User:Drorbn/06-1350-HW4

The Generators

Our generators are $T$, $R$, $Y=\Phi$ and $B^{\pm }$ (we might consider splitting $Y$ into two, $Y^{up}$ and $Y^{dn}$):

Picture
Generator	$T$	$R$	$Y^{up}$	$Y^{dn}$	$B^{+}$	$B^{-}$
Perturbation	$t$	$r$	$y^{u}$	$y^{d}$	$b^{+}$	$b^{-}$

The Relations

The Reidemeister Move R3

The picture is

In formulas, this is

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

Thus the R3 component of $d$ is

$(1230)^{\star }b^{+}+(1213)^{\star }b^{+}+(1023)^{\star }b^{+}-(1123)^{\star }b^{+}-(1203)^{\star }b^{+}-(1231)^{\star }b^{+}$.