Pensieve header: 1-Smidgen and Unknotting Numbers.

Loading Data Files

SetDirectory["C:\\drorbn\\AcademicPensieve\\2016-10"];
Once[<< KnotTheory`];
tab=Join[
{Knot[0,1]->{0., 𝔼[1,0,0,0]}}, (* Computed by hand *)
Get["../2016-09/tab.m"], (* Computed in http://drorbn.net/AcademicPensieve/2016-09/OneSmidgen.nb *)
Get["../2016-09/tab11.m"], (* Computed in http://drorbn.net/AcademicPensieve/2016-09/1co11.nb *)
Get["../2016-10/tab12.m"], (* Computed in http://drorbn.net/AcademicPensieve/2016-10/1co12.nb *)
Get["../2016-09/TKTable.m"], (* Computed in http://drorbn.net/AcademicPensieve/2016-09/1co4TorusKnots.nb *)
{GST[48] ->  {16149.016`, 𝔼[13-(1/(t^8))+(2/(t^7))-(1/(t^6))-(2/(t^4))+(5/(t^3))-(2/(t^2))-(7/t)-7 t-2 (t^2)+5 (t^3)-2 (t^4)-(t^6)+2 (t^7)-(t^8),0,0,224956+8628 c+(13/(t^32))-((16 c)/(t^32))-(110/(t^31))+((120 c)/(t^31))+(426/(t^30))-((385 c)/(t^30))-(1006/(t^29))+((676 c)/(t^29))+(1690/(t^28))-((784 c)/(t^28))-(2449/(t^27))+((1198 c)/(t^27))+(3698/(t^26))-((5793 c)/(2 (t^26)))-(5392/(t^25))+((9569 c)/(2 (t^25)))+(5807/(t^24))-((2918 c)/(t^24))-(3039/(t^23))-((3274 c)/(t^23))-(1952/(t^22))+((11923 c)/(2 (t^22)))+(7062/(t^21))-((646 c)/(t^21))-(13976/(t^20))+((5229 c)/(2 (t^20)))+(24036/(t^19))-((29531 c)/(t^19))-(27865/(t^18))+((58335 c)/(t^18))+(14568/(t^17))-((36499 c)/(t^17))+(1523/(t^16))-((31212 c)/(t^16))+(9189/(t^15))+((55648 c)/(t^15))-(32806/(t^14))+((9749 c)/(2 (t^14)))+(6572/(t^13))-((67387 c)/(2 (t^13)))+(70356/(t^12))-((91406 c)/(t^12))-(71698/(t^11))+((500701 c)/(2 (t^11)))-(87333/(t^10))-((361145 c)/(2 (t^10)))+(229431/(t^9))-((115093 c)/(t^9))-(53565/(t^8))+((575659 c)/(2 (t^8)))-(413089/(t^7))-((255051 c)/(2 (t^7)))+(688179/(t^6))-((190615 c)/(2 (t^6)))-(367037/(t^5))+((60617 c)/(2 (t^5)))-(313161/(t^4))+((394233 c)/(2 (t^4)))+(687947/(t^3))-((191058 c)/(t^3))-(442972/(t^2))-((58548 c)/(t^2))-(49189/t)+((162150 c)/t)+38417 t-((219015 c t)/2)-389878 (t^2)-((149991 c (t^2))/2)+442865 (t^3)+253710 c (t^3)-111675 (t^4)-126942 c (t^4)-381647 (t^5)-97032 c (t^5)+691047 (t^6)+51156 c (t^6)-589895 (t^7)+((344873 c (t^7))/2)+162331 (t^8)-((341955 c (t^8))/2)+248205 (t^9)-((233899 c (t^9))/2)-356293 (t^10)+((636481 c (t^10))/2)+194810 (t^11)-((389149 c (t^11))/2)-16686 (t^12)-46825 c (t^12)-15840 (t^13)+((221599 c (t^13))/2)-37762 (t^14)-((22403 c (t^14))/2)+49359 (t^15)-44534 c (t^15)+2371 (t^16)-((24259 c (t^16))/2)-46496 (t^17)+((141789 c (t^17))/2)+39311 (t^18)-51698 c (t^18)-7048 (t^19)-976 c (t^19)-9696 (t^20)+20285 c (t^20)+5130 (t^21)-6909 c (t^21)+2778 (t^22)-3177 c (t^22)-3453 (t^23)-((2503 c (t^23))/2)+467 (t^24)+6626 c (t^24)+708 (t^25)-((9737 