### M Theory Lesson 250

Recall that the Tutte polynomial for the trefoil knot is based on the trivalent vertex graph. On the other hand, we would like to consider categorified invariants such as in Khovanov homology.

The Khovanov cube for the trefoil knot contains the 8 ($= 2^3$) possible smoothing diagrams for the trefoil. For each crossing piece of the original knot there are two ways to smooth the crossing. In other words, there are two types of edge on the Tutte graphs. We might as well label these two colours by inward and outward arrows. Then the eight vertices of a cube are given by the diagrams

The Khovanov cube for the trefoil knot contains the 8 ($= 2^3$) possible smoothing diagrams for the trefoil. For each crossing piece of the original knot there are two ways to smooth the crossing. In other words, there are two types of edge on the Tutte graphs. We might as well label these two colours by inward and outward arrows. Then the eight vertices of a cube are given by the diagrams

## 2 Comments:

Similar orientations were also needed when one tries to axiomatize 3 MUBs on a qubit. We wrote something on that in http://arxiv.org/abs/0808.1029

Goodness, yes. Here is a proper link to your paper. By the way, here is also Carl's new paper on Koide mass formulas.

Post a Comment

<< Home