Experimentation on the size of x .

Ks = ReadList["http://drorbn.net/AcademicPensieve/People/Dunfield/nmd_random_knots"]

sampleKs = Ks[[#]] &/@ Range[5,200,5]

optSize5 = (#[[1]] + JonesResult[#[[2]]] * 0.000013) &/@ (Timing[OptimizedKnotPresentation[# , {GreedyCriteria -> JonesResult, InvariantCriteria -> PartialJonesOptimize, OptSize -> 5}]] &/@ sampleKs);

optSize6 = (#[[1]] + JonesResult[#[[2]]] * 0.000013) &/@ (Timing[OptimizedKnotPresentation[# , {GreedyCriteria -> JonesResult, InvariantCriteria -> PartialJonesOptimize, OptSize -> 6}]] &/@ sampleKs);

optSize7 = (#[[1]] + JonesResult[#[[2]]] * 0.000013) &/@ (Timing[OptimizedKnotPresentation[# , {GreedyCriteria -> JonesResult, InvariantCriteria -> PartialJonesOptimize, OptSize -> 7}]] &/@ sampleKs);

ListPlot[{optSize5, optSize6, optSize7}, PlotLegends->{"x = 5", "x = 6", "x = 7"}]

largeSampleKs = Ks[[#]] &/@ Range[205,400,5]

optSize5Back = (#[[1]] + JonesResult[#[[2]]] * 0.000013) &/@ (Timing[OptimizedKnotPresentation[# , {GreedyCriteria -> JonesResult, InvariantCriteria -> PartialJonesOptimize, OptSize -> 5}]] &/@ largeSampleKs);

optSize6Back = (#[[1]] + JonesResult[#[[2]]] * 0.000013) &/@ (Timing[OptimizedKnotPresentation[# , {GreedyCriteria -> JonesResult, InvariantCriteria -> PartialJonesOptimize, OptSize -> 6}]] &/@ largeSampleKs);

optSize7Back = (#[[1]] + JonesResult[#[[2]]] * 0.000013) &/@ (Timing[OptimizedKnotPresentation[# , {GreedyCriteria -> JonesResult, InvariantCriteria -> PartialJonesOptimize, OptSize -> 7}]] &/@ largeSampleKs);

ListPlot[{optSize5Back, optSize6Back, optSize7Back}, PlotLegends->{"x = 5", "x = 6", "x = 7"}]

ListPlot[{optSize5Back[[1;;20]], optSize6Back[[1;;20]], optSize7Back[[1;;20]]}, PlotLegends->{"x = 5", "x = 6", "x = 7"}]