Intelligence of low dimensional topology 2006: Hiroshima, by J. Scott Carter, Seiichi Kamada, Louis H. Kauffman, Akio

By J. Scott Carter, Seiichi Kamada, Louis H. Kauffman, Akio Kawauchi, Toshitake Kohno

This quantity gathers the contributions from the overseas convention "Intelligence of Low Dimensional Topology 2006," which happened in Hiroshima in 2006. the purpose of this quantity is to advertise study in low dimensional topology with the focal point on knot concept and similar issues. The papers comprise finished reports and a few most recent effects.

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Cn ± 1, ±1] Now we present our braiding algorithm. Let q/p = [2C1 , 2C2 , . . , 2Cn ], with C1 > 0 and Cj = 0 for all j. Modify it by inserting ±1 before and after each entry so that in the new continued fraction is of the following form, where σ(x) means the sign of x. 1 + [−1, 2C1 , −σ(C1 ), 2C2 , −σ(C2 ), . . , −σ(Cn−1 ), 2Cn , −σ(Cn ), ] Note that |Ck | = |Ck | − 1 if σ(Ck ) = σ(Ck−1 ), and otherwise Ck = Ck . The latter case happens between two blocks. Here, Ck may be 0. For example, 1849/10044 = [6, 2, 4, −6, −2, −6, 4] is modified to 1 + [−1, 4, −1, 0, −1, 2, −1, −6, +1, 0, +1, −4, +1, 4, −1].

Thus, L has the trivial Conway polynomial. 1 L is not self C2 -equivalent to trivial. Thus, the classification of 3-component links up to self C2 -equivalence will likely require Milnor numbers. Fig. 1. The Bing double of the Whitehead link. It is well known that Milnor’s link homotopy invariants vanish if and only if the link is link homotopic to the unlink. However, the 2-component link L in Figure 2 has vanishing Milnor numbers, and is not self C3 equivalent to a split link. The proof that this link is not split up to self C3 -equivalence depends on the fact that L is Brunnian, and that for an ncomponent Brunnian link, there is a relation between Ck+n−1 -equivalence and self Ck -equivalence.

We denote by mi (f ) the number of the critical points of f of index i. A Morse map f : CL → S 1 is said to be minimal if for each i the number mi (f ) is minimal on the class of all regular maps homotopic to f . Under these notations, the following basic theorem is shown ([10]). 1 ([10]). There is a minimal Morse map satisfying: (1) m0 (f ) = m3 (f ) = 0; March 4, 2007 11:41 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 36 (2) All critical values of the same index coincide; (3) f −1 (x) is a Seifert surface of L for any regular value x.

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