T^6 + 2615/T^4 - 3462/T^2 - 
   3462*T^2 + 2615*T^4 - 1793*T^6 + 1193*T^8 - 740*T^10 + 372*T^12 - 
   130*T^14 + 23*T^16, Knot[7, 4] -> 4994 + 112/T^8 - 720/T^6 + 2177/T^4 - 
   4064/T^2 - 4064*T^2 + 2177*T^4 - 720*T^6 + 112*T^8, 
 Knot[7, 5] -> 12090 + 23/T^16 - 168/T^14 + 632/T^12 - 1652/T^10 + 3389/T^8 - 
   5847/T^6 + 8697/T^4 - 11117/T^2 - 11117*T^2 + 8697*T^4 - 5847*T^6 + 
   3389*T^8 - 1652*T^10 + 632*T^12 - 168*T^14 + 23*T^16, 
 Knot[7, 6] -> 499 + 8/T^10 - 60/T^8 + 155/T^6 - 99/T^4 - 253/T^2 - 253*T^2 - 
   99*T^4 + 155*T^6 - 60*T^8 + 8*T^10, 
 Knot[7, 7] -> 929 - T^(-12) + 12/T^10 - 46/T^8 + 36/T^6 + 207/T^4 - 
   673/T^2 - 673*T^2 + 207*T^4 + 36*T^6 - 46*T^8 + 12*T^10 - T^12, 
 Knot[8, 1] -> 117 - 3/T^8 + 20/T^6 - 14/T^4 - 63/T^2 - 63*T^2 - 14*T^4 + 
   20*T^6 - 3*T^8, Knot[8, 2] -> 5698 + T^(-24) - 12/T^22 + 63/T^20 - 
   194/T^18 + 407/T^16 - 628/T^14 + 738/T^12 - 742/T^10 + 962/T^8 - 
   1826/T^6 + 3361/T^4 - 4979/T^2 - 4979*T^2 + 3361*T^4 - 1826*T^6 + 
   962*T^8 - 742*T^10 + 738*T^12 - 628*T^14 + 407*T^16 - 194*T^18 + 63*T^20 - 
   12*T^22 + T^24, Knot[8, 3] -> -2422 - 16/T^8 + 176/T^6 - 795/T^4 + 
   1848/T^2 + 1848*T^2 - 795*T^4 + 176*T^6 - 16*T^8, 
 Knot[8, 4] -> 1773 - T^(-16) + 10/T^14 - 60/T^12 + 268/T^10 - 761/T^8 + 
   1205/T^6 - 697/T^4 - 852/T^2 - 852*T^2 - 697*T^4 + 1205*T^6 - 761*T^8 + 
   268*T^10 - 60*T^12 + 10*T^14 - T^16, 
 Knot[8, 5] -> 10965 + T^(-24) - 12/T^22 + 65/T^20 - 214/T^18 + 501/T^16 - 
   927/T^14 + 1492/T^12 - 2298/T^10 + 3596/T^8 - 5579/T^6 + 8009/T^4 - 
   10117/T^2 - 10117*T^2 + 8009*T^4 - 5579*T^6 + 3596*T^8 - 2298*T^10 + 
   1492*T^12 - 927*T^14 + 501*T^16 - 214*T^18 + 65*T^20 - 12*T^22 + T^24, 
 Knot[8, 6] -> 6782 + 3/T^16 - 36/T^14 + 182/T^12 - 498/T^10 + 915/T^8 - 
   1593/T^6 + 3179/T^4 - 5544/T^2 - 5544*T^2 + 3179*T^4 - 1593*T^6 + 
   915*T^8 - 498*T^10 + 182*T^12 - 36*T^14 + 3*T^16, 
 Knot[8, 7] -> 3280 - T^(-20) + 10/T^18 - 58/T^16 + 238/T^14 - 692/T^12 + 
   1401/T^10 - 1908/T^8 + 1464/T^6 + 226/T^4 - 2319/T^2 - 2319*T^2 + 
   226*T^4 + 1464*T^6 - 1908*T^8 + 1401*T^10 - 692*T^12 + 238*T^14 - 
   58*T^16 + 10*T^18 - T^20, Knot[8, 8] -> 114 - T^(-16) + 12/T^14 - 
   82/T^12 + 326/T^10 - 790/T^8 + 1179/T^6 - 996/T^4 + 296/T^2 + 296*T^2 - 
   996*T^4 + 1179*T^6 - 790*T^8 + 326*T^10 - 82*T^12 + 12*T^14 - T^16, 
 Knot[8, 9] -> 7338 - T^(-20) + 12/T^18 - 58/T^16 + 156/T^14 - 236/T^12 + 
   60/T^10 + 735/T^8 - 2372/T^6 + 4578/T^4 - 6544/T^2 - 6544*T^2 + 4578*T^4 - 
   2372*T^6 + 735*T^8 + 60*T^10 - 236*T^12 + 156*T^14 - 58*T^16 + 12*T^18 - 
