User:Drorbn/06-1350-HW4

The Generators

Our generators are $T$, $R$, $\Phi$ and $B^{\pm}$:

Picture
Generator	$T$	$R$	$\Phi$	$B^+$	$B^-$
Perturbation	$t$	$r$	$\varphi$	$b^+$	$b^-$

The Relations

The Reidemeister Move R3

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

In formulas, this is

$(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+$.

Linearized and written in functional form, this becomes

$\rho_3(x_1, x_2, x_3, x_4) =$	$b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4)$
	$- b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3).$

The Syzygies

The "B around B" Syzygy

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