The rational Khovanov homology of 3-strand pretzel links

32 ANDREW MANION
t =
l + m − 1
l + m
n + m − 2
n + m −n 1+ m
Q
q = 3n + 3m − 1
Q Q
Q
q = 3n + 3m − 3
Q
q = 3n + 3m − 5
Q
Q
Q
q = 2l + 3m + n − 1
Q
Q
Q
Q
q = 2l + 3m + n − 1
q = 2l + 3m + n − 3
Figure 14. The case of odd n in Theorem 5.3. Red represents V and
blue represents W.
n+m. Also, X has no exceptional pair in this t-grading, and V has an exceptional pair
in t = n+m−1. Hence a standard cancellation must occur between t = n+m−1 and
t = n + m by Lemma 3.9.
As before, any further cancellations would be cancellations of U ′ ⊕ W ′ as in Remark
3.7, but U ′ and W ′ have no exceptional pairs and are contained in δ = n + m − 1
and δ = n+m−3, so Lemma 3.6 precludes any cancellations. We can now easily check
that we have completed the proof of our general formula.
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