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