# A system of geometry and trigonometry: Together with a by Abel Flint

By Abel Flint

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In a similar way, Milnor numbers can be used to study self Ck equivalence. Given a multi-index I, let r(I) denote the maximum number of times that any index appears in I. A Milnor invariant µ(I) is called realizable if µL (I) = 0 for some link L. Then our main result is the following. 1. Let µ(I) be a realizable Milnor number. Then µ(I) is an invariant of self Ck -equivalence if and only if r(I) ≤ k. Notice that a self Ck -equivalence can be realized by self Ck -moves when March 4, 2007 11:41 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 29 k < k, self C1 -equivalence is link homotopy, and self C2 -equivalence is self delta-equivalence.

Then we may March 4, 2007 11:41 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 39 regarded (M , γ) as a sutured manifold such that M is homeomorphic to D2 × S 1 . By the product decomposition as in Fig. 1, we have (M , γ) is a product sutured manifold. 1. Since it is known that L is not ﬁbred, we have MN (L) = 2 × h(R) = 2. α1 R+(γ) γ γ R-(γ) s(γ) R-(γ) α2 Fig. 2. s(γ) D D s(γ) s(γ') M' M'' Fig. 3. 2. Let L be 925 with the orientation as illustrated in Fig. 4. The oriented links in Fig. 4 are the same links.

Let i and j be two integers. We deﬁne C i (D) to be the free abelian group generated by all enhanced states with i(S) = i. Let C i,j (D) be the subgroup of C i (D) generated by enhanced states with j(S) = j. The Khovanov diﬀerential is deﬁned by: di,j : C i,j (D) −→ C i+1,j (D) S −→ (−1)t(S,S ) (S : S )S All states S’ where (S : S ) is • 1 if S and S diﬀer exactly at one crossing, call it v, where S has a +1 marker, S has a −1 marker, all the common circles in DS and DS have the same signs and around v, S and S are as in ﬁgure 2, • (S : S ) is zero otherwise and t(S, S ) is the number of −1 markers assigned to crossings in S labelled greater than v.

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