<< KnotTheory`

RotateToMinimal[l_]:=Module[
{bestl=l,rotatedl=RotateLeft[l]},
While[rotatedl=!=l,
bestl=First[Sort[{bestl,rotatedl}]];
rotatedl=RotateLeft[rotatedl]
];
bestl
];

pd=PD[Knot[11,Alternating,53]]

(Times@@(pd /. X[i_,j_,k_,l_] :> If[PositiveQ@X[i,j,k,l], c[i,j]c[l,k],c[i,l]c[j,k]]))

cycles=(Times@@(pd /. X[i_,j_,k_,l_] :> If[PositiveQ@X[i,j,k,l], cyc[i,j]cyc[l,k],cyc[i,l]cyc[j,k]]))//.cyc[i_,m1___,j_]cyc[j_,m2___,k_]:>cyc[i,m1,j,m2,k] /. c_cyc:>RotateToMinimal@Most[c]

cycles=SortBy[List@@cycles,First]

bl stands for “braid line”.

bl=cycles;Print[bl];
While[Length[bl]>1,
m=bl[[-1,1]];
bl=Most[bl]/.m-1->Sequence@@Prepend[Reverse@Last@bl,m-1];
Print[bl]
]

{1,2,3,4}/. 3->{5,6}

{1,2,3,4}/. 3->Sequence@@{5,6}