Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 2 / 6 / c / Solution

Read the supplied tangle from top to bottom. Its input word is $\downarrow \uparrow \downarrow$, and its output word is $\downarrow$. There are three successive events: the upper crossing of the rightmost two strands, the lower crossing of the leftmost two strands, and the right-hand oriented evaluation. At both crossings, the upper-left-to-lower-right strand passes over; both are therefore $X_{+}$ in the geometric convention of part (a). The evaluation joins a downward strand to an upward strand, so it is $\overleftarrow{\cup }$.

The elementary factorization is consequentlyIts boundary words give an explicit composition check:The third panel in part (b) draws these slices. Sliding the right-hand evaluation downwards and straightening the identity portions gives the PDF diagram without changing either crossing's overpass. If the names $X_{+},X_{-}$ are interchanged in another crossing convention, both displayed crossings must be renamed together; the stated overpasses determine the actual tangle independently of those names.