The rational Khovanov homology of 3-strand pretzel links

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$MATH 201 - FINAL EXAM Friday January 24 2003, 8:30am Problem 1$

8 ANDREW MANION exceptional pair q = 2 Q Q Q Q Q Q Q 3 b 6 Q 2 Q Q Q 2 Q 3 Q Q 2 Q 2 (2) c 2 q = 0 (1) t = 0 1 2 Figure 5. The space L 6,m,n for any m > 6, based on the formula in case 2 of Definition 2.5. Definition 2.5. The lower summand L l,m,n is defined below in various cases. See Figure 5 for an example of case 2 below. (1) If m ≠ l and l is odd, then L l,m,n := V [ a l−1 · ( l − 1 2 ) 2 · a l−1 ] . (2) If m ≠ l and l is even, then [ ( ) 3 ( l − 2 l L l,m,n := V (1) · c l−4 · · 2 2) ] · b l ,E ,
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 .
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The rational Khovanov homology of 3-strand pretzel links

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