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The reduced HOMFLY-PT homology for the Conway and the Kinoshita-Terasaka knots

In this paper we compute the reduced HOMFLY-PT homologies of the Conway and the Kinoshita-Terasaka knots and show that they are isomorphic.


INTRODUCTION
In this paper we use Rasmussen's results in [4] to compute the reduced HOMFLY-PT homologies, defined by Khovanov and Rozansky [2], of the Conway and the Kinoshita-Terasaka knots. It turns out that these homologies are isomorphic. We also show that our calculations imply that the Khovanov-Rozansky sl(N)-homologies of these two knots are isomorphic for all N ≥ 2. This result surprised us because the Floer knot homologies of these knots are non-isomorphic [3]. Since people have conjectured that for each knot there should exist a spectral sequence converging to the Floer knot homology with E 2 -page isomorphic to the HOMFLY-PT homology, our result shows that the differentials of the conjectured spectral sequences for the Kinoshito-Terasaka and the Conway knot should be different.
We did our calculations in the summer of 2006 and simply put them in a drawer. Since then several people, who knew about the result, asked us to write it up, which is why we finally decided to write this small note.
We claim no original theoretical insights. We have simply used Rasmussen's results. Since these calculations are hard and we had to use all sorts of tricks, this note might help other people to understand Rasmussen's results and to do calculations themselves and it also gives the reduced HOMFLY-PT homologies of the aforementioned knots, which to our knowledge were not available before.

RASMUSSEN'S TOOLKIT
In this section we explain Rasmussen's results [4] which enable us to do the computations. Note that we take the usual triple gradings (a, q,t) from the HOMFLY-PT homology, using the conventions in [2], rather than Rasmussen's gradings in [4]. We thank Rasmussen for explaining the conversion rules between the two gradings.
Given a knot K, denote its reduced HOMFLY-PT homology by H(K) and its reduced sl(N) homology by H N (K). The (a, q,t)-degree of the d k (N) is (−2k, 2Nk, 1). Note that for N big enough the differentials d k are zero.
. Another result that will be useful for us is the following.
We have not given all the maps explicitly, since we do not need them, but we have given their bidegrees.
Let K + be knot with a given positive crossing. Let K − be the same knot except for that particular crossing which is now negative and let K 0 be the two-component link obtained from K by the oriented resolution of the same crossing. Let us do a simple example to illustrate the exact sequences above. By abuse of notation we always identify the homology with the Poincaré polynomial. Thus, multiplying the homology by a polynomial means multiplying the Poincaré polynomial by that polynomial.
Recall (see [4]) that the positive Hopf link, H + , has reduced HOMFLY-PT homology equal to aq −1 + q(q N t −1 ) 2 (q −N+2 + q −N+4 + · · · + q N−2 ). Note that it does not matter which component we choose for reduction in this case. Multiplication by X maps the generator of degree q s to the generator of degree q s+2 in q −N+2 + q −N+4 + · · · + q N−2 . Therefore the kernel of is given by the generators aq −1 + q N−1 (q N t −1 ) 2 and the cokernel by the generators aq −1 + q −N+3 (q N t −1 ) 2 . Using the exact sequence (1) we see that Let K + be the positive trefoil T + , then (see [4]) Note that K 0 = H + and K − is the unknot. One now easily checks that the long exact sequence (2) holds.
There is also a useful variant of the exact sequence (1). Let L − and L + be the two-component link diagrams which differ by the sign of one crossing between different components and K the knot diagram obtained by resolving that crossing respecting the orientations. Lemma 2.6. There is a long exact sequence Notice that X acts as zero on H(K) and

