The rational Khovanov homology of 3-strand pretzel links

THE RATIONAL KHOVANOV HOMOLOGY OF 3-STRAND PRETZEL LINKS 9
where the exceptional pair is on the first index.
(3) If m = l and l is odd, then
[ ( ) 2 l − 1
L l,l,n := V a l−1 · · a l−1 · (0) n−l · (1),E]
,
2
where the exceptional pair is on the final index.
(4) If m = l, l is even, and n is odd, then
[ ( ) 3 ( ]
l − 2 l
L l,l,n := V (1) · c l−4 · · · b l · (0, 1)
2 2)
(n−l−1)/2 ,E ,
where the exceptional pair is on the first index.
(5) If m = l, l is even, n is even, and n ≠ l, then
[ ( ) 3 ( ]
l − 2 l
L l,l,n := V (1) · c l−4 · · · b l · (0, 1)
2 2)
(n−l)/2 ,E ,
with exceptional pairs on the first and last indices.
(6) If m = l, l is even, and n = l, then
[ ( ) 3 ( ]
l − 2 l
L l,l,l := V (1) · c l−4 · · · (b l + (...,0, 0, 1)),E ,
2 2)
with one exceptional pair on the first index and two on the last index. The
addition is done such that the last index of b l gets the extra 1.
Note that to obtain the formulas when m = l, you just add some extra generators,
in higher q- and t-gradings, to the formulas for m ≠ l.
2.4. Formula for the upper summand. The rest of the Khovanov homology of
P(−l,m,n) comes from the “upper summand” U l,m,n . It depends only on g := m − l
and h := n − m as well as the parities of l, m, and n. Each of the eight choices for
parities gives rise to a different formula, so the below definition has eight different cases.
Definition 2.6. Suppose 2 ≤ l ≤ m ≤ n are integers; let g = m − l and h = n − m.
(1) If l, m, and n are odd,
[ ( ) h g
U l,m,n := V a g · · a g ,E]
,
2
where the one exceptional pair is on the final index.
(2) If l and m are odd but n is even,
[ ( ) h−1 g
U l,m,n := V a g · · c g ,E]
,
2
where the exceptional pair is on the first index of c g .