Pensieve header: Mathemaica notebook accompanying “A Note on the Unitarity Property of the Gassner Invariant” by Dror Bar-Natan, http://drorbn.net/AcademicPensieve/2014-06/UnitarityOfGassner/.

Definitions.

(Subscript[U, i_])[t_] := ReplacePart[IdentityMatrix[n],
{{i,i}->1-t, {i,i+1}->1,
{i+1,i}->t, {i+1,i+1}->0}];
(Subscript[U, i_]^(-))[t_] := Inverse[(Subscript[U, i])[t]];
Ω[τ___] :=  Table[
Which[i<j, 0,i==j,(((1-(Subscript[t, {τ}[[i]]])))^(-1)),i>j, 1],
{i,n}, {j,n}];
(OverBar[X_]) := X /. (Subscript[t, i_]):>1/(Subscript[t, i]);
(Subscript[U, i_,j_]) := ReplacePart[IdentityMatrix[n],
{{i,i}->1, {i,j}->1-(Subscript[t, i]),
{j,i}->0, {j,j}->(Subscript[t, i])}];
(Subscript[V, i_,j_]):= ReplacePart[IdentityMatrix[n],
{{i,i}->1, {i,j}->1-(Subscript[t, j]),
{j,i}->0, {j,j}->(Subscript[t, i])}];
DD := DiagonalMatrix[Table[1-(Subscript[t, i]), {i,n}]];

The named matrices.

n=5; MatrixForm /@ Simplify /@ {(Subscript[U, 3])[t], (Subscript[U, 3]^(-))[t]}

n=3; MatrixForm /@ Simplify /@ {Ω[2,3,1], Inverse[Ω[2,3,1]]}

n=5; MatrixForm/@ {(Subscript[U, 4,1]), (Subscript[V, 4,1]), DD}

The R3 move  .

n=3; MatrixForm /@ Simplify /@ {(Subscript[U, 1])[(Subscript[t, 1])].(Subscript[U, 2])[(Subscript[t, 1])].(Subscript[U, 1])[(Subscript[t, 2])], (Subscript[U, 2])[(Subscript[t, 2])].(Subscript[U, 1])[(Subscript[t, 1])].(Subscript[U, 2])[(Subscript[t, 1])]}

The unitarity property for the generators.

n=5; γ=(Subscript[U, 3])[(Subscript[t, 3])];
MatrixForm /@ Simplify /@ {Ω[1,2,4,3,5].Inverse[γ], Transpose[(OverBar[γ])].Ω[1,2,3,4,5]}

n=5; γ=(Subscript[U, 3]^(-))[(Subscript[t, 4])];
MatrixForm /@ FullSimplify /@ {Ω[1,2,4,3,5].Inverse[γ], Transpose[(OverBar[γ])].Ω[1,2,3,4,5]}

The braid .

n=4; MatrixForm[(Subscript[γ, 0])=(Subscript[U, 1])[(Subscript[t, 1])].(Subscript[U, 3]^(-))[(Subscript[t, 4])].(Subscript[U, 2])[(Subscript[t, 1])]]

The unitarity property for (Subscript[b, 0]).

MatrixForm /@ Simplify /@ {Ω[2,4,1,3].Inverse[(Subscript[γ, 0])], Transpose[(OverBar[(Subscript[γ, 0])])].Ω[1,2,3,4]}

On to w-braids

n=3; MatrixForm /@ Simplify /@ {(Subscript[U, 1,2]).(Subscript[U, 1,3]).(Subscript[U, 2,3]),(Subscript[U, 2,3]).(Subscript[U, 1,3]).(Subscript[U, 1,2])}

n=3; MatrixForm /@ Simplify /@ {(Subscript[U, 1,2]).(Subscript[U, 1,3]),(Subscript[U, 1,3]).(Subscript[U, 1,2])}

The “Other Gassner” Γ’

n=4; MatrixForm /@ Simplify /@ {(Subscript[V, 4,1]), Inverse[DD].(Subscript[U, 4,1]).DD}

n=4; MatrixForm /@ Simplify /@ {
Transpose[(OverBar[DD])].Ω[1,2,3,4].DD,
Ψ' = I Transpose[(OverBar[DD])].(Ω[1,2,3,4]-Transpose[(OverBar[Ω[1,2,3,4]])]).DD
}