User:Drorbn/06-1350-HW4

The Generators

Our generators are [math]\displaystyle{ T }[/math], [math]\displaystyle{ R }[/math], [math]\displaystyle{ \Phi }[/math] and [math]\displaystyle{ B^{\pm} }[/math]:

Picture
Generator	[math]\displaystyle{ T }[/math]	[math]\displaystyle{ R }[/math]	[math]\displaystyle{ \Phi }[/math]	[math]\displaystyle{ B^+ }[/math]	[math]\displaystyle{ B^- }[/math]
Perturbation	[math]\displaystyle{ t }[/math]	[math]\displaystyle{ r }[/math]	[math]\displaystyle{ \varphi }[/math]	[math]\displaystyle{ b^+ }[/math]	[math]\displaystyle{ b^- }[/math]

The Relations

The Reidemeister Move R3

The picture (with three sides of the shielding removed) is

In formulas, this is

[math]\displaystyle{ (1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+ }[/math].

Linearized and written in functional form, this becomes

[math]\displaystyle{ \rho_3(x_1, x_2, x_3, x_4) = }[/math]	[math]\displaystyle{ b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4) }[/math]
	[math]\displaystyle{ - b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3). }[/math]

The Syzygies

The "B around B" Syzygy