COMPUTATIONS
In this section we compute the HOMFLY-PT homologies of the Conway and the Kinoshita-Terasaka knots. Resolving and changing a particular crossing of a diagram results in simpler diagrams where homology can be computed more easily. We then use the long exact sequence (2) recursively to obtain the homology of the initial diagram. The spectral sequences of Theorems 2.1 and 2.2 and the knowledge of the sl(N) homology of the initial diagram help us to unambiguously identify isomorphisms and zero maps in the long exact sequences (2) and (3).
The HOMFLY-PT polynomial of the Conway and Kinoshita-Terasaka knots is and their reduced Khovanov homologies are isomorphic and given by From Theorem 2.1 it follows that dim H(K Conway ) ≥ 33 and dim H(K KT ) ≥ 33.
3.1. The Kinoshita-Terasaka knot. A diagram of the KT knot is given in Figure 1. Changing  Figure 1. The (negative) encircled crossing is used for the long exact sequence (3) phicheiral (K 0 = K ! 0 ). This diagram corresponds to the (non-alternating) link L 10n36 in Thistlethwaite's link table [1].
Taking the oriented resolution and changing the encircled crossing of K 0 we obtain the diagrams M 0 and M + respectively, depicted in The diagram M + is the connected sum of the (positive) Hopf link with the connected sum of the positive trefoil T + and the negative trefoil T − . From Lemma 7.8 of [4] for connected sums it follows that We can now determine H(K 0 ) using the long exact sequence (3) where the diagrams K 0 , M 0 and M + correspond to L − , K and L + respectively. It reads From Equations (5) and (6) and comparing degrees we have that t −11/2 a 4 q 3 , t −7/2 a 4 q −1 and t −1/2 q are in the kernel of the map H(M 0 ) , ta −1 q, t 2 aq −5 and t 3 a −1 q −3 are in the cokernel. Using this can we form a first list of guaranteed and possible generators of H(K 0 ). Since K ! 0 = K 0 the polynomial of H(K 0 ) has to be invariant under the transformation ψ(a, q,t) = (a −1 , q −1 ,t −1 ). To have this symmetry we need to promote some possible generators to generators of H(K 0 ) and discard possible generators not paired by ψ. Then we apply the exact sequence (3) again to the newly promoted generators to obtain a new list which is in Table 1.
The dimension of E ∞ (1) has to be 1, living in homological degree 0. A straightforward computation shows that we already have this convergence in the column of guaranteed generators of H(K 0 ). By inspection we see that if we promote the generator t −3 aq 3 in the column of possible generators from H(M 0 ) than it would survive in E ∞ (1). Therefore the exact sequence (3) and the symmetry under ψ imply that t −3 aq 3 , t −2 aq 3 , t 2 a −1 q −3 and t 3 a −1 q −3 must be discarded from the list of possible generators of H(K 0 ). We present the updated list in Table 2.
To determine whether the remaining possible generators are generators of H(K 0 ) we use the spectral sequence E k (2). The reduced Khovanov homology of K 0 (computed from KhoHo [5] with q → q −1 to agree our conventions) is To have E k (2) ⇒ Kh(K 0 ) we have to promote the generators t −2 aq, t −1 aq, t −1 a −1 q 3 , a −1 q 3 , aq −3 , taq −3 , ta −1 q −1 and t 2 a −1 q −1 (recall that in our conventions d k (N) = a −2k q 2Nk t). A simple computation shows that E k (2) collapses after the second page i.e. E 2 (2) = E ∞ (2) = Kh(K 0 ) and that by promoting the four remaining generators the spectral sequence E k (2) would not converge to Kh(K 0 ). We leave the details to the reader. After rearranging some terms we have that Using the exact sequence (1) we obtain Finally, comparing at −1/2 H(K 0 ) to H(unknot) gives 3.2. The Conway knot. We follow the same method as in the calculation for the Kinoshita-Terasaka knot. A diagram of the Conway knot is given in Figure 4. Changing the (negative) encircled crossing we obtain the unknot, while resolving it results in the two-component link diagram L 0 of Figure 5 and corresponds to the Pretzel link P(3, −2, −3, 2) which corresponds to the link L 10n59 in Thistlethwaite's table [1] and is amphicheiral. Using KhoHo [5] we find that with Kh(L 10n36 ) given in Equation 7.
Note that d k (N) = 0 for N ≥ 3 for H(K Conway ) and H(K KT ) for degree reasons. This implies that H N (K Conway ) = H N (K KT ) for all N ≥ 3. We already knew that the same holds for N = 2 by direct computation. Since the knot Floer homology of these two knots differ [3], this fact might be interesting for someone trying to find a relation (e.g. a spectral sequence) between Khovanov-Rozansky homology and knot Floer homology.