Tuesday, August 16, 2011

Theory Update 103

Three loops in the Hopf fibration form the $(3,3)$ torus knot. This is one of three links of three components that are described by six crossings. The other two are the Borromean rings and the loop chain. For the torus knot, where all three components are linked together, breaking one loop does not unlink the other two loops. Links are often used to discuss quantum entanglement. Note that another way to draw the torus knot is as a three stranded boson ribbon graph, with double twists on each strand in the directions $(+--)$. There is a simple $B_3$ representation for this knot, given by $B^{-1} A B^{-1} A^{-1} B A^{-1}$ in the generators $A$ and $B$.

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