Code to find a crossing order that would produce a minimum virtual width permutation as well as the minimum virtual width of a knot.

<<KnotTheory`

maxWidthPresentationList[pdList_] := Max[Length/@FoldList[Complement [#1 ⋃ #2, #1 ⋂ #2] &, {}, List  @@@ pdList]]

maxWidthPresentationList[List @@ PD[Knot[5,2]]]

RandomSample[List @@ PD[Knot[5,2]]]

Permutations[List @@ PD[Knot[5,2]]][[2]]

MinVWidthPermutation[pd_PD] := SortBy[Permutations[List @@ pd], maxWidthPresentationList[#]&][[1]]

MinVWidthPermutation[PD[Knot[5,2]]]

MinVWidth[pd_PD] := maxWidthPresentationList[MinVWidthPermutation[pd]]

MinVWidth[PD[Knot[5,2]]]

quantitiesGivenCrossingNum = <|3 -> 1, 4 -> 1, 5 -> 2, 6 -> 3, 7 -> 7, 8 -> 21, 9 -> 49, 10 -> 165|>

Table[Echo[Knot[i,j] -> MinVWidthPermutation[PD[Knot[i,j]]]], {i, 3,10}, {j, 1, quantitiesGivenCrossingNum[i]}]

Table[Echo[Knot[i,j] -> MinVWidth[PD[Knot[i,j]]]], {i, 3,10}, {j, 1, quantitiesGivenCrossingNum[i]}]