   T^20, Knot[8, 10] -> 3287 - 2/T^20 + 19/T^18 - 99/T^16 + 351/T^14 - 
   892/T^12 + 1634/T^10 - 2074/T^8 + 1502/T^6 + 270/T^4 - 2351/T^2 - 
   2351*T^2 + 270*T^4 + 1502*T^6 - 2074*T^8 + 1634*T^10 - 892*T^12 + 
   351*T^14 - 99*T^16 + 19*T^18 - 2*T^20, 
 Knot[8, 11] -> 9879 + 3/T^16 - 38/T^14 + 199/T^12 - 552/T^10 + 1029/T^8 - 
   1949/T^6 + 4339/T^4 - 7971/T^2 - 7971*T^2 + 4339*T^4 - 1949*T^6 + 
   1029*T^8 - 552*T^10 + 199*T^12 - 38*T^14 + 3*T^16, 
 Knot[8, 12] -> 14553 + 4/T^12 - 72/T^10 + 557/T^8 - 2436/T^6 + 6669/T^4 - 
   12000/T^2 - 12000*T^2 + 6669*T^4 - 2436*T^6 + 557*T^8 - 72*T^10 + 4*T^12, 
 Knot[8, 13] -> 433 - T^(-16) + 14/T^14 - 98/T^12 + 410/T^10 - 1057/T^8 + 
   1647/T^6 - 1373/T^4 + 242/T^2 + 242*T^2 - 1373*T^4 + 1647*T^6 - 1057*T^8 + 
   410*T^10 - 98*T^12 + 14*T^14 - T^16, 
 Knot[8, 14] -> 18714 + 3/T^16 - 40/T^14 + 248/T^12 - 948/T^10 + 2618/T^8 - 
   5829/T^6 + 10838/T^4 - 16247/T^2 - 16247*T^2 + 10838*T^4 - 5829*T^6 + 
   2618*T^8 - 948*T^10 + 248*T^12 - 40*T^14 + 3*T^16, 
 Knot[8, 15] -> 230228 + 129/T^16 - 1271/T^14 + 6263/T^12 - 20644/T^10 + 
   50993/T^8 - 99815/T^6 + 159284/T^4 - 210051/T^2 - 210051*T^2 + 
   159284*T^4 - 99815*T^6 + 50993*T^8 - 20644*T^10 + 6263*T^12 - 1271*T^14 + 
   129*T^16, Knot[8, 16] -> 8109 - 2/T^20 + 25/T^18 - 156/T^16 + 621/T^14 - 
   1709/T^12 + 3325/T^10 - 4416/T^8 + 3259/T^6 + 820/T^4 - 5821/T^2 - 
   5821*T^2 + 820*T^4 + 3259*T^6 - 4416*T^8 + 3325*T^10 - 1709*T^12 + 
   621*T^14 - 156*T^16 + 25*T^18 - 2*T^20, 
 Knot[8, 17] -> 35247 - T^(-20) + 12/T^18 - 56/T^16 + 108/T^14 + 135/T^12 - 
   1564/T^10 + 5571/T^8 - 13056/T^6 + 22931/T^4 - 31704/T^2 - 31704*T^2 + 
   22931*T^4 - 13056*T^6 + 5571*T^8 - 1564*T^10 + 135*T^12 + 108*T^14 - 
   56*T^16 + 12*T^18 - T^20, Knot[8, 18] -> 77731 + 13/T^16 - 180/T^14 + 
   1168/T^12 - 4760/T^10 + 13742/T^8 - 29964/T^6 + 51208/T^4 - 70092/T^2 - 
   70092*T^2 + 51208*T^4 - 29964*T^6 + 13742*T^8 - 4760*T^10 + 1168*T^12 - 
   180*T^14 + 13*T^16, Knot[8, 19] -> 215 + 3/T^24 - 12/T^22 + 18/T^20 - 
   7/T^18 - 17/T^16 + 30/T^14 - 25/T^12 + 20/T^10 - 40/T^8 + 65/T^6 - 
   10/T^4 - 130/T^2 - 130*T^2 - 10*T^4 + 65*T^6 - 40*T^8 + 20*T^10 - 
   25*T^12 + 30*T^14 - 17*T^16 - 7*T^18 + 18*T^20 - 12*T^22 + 3*T^24, 
 Knot[8, 20] -> 416 - 7/T^12 + 46/T^10 - 120/T^8 + 138/T^6 + 14/T^4 - 
   278/T^2 - 278*T^2 + 14*T^4 + 138*T^6 - 120*T^8 + 46*T^10 - 7*T^12, 
 Knot[8, 21] -> 2670 + T^(-14) + 4/T^12 - 60/T^10 + 276/T^8 - 775/T^6 + 
   1550/T^4 - 2331/T^2 - 2331*T^2 + 1550*T^4 - 775*T^6 + 276*T^8 - 60*T^10 + 
   4*T^12 + T^14}
