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By Dieudonne J.
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This means that Heq is invariant under equivariant Reidemeister moves ER1, ER2 and ER3. March 27, 2007 11:10 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 24 Proof. As we have mentioned earlier, we only give an outline of the proof. 1,8 Let D and D′ be two periodic diagrams related by an equivariant Reidemeister move ERi, the idea is to construct a chain map ρ from (C ∗,∗ (D, F2 ), d) to (C ∗,∗ (D′ , F2 ), d) such that: • ρ induces an isomorphism ρ∗ in homology. • ρ is equivariant under the action of < ϕ >, thus it induces a map ρ between the quotient sub-complexes.
Birman and W. Menasco  extended the result of D. Bennequin  and showed that for 3-braid links, any minimal genus Seifert surface is realized as a Bennequin surface on a 3-braid. In , we showed that this result on 3-braids can not be extend to 4-braids (also done in ), by explicitly giving examples of 4-braid links having no Bennequin surface on 4-braid presentations. Our 4-braid examples had Bennequin surfaces carried on 5-braids. ) ∗ Partially supported by MEXT, Grant-in-Aid for Young Scientists (B) 1874035 March 4, 2007 11:41 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 44 Fig.
Consider the generalization of link homotopy known as self Ck -equivalence, introduced by Shibuya and the second author in . Two links L and L are self Ck -equivalent if L is obtained from L by ambient isotopies and Ck moves, where all the arcs in the Ck -move belong to the same component of L. Milnor’s link homotopy invariants are very useful for studying link homotopy. 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.
Algebre lineare et geometrie elementaire by Dieudonne J.