By Edoardo Ballico, Ciro Ciliberto
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7. R. Fenn, C. Rourke, and B. uk/~bjs/ 8. , A classifying invariant of knots, the knot quandle, J. Pure Appl. Alg. 23, 37–65. 9. Majid, S. ” London Mathematical Society Lecture Note Series, 292. Cambridge University Press, Cambridge, 2002. 10. , Distributive groupoids in knot theory, (Russian) Mat. Sb. ) 119(161) (1982), no. 1, 78–88, 160. 11. , Abstraction of symmetric Transformations: Introduction to the Theory of kei, Tohoku Math. J. 49, (1943). 145–207 (a recent translation by Seiichi Kamada is available from that author).
For a 2-bridge link L of µ components, the band length w(L) satisﬁes the following: w(L) = 2g(L) + b(L) + µ − 2, where g(L) and b(L) respectively denote the genus and braid index of L. Proof. Let β denote the minimal ﬁrst betti number of all Seifert surfaces for L. Then we have w − (b − 1) = β = 2g + (µ − 1) We see that for 2-bridge links, braid presentations with minimal band length have minimal number of strings. Thus the braids we obtain (in Section 4) for 2-bridge links are of minimal string and at the same time, of minimal band length.
The last operation happens to distant elements. This corresponds to the 3-diﬀerential, for 3-cochains ξ1 ∈ Hom(X ⊗3 , X) and ξ2 ∈ Hom(X ⊗2 , X ⊗2 ), d3,1 (ξ1 , ξ2 )(x ⊗ y ⊗ z ⊗ w) = q(ξ1 (x ⊗ y ⊗ z) ⊗ w) + ξ1 (q(x ⊗ z(1) ) ⊗ q(y ⊗ z(2) ) ⊗ w) +q(ξ1 (x ⊗ z(1) ⊗ w(1) ) ⊗ q(q(y ⊗ z(2) ) ⊗ w(2) )) +q(q(q(x ⊗ w(1)(1) ) ⊗ q(z(1) ⊗ w(1)(2) )) ⊗ ξ1 (y ⊗ z(2) ⊗ w(2) )) −ξ1 (q(x ⊗ y) ⊗ z ⊗ w) − q(ξ1 (x ⊗ y ⊗ w(1) ) ⊗ q(z ⊗ w(2) )) −ξ1 (q(x ⊗ w(1)(1) ) ⊗ q(y ⊗ w(1)(2) ) ⊗ q(z ⊗ w(2) )) −q(q(q(x ⊗ w(1)(1) ) ⊗ ξ2,1 (z ⊗ w(2) )) ⊗ q(q(y ⊗ w(1)(2) ) ⊗ ξ2,2 (z ⊗ w(2) ))) where ξ2 : X ⊗X → X ⊗X is denoted by ξ2 (x⊗y) = ξ2,1 (x⊗y)⊗ξ2,2 (x⊗y) by suppressing the sum.