c (t^25))/2)+110 (t^26)+728 c (t^26)-775 (t^27)+748 c (t^27)+598 (t^28)-((447 c (t^28))/2)-194 (t^29)-((241 c (t^29))/2)+6 (t^30)-56 c (t^30)+14 (t^31)+184 c (t^31)-3 (t^32)-120 c (t^32)+35 c (t^33)-4 c (t^34)-((138459 u w)/2)+((16 u w)/(t^32))-((104 u w)/(t^31))+((281 u w)/(t^30))-((395 u w)/(t^29))+((389 u w)/(t^28))-((809 u w)/(t^27))+((4175 u w)/(2 (t^26)))-((2697 u w)/(t^25))+((221 u w)/(t^24))+((3495 u w)/(t^23))-((4933 u w)/(2 (t^22)))-((3641 u w)/(2 (t^21)))-((4435 u w)/(t^20))+((25096 u w)/(t^19))-((33239 u w)/(t^18))+((3260 u w)/(t^17))+((34472 u w)/(t^16))-((21176 u w)/(t^15))-((52101 u w)/(2 (t^14)))+((7643 u w)/(t^13))+((99049 u w)/(t^12))-((302603 u w)/(2 (t^11)))+((29271 u w)/(t^10))+((144364 u w)/(t^9))-((286931 u w)/(2 (t^8)))-((15940 u w)/(t^7))+((158735 u w)/(2 (t^6)))+((49059 u w)/(t^5))-((296115 u w)/(2 (t^4)))+((86001 u w)/(2 (t^3)))+((203097 u w)/(2 (t^2)))-((121203 u w)/(2 t))+40278 t u w+(230547/2) (t^2) u w-(276873/2) (t^3) u w-(22989/2) (t^4) u w+(171075/2) (t^5) u w+(68763/2) (t^6) u w-138055 (t^7) u w+(65845/2) (t^8) u w+149872 (t^9) u w-(336737/2) (t^10) u w+26206 (t^11) u w+73031 (t^12) u w-(75537/2) (t^13) u w-26567 (t^14) u w+17967 (t^15) u w+(60193/2) (t^16) u w-40798 (t^17) u w+10900 (t^18) u w+11876 (t^19) u w-8409 (t^20) u w-1500 (t^21) u w+1677 (t^22) u w+(5857/2) (t^23) u w-(7395/2) (t^24) u w+1171 (t^25) u w+443 (t^26) u w-305 (t^27) u w-(163/2) (t^28) u w+39 (t^29) u w+95 (t^30) u w-89 (t^31) u w+31 (t^32) u w-4 (t^33) u w]}} (* Computed in http://drorbn.net/AcademicPensieve/2016-09/GST48.nb *)
];
tab /. (K_->{_,z_}) :> (z1[K]=z);
AllKs=First /@ tab;
Ribbons={Knot[0,1],Knot[6,1],Knot[8,8],Knot[8,9],Knot[8,20],Knot[9,27],Knot[9,41],Knot[9,46],Knot[10,3],Knot[10,22],Knot[10,35],Knot[10,42],Knot[10,48],Knot[10,75],Knot[10,87],Knot[10,99],Knot[10,123],Knot[10,129],Knot[10,137],Knot[10,140],Knot[10,153],Knot[10,155]};  (* From Kawauchi via Andrey Khesin *)

z1[Knot[3,1]]

The Essence

al[K_] := z1[K][[1]];
ap[K_] :=  al[K]/. (t^(n_.))/; n<0 :> 0;

e[K_] := e[K]=Expand[Together[(((t z1[K][[4]] /. c|u|w->0)+((al[K])^3)(t^2) D[al[K],t])/((((t-1))^2)((al[K])^2)))]];
ep[K_] :=  e[K]/. (t^(n_.))/; n<0 :> 0;

Work

Column@Sort@DeleteDuplicatesBy[
Table[{UnknottingNumber[K], Exponent[ap[K], t], Exponent[ep[K], t], K},{K,AllKnots[{0,11}]}],
Most
]

Column@Sort@DeleteDuplicatesBy[
Table[{Exponent[ap[K], t], Exponent[ep[K], t], K},{K,AllKs}],
Most
]

ListPlot[Union[
Table[{Exponent[ap[K], t], Exponent[ep[K], t]},{K,AllKnots[{0,12}]}]
] /. -∞ -> -1]

ListPlot[Union[
Table[{Exponent[ap[K], t], Exponent[ep[K], t]},{K,AllKs}]
] /. -∞ -> -1]