The rational Khovanov homology of 3-strand pretzel links

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

26 ANDREW MANION t = n − l − 2n − l − 1 n − l Q Q Q q = 3n − 3l − 3 q = 3n − 3l − 5 q = 3n − 3l − 7 Q Figure 11. The case n > l + 1 in Lemma 4.4. Red denotes V and blue denotes W. the Khovanov homology of this knot in Lemma 4.1. So we still know what V is, by looking at the formula above for P(−l,l,l + 1). In fact, there is still no cancellation in the sequence. We have dimV ∗,0 = dimW ∗,0 = 2, but X needs two exceptional pairs in t = 0. So X is just V ⊕ W. One can check that this result agrees with our general formula. □ 4.4. P(−l,m,n) for odd l and even n. Theorem 4.5. When l is odd and n is even, Kh(P(−l,m,n) LR ) is given by the formula in Theorem 2.9. Proof. Our base case is m = l + 1, where we can appeal to Lemma 4.4. Note that for P(−l,m,n) with l odd, m even, and n even, switching orientations from LL to LR picks up a factor of q −6((n−m)/2) t −2((n−m)/2) = q 3m−3n t m−n , since the linking number of the relevant component with the rest of the link is (n − m)/2. Comparing this shift with the data in Table 1, we see that our formulas hold for P(−l,m,n) LR as soon as they hold for P(−l,m,n) LL . In particular, our formulas hold for P(−l,l + 1,n) LR . Now assume m > l+1. The diagram for P(−l,m,n) LR has l+n negative crossings. We will resolve the top crossing on the middle strand, a positive crossing. The unoriented resolution, a diagram for the right-handed torus knot T n−l,2 , has l + m − 1 negative crossings. Thus ǫ = m−n−1. The oriented resolution is a diagram for P(−l,m−1,n) LR , and the skein sequence is f q 3m−3n−1 t m−n Kh(T n−l,2 ) Kh(P(−l,m,n)) qKh(P(−l,m − 1,n)) f .
THE RATIONAL KHOVANOV HOMOLOGY OF 3-STRAND PRETZEL LINKS 27 Call the left-hand term W, the middle term X, and the right-hand term V . W has one exceptional pair in t = m − n. If m is even, V has one exceptional pair in t = 0 and X has exceptional pairs in t = 0 and t = m − n. Similarly, if m is odd, V has exceptional pairs in t = 0 and t = m − n − 1, and X has one exceptional pair in t = 0. Write V = L ⊕ U. It is easy to see, by looking at the LR row of Table 1 and at the shift of q in the above sequence, that L and U are based at the correct points. The exceptional pair, or pairs, of V fall in the U summand. We need only consider cancellations of U ⊕ W. The exceptional pair of W is in t = m − n. When m is odd, U has two exceptional pairs (t = 0 and t = m − n − 1). When m is even, U has one exceptional pair in t = 0. Hence, when m is even, no cancellations can occur by Lemma 3.6 (note that we only need to consider the case m ≤ n), and it is easy to check our formula. When m is odd, Lemma 3.8 guarantees a standard cancellation between t = m − n − 1 and t = m − n, while Remark 3.7 and Lemma 3.6 preclude any further cancellations. Again, one can now check our formula, finishing the computation. □ 5. Proof of the general formula for even l 5.1. P(−l,l,n) for even l. We now carry out the strategy of Section 3.3 for even l. We first compute Kh(P(−l,l,n)); recall that we are using the RL orientation throughout Section 5. This computation will not be used as a base case for induction, but we need to deal with it anyway since it is not covered by our other computations. Theorem 5.1. Let l ≥ 2 be even and let n ≥ 0. Let c denote the sequence (1) · c l−4 · ( l−2 2 )6 · c l−4 · (0, 1). For even n ≥ 0, let d (n) denote the sequence (1, −1,...,1, −1, 2) of length n + 1 (when n = 0, d (0) is just the sequence (2)). For odd n ≥ 1, let d (n) be the sequence (1, −1,...,1) of length n. Define c (n) = c + d (n) , where the addition to c starts in the (l + 1) st spot. Then if n < l, Kh(P(−l,l,n) RL ) = q l+n−1 V [c (n) ,E], where there are exceptional pairs in the first index and the last index, plus two exceptional pairs in the (l + n + 1) st index if n is even. If n ≥ l, then Kh(P(−l,l,n) RL ) is as described in Theorem 2.9. Proof. The proof is by induction on n, as before. When n = 0, we know P(−l,l, 0) RL is quasi-alternating, and its Jones polynomial was given in Equation (2). A check similar to that in Lemma 4.1 verifies the formula. Suppose the lemma holds for n − 1, and consider P(−l,l,n). Again, we will use the skein sequence from resolving the top crossing on the rightmost strand, a positive crossing. Our diagram for P(−l,l,n) has no negative crossings. The unoriented resolution, a diagram for the 2-component unlink, has l + n − 1 negative crossings (regardless of
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The rational Khovanov homology of 3-strand pretzel links

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