We give a combinatorial description of the ``$D_{2n}$ planar algebra,'' bygenerators and relations. We explain how the generator interacts with theTemperley-Lieb braiding. This shows the previously known braiding on the evenpart extends to a `braiding up to sign' on the entire planar algebra. We give a direct proof that our relations are consistent (using this`braiding up to sign'), give a complete description of the associated tensorcategory and principal graph, and show that the planar algebra is positivedefinite. These facts allow us to identify our combinatorial construction withthe standard invariant of the subfactor $D_{2n}$.
展开▼
