A3[Knot[10,121]]

M//MatrixForm

NullSpace[M]//MatrixForm

FirstPosition[1]/@NullSpace[M]

keeps=Complement[Range@Length@M,Union@@FirstPosition[1]/@NullSpace[M]]

Factor@Det[M[[keeps,keeps]]]

A4[Knot[10,121]]

M//MatrixForm

NullSpace[M]//MatrixForm

RowReduce@NullSpace[M]//MatrixForm

pivots=Union@@FirstPosition[1]/@RowReduce@NullSpace[M]

keeps=Complement[Range@Length@M,Union@@FirstPosition[1]/@RowReduce@NullSpace[M]]

Factor@Det[M[[keeps,keeps]]]

A5[Knot[4,1]]

M//MatrixForm

NullSpace[M]//MatrixForm

PowerExpand@RowReduce@NullSpace[M]//MatrixForm

Det[NullSpace[M][[{1,2},{1,2}]]]

rad=RowReduce@NullSpace[M]

Subsets[Range[5],{2}]